MORPHOLOGICAL PROPERTIES DETERMINATION OF MAIN BELT

by Guillermo Cerdán Hernandez Thesis submitted as a partial requirement to obtain the degree of MASTER OF SCIENCE IN SPACE SCIENCE AND TECHNOLOGY at Instituto Nacional de Astrofísica, Óptica y Electrónica April 2019 Tonantzintla, Puebla Supervised by: Dr. José Ramón Valdés Parra Dr. José Silviano Guichard Romero

c INAOE 2019

The author hereby grants to INAOE permission to reproduce and distribute copies of this thesis document in whole or in part.

Abstract

We used lightcurves from the Lightcurve Database (ALCDEF), observations made at Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE, México) Schmidt telescope, and Observer Lightcurve Inversion software (MPO LC Invert) to make the lightcurve inversion process, deriving shape, and pole orientation for the asteroids: (22) Kallipe, (287) Nepthys, (711) Marmulla, (1117) Reginita, (1318) Nerina, (1346) Gotha, (1492) Oppozler, (3028) Zhangguoxi, (3800) Karayusuf, (4713) Steel, and (5692) Shirao.

Fueron usadas curvas de luz de la base de datos de la Asteroid Photometry Database (ALCDEF), observaciones hechas en el telescopio Schmidt del Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE, México) y el software del Minor Planet Observer Lightcurve Inversion (MPO LC Invert, para aplicar el proceso de inversión de cruvas de luz, obteniendo la forma y la orientación del polo para los asteroides: (22) Kallipe, (287) Nepthys, (711) Marmulla, (1117) Reginita, (1318) Nerina, (1346) Gotha, (1492) Oppozler, (3028) Zhangguoxi, (3800) Karayusuf, (4713) Steel y (5692) Shirao.

Acknowledgments

I thank CONACYT for granting the financial, intellectual capital, and infrastructure support to carry out this thesis work.

Likewise, I thank all the people who have contributed in my life and who have allowed me to live this experience, people like: Dolores Hernandez

Gonzalez, Juan Carlos Herrera Tavera, Jose Ramón Valdes Parra, Karem

Contreras Aguilera, trench companions, and the teachers with vocations who have instructed me.

Also, I want to thank the people who have hindered my life, without them I could not have reached this moment, people like my fifth grade teacher Cristino, Narro, Dulce Cruz, the Refrescos King administrators, and many others, without the rocks that they threw at me

I could not have built my temple to virtue and science.

V

Contents

Contents VII

1. INTRODUCTION AND BACKGROUND 1 1.1. INTRODUCTION ...... 1 1.2. SPACE ENVIRONMENT DELIMITATION AND DEFINITION . . . 3 1.3. ASTEROIDS MORPHOLOGY AND DEFINITION ...... 3 1.3.1. DEFINITION ...... 3 1.3.2. OR MONOLITHS ...... 5 1.3.3. BINARIES OR MICRO-SYSTEM ASTEROIDS ...... 5 1.3.4. SPECTRA ...... 6 1.3.5. ASTEROIDS NOMENCLATURE ...... 6 1.3.6. CURRENT PROVISIONAL DESIGNATION FOR ASTEROIDS 7 1.3.7. ...... 9 1.3.8. PERTURBATIONS ...... 9 1.4. MAIN BELT ASTEROIDS ...... 13 1.4.1. MAIN BELT FORMATION AND DISTRIBUTION OF ASTEROIDS ...... 14 1.5. OTHER MORPHOLOGY PARAMETERS AND CHARACTERISTICS 17 1.5.1. (H) ASTEROID ABSOLUTE ...... 17 1.5.2. SLOPE PARAMETER (G) ...... 18 1.5.3. H-G ...... 18 1.5.4. ASTROMETRY AND PHOTOMETRY ...... 19 1.5.5. H-G PARAMETER AND H-G1G2 ...... 21 1.6. DETERMINATION OF ASTEROID CHARACTERISTICS FROM LIGHTCURVES ...... 23

2. Problem Statement 25 2.1. BRIEF IMPLICIT PHILOSOPHICAL JUSTIFICATION ...... 25 2.2. WHY DETERMIN ASTEROIDS PROPERTIES? ...... 26 2.3. DISCOVERIES MADE BY STATISTICAL ANALYSIS ...... 27 2.3.1. RUBBLE PILE SPIN BARRIER ...... 27 2.4. EARTH DEFENCE ...... 28 2.5. ASTEROID MINING ...... 29

VII 2.5.1. GROUND-BASED VISUAL OBSERVATIONS AND ANALYSIS ...... 30 2.6. FUTURE SPACIAL PROBES TO ASTEROIDS ...... 30

3. METHODOLOGY 33 3.1. FINDING MORPHOLOGICAL PARAMETERS ...... 33 3.2. TONANTZINTLA SCHMIDT TELESCOPE ...... 34 3.3. ELECTRONIC DETECTOR ...... 38 3.4. SCHMIDT TELESCOPE AND CCD CHARACTERISTICS ...... 39 3.5. ASTRONOMICAL OBSERVATIONS ...... 41 3.5.1. PLANING AND SELECTING OBJECTS ...... 41 3.5.2. CCD SETTINGS ...... 43 3.5.3. IMAGE REDUCTION BY IRAF ...... 44 3.6. MPO CANOPUS ...... 46 3.6.1. CONFIGURATION SETTINGS ...... 46 3.6.2. APERTURES ...... 46 3.6.3. ASTROMETRY ...... 46 3.6.4. MPO CANOPUS PHOTOMETRY SESSION ...... 46 3.6.5. VERIFYING COMPARISON ...... 50 3.6.6. MERGE SESSIONS ...... 50 3.6.7. PERIOD SPECTRUM ANALYZIS ...... 51 3.6.8. CSmin/CSmax ...... 54 3.7. LIGHTCURVE INVERSION ...... 54 3.7.1. MINKOWSKI REDUCTION ...... 61 3.8. INVERSION WITH LC INVERT ...... 61 3.8.1. DATA SELECTION ...... 62 3.8.2. IMPORTING LIGHTCURVE FORMATS ...... 62 3.8.3. SYNODIC PERIOD FINDING ...... 63 3.8.4. SEARCHING POLE ORIENTATION ...... 64 3.8.5. MINKOWSKI MODELING ...... 65

4. OBSERVATIONS 67 4.1. LIGHTCURVES REPORT ...... 67 4.2. OBSERVATIONS REPORT ...... 67 4.2.1. DATABASES REPORT ...... 71 4.3. OBSERVATIONS AND DATABASE GRAPHIC REPORTS ...... 75 4.3.1. SCHMIDT TELESCOPE OBSERVATIONS LIGHTCURVES . 75

5. RESULTS 95 5.1. (22) KALLIOPE ...... 98 5.1.1. DAMIT LIGHTCURVES OF (22) KALLIOPE ...... 98 5.1.2. PERIODS REPORTED FOR (22) KALLIOPE ...... 100

VIII 5.1.3. DAMIT SHAPE RECREATION FOR (22)KALLIOPE USING DAMIT PARAMETERS ...... 100 5.1.4. DAMIT SHAPE RECREATION FOR (22) KALLIOPE FOLLOWING THE PROCESS OF SEARCH PERIOD AND POLES WITH LC INVERT ...... 101 5.1.5. LIGHT CURVE INVERSION FOR (22)KALLIOPE USING ALCDEF AND INAOE OBSERVATIONS ...... 105 5.2. (287) NEPTHYS ...... 112 5.2.1. DAMIT LIGHTCURVES OF (287) NEPTHYS ...... 112 5.2.2. PERIODS REPORTED FOR (287) NEPTHYS ...... 114 5.2.3. DAMIT SHAPE RECREATION FOR (287) NEPHTYS USING DAMIT PARAMETERS ...... 114 5.2.4. SHAPE FOR (287) NEPHTYS USING DAMIT AND INAOE LIGHTCURVES ...... 116 5.3. (711) MARMULLA ...... 122 5.3.1. LIGHTCURVES OF (711) MARMULLA ...... 122 5.3.2. PERIOD SEARCH INTERVAL ...... 122 5.3.3. SEARCHING PERIOD FOR (711) MARMULLA USING ALCDEF AND INAOE LIGHTCURVES...... 122 5.3.4. SEARCH POLES AND MODEL FOR (711) MARMULLA . . 124 5.3.5. GENERATING 3D SHAPE FOR (711) MARMULLA . . . . . 126 5.4. (1117) REGINITA ...... 127 5.4.1. LIGHTCURVES OF (1117) REGINITA ...... 127 5.4.2. PERIOD SEARCH INTERVAL ...... 127 5.4.3. SEARCHING PERIOD FOR (1117) REGINITA USING ALCDEF AND INAOE LIGHTCURVES...... 127 5.4.4. SEARCH POLES AND MODEL FOR (1117) REGINITA . . . 129 5.4.5. GENERATING 3D SHAPE FOR (1117) REGINITA ...... 131 5.5. (1318) NERINA ...... 132 5.5.1. LIGHTCURVES OF (1318) NERINA ...... 132 5.5.2. PERIOD SEARCH INTERVAL ...... 132 5.5.3. SEARCHING PERIOD FOR (1318) NERINA USING ALCDEF AND INAOE LIGHTCURVES...... 132 5.5.4. SEARCH POLES AND MODEL FOR (1318) NERINA . . . . 134 5.5.5. GENERATING 3D SHAPE FOR (1318) NERINA ...... 136 5.6. (1346) GOTHA ...... 137 5.6.1. LIGHTCURVES OF (1346) GOTHA ...... 137 5.6.2. PERIOD SEARCH INTERVAL ...... 137 5.6.3. SEARCHING PERIOD FOR (1346) GOTHA USING ALCDEF AND INAOE LIGHTCURVES...... 137 5.6.4. SEARCH POLES AND MODEL FOR (1346) GOTHA . . . . 139 5.6.5. GENERATING 3D SHAPE FOR (1346) GOTHA ...... 141 5.7. (1492) OPPOLZER ...... 142

IX 5.7.1. LIGHTCURVES OF (1492) OPPOLZER ...... 142 5.7.2. PERIOD SEARCH INTERVAL ...... 142 5.7.3. SEARCHING PERIOD FOR (1492) OPPOLZER USING ALCDEF AND INAOE LIGHTCURVES...... 142 5.7.4. SEARCH POLES AND MODEL FOR (1492) OPPOLZER . . 144 5.7.5. GENERATING 3D SHAPE FOR (1492) OPPOLZER . . . . . 146 5.8. (3028) ZHANGGUOXI ...... 147 5.8.1. LIGHTCURVES OF (3028) ZHANGGUOXI ...... 147 5.8.2. PERIOD SEARCH INTERVAL ...... 147 5.8.3. SEARCHING PERIOD FOR (3028) ZHANGGUOXI USING ALCDEF AND INAOE LIGHTCURVES...... 147 5.8.4. SEARCH POLES AND MODEL FOR (3028) ZHANGGUOXI 149 5.8.5. GENERATING 3D SHAPE FOR (3028) ZHANGGUOXI . . . 151 5.9. (3800) KARAYUSUF ...... 152 5.9.1. LIGHTCURVES OF (3800) KARAYUSUF ...... 152 5.9.2. PERIOD SEARCH INTERVAL ...... 152 5.9.3. SEARCHING PERIOD FOR (3800) KARAYUSUF USING ALCDEF AND INAOE LIGHTCURVES...... 152 5.9.4. SEARCH POLES AND MODEL FOR (3800) KARAYUSUF . 154 5.9.5. GENERATING 3D SHAPE FOR (3800) KARAYUSUF . . . . 156 5.10. (4713) Steel ...... 157 5.10.1. LIGHTCURVES OF (4713) STEEL ...... 157 5.10.2. PERIOD SEARCH INTERVAL ...... 157 5.10.3. SEARCHING PERIOD FOR (4713) STEEL USING ALCDEF AND INAOE LIGHTCURVES...... 157 5.10.4. SEARCH POLES AND MODEL FOR (4713) STEEL . . . . . 159 5.10.5. GENERATING 3D SHAPE FOR (4713) STEEL ...... 161 5.11. (5692) SHIRAO ...... 162 5.11.1. LIGHTCURVES OF (5692) SHIRAO ...... 162 5.11.2. PERIOD SEARCH INTERVAL ...... 162 5.11.3. SEARCHING PERIOD FOR (5692) SHIRAO USING ALCDEF AND INAOE LIGHTCURVES...... 162 5.11.4. SEARCH POLES AND MODEL FOR (5692) SHIRAO . . . . 164 5.11.5. GENERATING 3D SHAPE FOR (5692) SHIRAO ...... 166

6. CONCLUSIONS AND FUTURE WORK. 167 6.1. GENERAL CONCLUSIONS ...... 167 6.2. RESULTS CONCLUSIONS ...... 167 6.2.1. ASTEROIDS WITH PARAMETERS DETERMINED BY DAMIT DATABASE...... 168 6.3. ASTEROIDS WITH OUT ENTRIES IN DAMIT...... 171 6.4. ASTEROIDS WITH NOT ENOUGH COVERAGE OF α or PABL . . . 171 6.5. FUTURE WORK ...... 172

X 6.5.1. RECOMMENDATIONS TO MAKE THE FUTURE OBSERVATIONS ...... 172

List of Figures 173

List of Tables 181

XI

Chapter 1

INTRODUCTION AND BACKGROUND

Measure what is measurable, and make measurable what is not so. Galilei

1.1. INTRODUCTION

Scientists and philosophers have been trying to measure, sort out classes, establish ratios and to find reasons why the objects in the physical world are the way they are. To achieve this it is necessary to use instruments but even more, mathematical models that are more sensitive and precise. In 1609, Galileo constructed and pointed skyward his telescope with a power to magnify 20 times. Since that , the humankind was able to make out mountains and craters on the , rings around , sunspots and of .

The history of the study of asteroids (or minor planets) began with the discovery of . Located in the main belt, it was observed for the first time in January 1st, 1801 by Guisseppe Piazzi. Ceres is the biggest object in the main belt orbiting around the . Since then we have found more than 746412 1 objects orbiting around the sun and this number increases every week. In spite of asteroids being relatively smaller, in comparison to other objects in the solar system, they can cause extensive damage if one reaches a planetary atmosphere or surface. We can observe the devastation caused by the impact of an asteroid on a planet in the events of: Tunguska 1908, Chelyabinsk fireball in 2013 or Shoemaker-Levy 9 ; this last one caused an explosion the size of the Earth when it entered the atmosphere of Jupiter in 1993. In addition to this danger, there is another strong reason to study asteroids, this reason is that they are

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1 also sources of valuable minerals and metals, only if we can reach them, and we will reach them in the future.

In addition to the danger they cause and the valuable materials inherent in asteroids, another strong motivation for studying asteroids is the scientific curiosity. Because the asteroids remain in the same geological conditions since the formation of the solar system 4.6 billion ago (Bouvier, 2010); the are in constant evolution, because of gravitational interactions with planets and YORP effect. The current state of all objects orbits of the entire solar system is the result of this process. Understanding the evolution of our solar system will help us understand the evolution of other solar systems.

Asteroids have gained relative importance in recent decades, and that is why our knowledge of asteroids is constantly increasing. The databases of discovered asteroids expand every day with new objects that are being discovered, even beyond Neptune. That is why there is a goal to be reached; to find, catalog, classify and parametrize all the objects that can be observed with current instruments.

The achievement of this objective is only possible through an international effort. A large number of asteroid discoveries and characterization has been possible using automated telescopes like Pan-STARRS 2 and Catalina Sky Survey(CSS)3; or space missions like Dawn 4 or Hayabusa 5 that provide new data. Studying, and classifying this data reveals the nature of dwarf planets and asteroids. This information is used to know how our Solar System was formed and how it arrived at the current "stable state" of all its components. Consequently, we could find a correct model of how other planetary systems evolve. In the following pages, we will deal with how to observe the asteroids and unveil information, to eventually perform space missions and perhaps do a future mining business.

There are different ways to obtain asteroid characteristics. The most expensive techniques are the space missions that take direct images and information in-situ; missions like: Galileo; on its way to Jupiter; it flew over (951) Gaspra at a distance of 1900 Km, it was the first overflight of an asteroid, also passed near of (243) Ida. NEAR overflew (253)Matilde, then made the first orbit over (433) Eros asteroid and landed it . Cassini-Huygens overflew (2685) Masursky. overflew (5535) Annefrank. Hayabusa overflew (25143) Itokawa, it was the first mission that brought back

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2 samples of an asteroid to the Earth. New Horizons overflew (132524)APL. Rosetta overflew (28767)Steins and (21)Lutetia. Dawn overflew (4)Vesta. Chang’e 2 overflew (4179)Toutatis. OSIRIS-REX will bring back samples from (101955)Bennu to Earth. The coverage of these space missions compared to the asteroid population is almost null.

Ground-based techniques like photometry, spectroscopy, polarimetry, high-resolution imaging, and asteroid radar research are applied to the study of characteristics of near-Earth asteroids and main belt asteroids.

1.2. SPACE ENVIRONMENT DELIMITATION AND DEFINITION

For scientific purposes the space environment is the space beyond the last atmospheric layer of the Earth; the exosphere, which ends approximately at 10,000 kilometers away from the Earth. This "unlimited" space includes the interplanetary, and interstellar.

The interplanetary environment of our solar system is crowded with rocky, silicates, metallic and objects; there are even some dwarf planets or minor planets like Ceres, and the (14674)INAOE asteroid. In the space environment we can find electromagnetic radiation which can be solar wind, cosmic rays, magnetic fields, gamma rays and visible light.

1.3. ASTEROIDS MORPHOLOGY AND DEFINITION

1.3.1. DEFINITION From the Greek etymology of the word ’asteroid’ is ’-like’ object. According to resolutions 5 and 6 of the XXVIth General Assembly of the International Astronomical Union (IAU), which was held in Prague, Czech Republic, during August 2006, it was determined that the solar system is now made up of planets, dwarf planets, and small solar system bodies (e.g., asteroids and ).

OFFICIAL DEFINITION Resolution 5 of International Astronomical Union (Dymock, 2010).

3 A planet is a celestial body that: is in orbit around the Sun. • has sufficient for its self-gravity to overcome rigid body forces so that • it assumes a hydrostatic equilibrium (nearly round) shape. has cleared the neighborhood around its orbit. A dwarf• planet is a celestial body that is in orbit around the Sun, • has sufficient mass for its self-gravity to overcome rigid body forces so that • it assumes a hydrostatic equilibrium (nearly round) shape. (Figure 1.1) has not cleared the neighborhood around its orbit. • is not a satellite. When asteroids• and comets are seen through a telescope they appear almost the same, but they have characteristics that differentiate them, the main one is the coma of comets. Comets can have two tails, one made of dust and one made of gas. On the other hand, an inactive asteroid or one that is far from the sun lacks a coma. Because of this, when scientists are trying to classify objects into groups, some asteroids are difficult to categorize. Some cases are asteroids-comets, such as the object (118401) Linear which has a dust tail similar to a comet; there are comets that move as asteroids, such is the case of P / 2005 U1, (7968) Elst-Pizarro, P / 2008 J2 (Belshore) and P / 2008 R1 (Garradd), which move in a typical circular orbit similar to an main belt asteroid (Dymock, 2010).

To be able to differentiate a comet from an asteroid is necessary to consider: Comets and asteroids are celestial bodys that, (Figure 1.2 ) (Dymock, 2010) Comets have coma. • Asteroids tend to be in low inclination has low eccentricity prograde orbits. • Comets are in high inclination and high eccentricity orbits. The encyclopaedia• of the solar system, defines a minor planet as "a rocky, carbonaceous or metallic body, smaller than a planet and orbiting the Sun." (McFadden, 2006)

The topic of this thesis, are asteroids, so comets will not be mentioned again in this work.

4 Figure 1.2: Breakup of comet 73/P. The Figure 1.1: Dwarf Planet Cross-section original fragments were created during a Model (Dymock, 2010) splitting event in 1995. Figure displays a wider field of view, showing several of the original fragments. (Dymock, 2010)

1.3.2. RUBBLE PILE OR MONOLITHS The internal structure of asteroids can be classified into two types: Rubble Pile Monolith or solid body The "Rubble Pile" asteroids: are actually fragments that are slightly linked by their gravity. For example, the asteroid (25143) Itokawa, which was formed by the reassembly of debris generated in a collision of larger objects, (Figure 1.3). Something characteristic of the rubble piles is that they have a rotational period less than or equal to 2.5 , a faster rotation would cause the fragments that form it to separate. The diameters observed in this type of objects have a range from 150 to 300 meters.

An asteroid with a diameter between 100 and 150 meters is most likely a monolithic asteroid, or if it has a of more than 2.5 hours, it is surely a solid body.

To sum up a definition of an asteroid: Asteroids are celestial objects that lack a coma, that shine by reflected sunlight and they are not satellites, they can be a rubble pile or a solid body, and it has not cleared the neighbourhood around its orbit.

1.3.3. BINARIES OR MICRO-SYSTEM ASTEROIDS A large number of asteroids are binary; even an asteroid may possess one or more small moons. The binary asteroids can begin their existence as a rubble pile, which by increasing their rotation speed due to non-gravitational forces such as the YORP effect (Yarkovsky-O’keefe-Radzievskii-Paddack), their members are separated orbiting each other. Examples of this are the asteroid (243) Ida the first system found in 1994 by the Galileo spacecraft, or (65803) Didymos which is targeted by the AIDA

5

optional number for the discovery order within that period. (Dimock, 2010).

For example, the asteroid 2005 YF127 was discovered in 2005, the letter "Y" indicates that it was discovered during the period between December 16 and December 31; the letter "F" and the the number 127 indicates that it was the 3181st object discovered of that period; due to the alphabet, with out the "I", having "looped 127 times" and that the letter F is the sixth letter of the alphabet, we have (127 x 25) + 6 = 3181. (Badescu, 2013) Table 1.1 shows shows the Asteroid designation of Bower for new asteroids.(Peebles, 2000).

Table 1.1: Bower asteroid discovery periods.(Peebles, 2000)

Half-Month of Discovery Letter Designation Half-Month of Discovery Letter Designation January 1-15 A July 1-15 N January 16-31 B July 16-31 O February 1-15 C August 1-15 P February 16-29 D August 15-31 Q March 1-15 E September 1-15 R March 16-31 F September 16-31 S April 1-15 G October 1-15 T April 16-31 H October 16-31 U May 1-15 J November 1-15 V May 16-31 K November 16-31 W June 1-15 L December 1-15 X June 16-31 M December 16-31 Y

The current designation was determined by the MPC. This designation follows the process detailed below: When an asteroid it is discovered is denoted with a provisional name. This name could be given by the first observer. Then, the new object is confirmed; This confirmation is made by further observations or observations made by another observatory. After the confirmation is made, MPC gives a standard provisional designation. This provisional designation matches the designation of Bower. When the orbital parameters are well defined, then the asteroid is designated with a permanent number and name. An example of a permanent designation is 1996 XW18 discovered in 1996 by M. Tichý and Z. Moravec, after parameters were well determined became (12448) Mr. Tompkins. (Shmadel, 2003) It is important to note, that nowadays almost all the new asteroids are discovered by automatic surveys systems that are placed on Earth and space.

8 1.3.7. ORBITAL ELEMENTS From the geocentric model of the universe and back in the days of Johannes Kepler and Tycho Brahe, going through Carl F. Gauss and the determination of the orbit of (1)Ceres, the idea of our place in the universe has evolved. The Orbital Parameters are the current mathematical language to uniquely identify a specific orbit. They are described in the table 1.2 and figure 1.6:

Figure 1.5: Schematic definition of orbital elements of asteroids. (Badescu, 2013)

These Keplerian elliptical orbital elements, also called "proper orbital elements", are a set of independent parameters. These describe the orbital motion of a body by its instantaneous position and velocity in space at a particular time. These elements are obtained from observations, and they can predict very accurately the position of a body in the near future or past.

The predictions of positions are called ephemerides. The ephemerides are accurate only in near future or past because asteroids experience gravitational perturbations from gravitational and others non-gravitational forces like YORP effect, but the later kind of forces change the orbit very slowly, on the order of 104 years.

1.3.8. ORBIT PERTURBATIONS There are diverse mechanisms that modify the orbit of an object in the space, and if the object is small, like asteroids these mechanisms perturb its orbit faster. The orbit

9 Table 1.2: Orbital elements

Name Symbol MPC Description Notation Mean M M Although an accurate definition anomaly is a little more complicated, this is essentially the current angular distance from perihelion to the present position of the asteroid measured in the direction of motion. Semi- a a Half of the length of the long axis mayor of the ellipse. axis Eccentricity e e A measure of the deviation of the orbit from a circle (all asteroid orbits are ellipses) e = c/a. Typical main belt asteroid e =0.1 0.2 Inclination i Incl The angle between the planet− plane and the orbit of the asteroid and the . If the inclination is > 90◦ then the motion of the object is considered to be retrograde. Longitude Ω Node The direction in space of the line of the where the orbital plane intersects ascending the plane of the ecliptic. Its node measured eastwards (increasing RA) from the vernal equinox (First point of Aries) Argument ω P eri It defines how the mayor axis of of the orbit is oriented in the orbital perihelion plane and it is the angle between the ascending node and the perihelion point measured in the direction of motion. Epoch The date on which a set of orbital elements were calculated. perturbations could provoke an asteroid impact on a planet or, just send it out of the solar system.

Sometimes asteroids, fall in a region of gravitational stability keeping the asteroid in the same orbit for long periods of time. These regions are the lagrangian planetary

10

What causes this effect is re-emission of radiation into the space, generating a thrust that changes the orbit and spin of the asteroid.

Figure 1.7: How the Yarkovsky effect changes the orbit of an asteroid. (Dimock, 2010)

The change of the semi-major axis can cause the fall of an asteroid into the Kirkwood gaps; the asteroid would be kicked out of the main to outer space or into the inner solar system, where the rocky planets are. (Rubincam, 2007) What causes this effect is the thermal re-emission of radiation received from the Sun back into the space, generating a thrust that changes the orbit and spin of the asteroid. An example of this effect is the one measured at (6489) Golevka which, from 1991 to 2003, drifted 15km from its predicted position. The orbit was established with great precision by a series of radar observations (in 1991,1995, 1999 and finally 2003) made with the Arecibo radio telescope. (Chasley, 2003).

Yarkovsky-O’Keefe-Radzievskii-Paddack, or YORP effect is a second order variation of the Yarkovsky effect that provokes a change of asteroid spinning. Although its a weak and almost immeasurably force, its accumulation during periods of the order of 108 years causes changes in the spin of an asteroid . Small asteroids are more susceptible to this effect.

12 A theoretical example of this is (951)Gaspra; it has a diameter of 6km, and a semi-major axis of 2.21 UA. It will take about 240 106 years to change its rotation period from 12 to 6 hrs. (Rubincam, 2010) ×

If a rubble pile asteroid increases its rotational period above 2.5h, its material will be shed, which may accumulate to form one or more small, nearby satellites. (binary or a system of three asteroids).

HIGH SPEED COLLISIONS

The population of the main asteroid belt is the result of millions of years of evolution, which began with the birth of the solar system. This evolution includes high-speed collisions that were more usual at the birth of the solar system than nowadays. Collisions caused the fragmentation of original bodies. (Lisauer, 2018)

Asteroid Families were originated by collisions of bigger bodies, which caused the fragmentation and the scatter of the debris with orbits that evolved in different ways, producing what we call today asteroid families. A collision between asteroids and/or dwarf planets is erosive or fragmentary. The outcome of a collision depends on the energy and relative velocities of the objects.

1.4. MAIN BELT ASTEROIDS

The main asteroid belt is the circumstellar disc in the solar system located between the orbits of planets Mars and Jupiter from 2.0 and 3.5 AU from the Sun. It is distinguished because of its large asteroid population in the solar system. It was originated from the primordial solar nebula as a group of planetesimals. The total mass of the main asteroid belt is approximately 4% that of the moon, and about half of that mass is contained in the four largest asteroids: Ceres, Vesta, Pallas, and Hygiea. (Williams, 2015).

Main belt asteroids have low orbital velocities (several km s−1), these speeds exceed −1 their escape velocities, for example, Ceres has a ve 0.5kms . ≥ In addition to the large quantity of asteroids and minor planets that are in the main asteroid belt, we can find asteroids all over the solar system.

13 1.4.1. MAIN BELT FORMATION AND DISTRIBUTION OF ASTEROIDS The main asteroid belt is a relatively stable dynamic disk. The orbits of its asteroids were modified by the gravity of Jupiter, then these changes caused collisions and the original asteroids were fragmented. These high-speed collisions shattered the asteroids creating fragments that are known as asteroid families. We know to which family different asteroids belong from the study of their orbital elements, performing time regressions that make it possible to calculate when the collision occurred and from the correlation of some of their physical properties, morphological and chemical composition. The figure 1.8 shows a correlation of orbital elements that denotes some families (Nesvorny et al., 2015).

Figure 1.8: (a) Clustering algorithm applied to he asteroid belt separates dynamical families (yellow) from the background (red). (b) Variation in reflectance properties of main belt asteroids. Here we plot 25,000 asteroids that were observed by both SDSS and WISE, The color code was chosen to highlight the /color contrast of different families. (Nesvorny et al., 2015)

The study of asteroid families began with the work of Hirayama in identifying groups of asteroids with similar orbital parameters. His study suggested that they had an origin as fragments of a larger common object(Nesvorny et al., 2015). The Koronis, Eos and Themis families were the first to be declared, initially with few members, but these members increased over time, reaching thousands of members and finding new families. (Nesvorny et al., 2015).

The current distribution of the mass and number of asteroids in the solar system is the

14 consequence of its own evolution, so studying the asteroids provides information of this evolution and, in a broad point of view, the structure and development of all planetary systems.

The formation of a planetary system could be considered in three stages: 1) The formation of the solar nebula by gravitational collapse, this consists of a rotating disk of dust and gas; 2) In the center of the disk, a protostar, in the intermediate zone, ice and dust begin to grow by an accretion process; this is where planetesimals are generated; 3) Gravitational interactions that cause exchanges of angular momentum magnitudes among all objects. Some of these objects became planets and satellites, and others will be the so-called small bodies of the planetary system, such as asteroids and comets. The last stage continues for a long period until the system reaches stability.

A study of Nesvorný, D.; Broz, M. and Carruba, V. identified 122 remarkable families with about 100,000 asteroids; this study uses the hierarchical clustering method (Zappalà et al., 1990):

Table 1.3: Main Belt Asteroid Familes(Nesvorny et al., 2015)

Inner Main Belt, 2.0

16 Table 1.3 – Continued from previous page 158 Koronis(2) 246 3438 Inarradas 38 81 Terpsichore 138 7468 Anfimov 58 709 Fringilla 134 1332 Marconia 34 5567 Durisen 27 106302 2000 UJ87 64 5614 Yakovlev 67 589 Croatia 93 7481 San Marcello 144 926 Imhilde 43 15454 1998 YB3 38 - P/2012 F5 (Gibbs) 8 15477 1999 CG1 248 816 Juliana 76 Outer Main Belt, 2.82 < a < 3.5 AU, i > 17.5◦ Num. Name Members Num. Name Members 31 Euphrosyne 2035 1303 Luthera 163 702 Alauda 1294 780 Armenia 40 909 Ulla 26 - - -

1.5. OTHER MORPHOLOGY PARAMETERS AND CHARACTERISTICS

Besides the characteristics we have already discussed, there are some others morphological characteristics which are of crucial importance: G-H magnitude, rotational period, axis or axes of rotation, form, shape, and size.

1.5.1. (H) ASTEROID The magnitude of celestial objects was classified millennia ago by an ancient Greek man named Hipparchus of Nicaea (190BC - 120BC). He was an astronomer and mathematician, famous for the making of an accurate model of the Moon and Sun orbits. He calculated as well the precession of the Earth axis.

Hipparchus also introduced the concept of stellar magnitude. He set what we call today apparent visual magnitude (mv) or just V, a number that indicates the brightness of an object seen by an observer on Earth. Hipparchus set the magnitudes of stars in a scale from 1 to 6, being 1 for the brightest stars and 6 for the faintest, according to this, Hipparchus classified nearly 1000 stars in this magnitude scale. The range of this "ancient photometry method" does not take into account the Sun, the moon, stars like Sirius, or the planets and objects that the human eye could not see.

In 1856, Norman Pogson implemented a new magnitude scale; he used early photometric studies that indicates that the brightness of a star with mv = 1 was 100 times brighter than a star with mv = 6. Pogson noticed that the naked eye has a logarithmic response to the light because Hipparchus used the naked eye to make the

17 original magnitude system. This discovery was used by Pogson to readjust and set a new stellar magnitude set point, he proposed Polaris to be the set point as mv =0, and a logarithmic scale of √5 100 2.512 to be considered between magnitudes; in this way there is precisely a factor≈ of 100 in brightness. (Hoskin, 1999). Astronomers later discovered that Polaris is slightly variable, so they switched to Vega as the standard reference star assigning the brightness of Vega as the definition of zero magnitude at any specified wavelength. The new magnitude scale extends to all of the objects we can see with the modern technology, or even artificial objects like satellites, ISS and Hubble telescope.

Although this concept is used to set asteroids visual magnitude at one determined position and time, the magnitude of an asteroid is determined by direct observation or calculated from the H magnitude, or asteroid absolute magnitude, (in other objects like planets absolute visual magnitude is (Mv or just (V).

To summarize, a formal definition of absolute magnitude of an asteroid: "Absolute magnitude of an asteroid (H) is the visual magnitude an observer would record if the asteroid were placed 1 (AU) away, and 1 AU from the Sun and at a zero phase angle" 6. A technical definition is: The absolute magnitude of an asteroid (H) at zero phase angle and at 1 AU heliocentric and geocentric distances. 6

1.5.2. SLOPE PARAMETER (G) "It is related to the opposition effect. This is a surge in brightness, typically 0.3 magnitudes, observed when the object is near opposition. Its value depends on the way light is scattered by particles on the asteroid’s surface. It is known accurately for only a small number of asteroids, hence for most asteroids has an assumed value of 0.15." (Dymock, 2007)

1.5.3. H-G The parameter H-G magnitude system was adopted by the International Astronomical Union in 1985, it was developed to predict the magnitude of an asteroid as a function of solar phase angle. (Dymock, 2010)

The G parameter is assumed, because it has been derived only for 0.1% of all asteroid population.(Bucheim et al., 2010)

A considerable amount of observations must be performed to obtain the G slope; these

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18 observations must be analyzed by astrometry and photometry light curves.

1.5.4. ASTROMETRY AND PHOTOMETRY Astrometry and photometry are necesary to derive H-G parameters.

ASTROMETRY The etymology of this word is "measure of stars"; it refers to measuring the position of stars (or any object) in the sky. Astrometry is the measurement of positions, parallaxes, and proper motions of an astronomical body in the sky. (Dimock, 2010). It provides information about the asteroid location in the solar system, the distance of the object from Earth and Sun, and the angle formed by these two seen from the object (phase angle), when phase angle is at 0◦ is called opposition, figure 1.9(b).

PHOTOMETRY It is the branch of science that deals with measuring the light flux intensity of an astronomical object. (Casagrande, 2014). Photometry is used to calculate the absolute magnitude (H) of asteroids. But the absolute magnitude it is not quite absolute. The absolute magnitude of an asteroid can have more than one value. The values quoted in papers or internet data bases are usually an average over several oppositions. The absolute magnitude of an asteroid and the slope parameter (G) can have more than one value; these are affected by the position of the rotation axis of the asteroid (“aspect angle” or “rotational phase angle”), and its obliquity. So, to determine H-G values, several observations must be done at several oppositions. (Dymock, 2010)

Table 1.4: Variation in values of H and G for (1)Ceres and (8) Flora. (Dimock, 2010).

Asteroid H G Mean H/G (1) Ceres 1990 3.29 0.08 1991 3.31 0.07 3.33/0.09 1992 3.39 0.20 (8) Flora 1990 6.42 0.27 1992 6.52 0.37 6.51/0.36 1993 6.60 0.36

Table 1.4 shows how H and G varied year by year for two asteroids. Figure 1.9 (a) shows how reduced magnitude and absolute magnitude are calculated. Absolute magnitude is calculated from the apparent visual magnitude (V) and

19

Absolute photometry determines the flux of an object, it finds absolute magnitudes subtracting the instrumental bias, sky flux contribution and air mass extinction. (Dymock, 2010) Differential photometry measures the differential flux of an object using another comparison object with constant flux, it subtracts the instrumental bias. (Dymock, 2010)

OPPOSITION EFFECT

Considering that opposition effect is built on what we call slope parameter G, the opposition effect increases the brightness of the asteroid in about 0.3mag. This change of an asteroid brightness (∆mag) due to opposition effect depends on its size, shape, and surface roughness. These parameters generate light scattering reflected from the sun.

1.5.5. H-G PARAMETER AND H-G1G2

GEOMETRIC ALBEDO (Pv)

" is the ratio between actual reflected light and what would be reflected by a perfectly white sphere, which reflects all incident light, e.g., its geometric albedo is 1.0. It is common on normalized optical spectra of asteroids for the vertical axis to be adjusted so that the albedo at a wavelength of 0.55 mm has the value of 1." (Dymock, 2010). Albedo is a number that denotes the light reflectance of a planetary object. Most main belt asteroids have between 0.05 and 0.25 (Dimock, 2010). The bond Albedo (A), diameter, G, H and Pv, are correlated in equations 1.3 and 1.4 (Bowell et al.,2008).

− H 1329 10 5 A =( × )2 (1.3) D

A = pvq; q =0.290+0.684G (1.4) Where: A = bond albedo; is consider the reflectance of all electromagnetic spectrum frequencies. Pv = geometric albedo (related to the V Johnson photometric light frequencies interval). q= constant that relates Pv and A. G= G parameter or slope between magnitude and phase angle of an asteroid. D= asteroid diameter.

21 SIZE OF THE ASTEROIDS

Talk about the diameter of an irregular object is not entirely precise. However, there is a relationship between absolute magnitude and diameter. Thus we can calculate the diameter of an asteroid from equation 1.5, (Muinonen et al.,2010).

log D =3.1236 0.2H 0.5log Pv (1.5) 10 − − 10 Rewriting equation 1.5:

− − D = 10(3.1236 0.2H 0.5log10 Pv) (1.6) Or if only magnitude (H) is known then we can use 1.7 to derived the asteroid diameter. (Harris 2014).:

17.75−H D = (1km) 10 5 (1.7) × Asteroid collisions shaped the population of the main belt asteroids. It is not rare to find that the number and the size distribution correspond to a power law. We can approximate de sizes distribution by a power law. The power law is just valid in an radius (R) value interval, equation 1.8 (Lissauer, 2013).

N R ζ N(R)dR = 0 (1.8) R0 R0  Where:

R differential interval (Rmin

FORM OF THE ASTEROIDS

Asteroids are irregular objects, except for those that got together sufficient mass for their self-gravity to overcome rigid body forces so that they assume a nearly round form, like (1) Ceres. “Potato” form is very common in these objects. We show some photographs at close-range of nine asteroids in figure 1.10. Images at close ranges with space probes, radar observations, or the inversion of photometric light curves can reveal the shape and form of an asteroid. The lightcurve inversion technique is the main topic of this " Morphological Properties Determination of Main Belt Asteroids " thesis.

22 Figure 1.10: Views of the first four comets (lower right) and nine asteroid systems that were imaged close-up by interplanetary spacecraft, shown at the same scale. The object name and dimensions, as well as the name of the imaging spacecraft and the year of the encounter, are listed below each figure. Note the wide range of sizes. Dactyl is a moon of Ida. (Lissauer, 2013)

1.6. DETERMINATION OF ASTEROID CHARACTERISTICS FROM LIGHTCURVES

The lightcurve method is used to measure flux variations (∆mag) of an asteroid (or any object). The lightcurve is the plot of ∆mag versus time or asteroid spin phase. Lightcurves allow to derive the rotation period of an asteroid and the lightcurve amplitude.

When we construct the lightcurve of an asteroid, the first plot we obtained is the "raw plot" that shows the change of asteroid magnitude by comparing it with cataloged comparisons stars. This plot shows the relationship between magnitude and time expressed in Julian date.

The phase plot, is the result of a Fourier analysis to determine the period of rotations, it shows the relationship between magnitude change and rotation asteroid phase, with the phase expressed as one, and fractions of the stage of asteroid rotation.

23 Figure 1.11: Light curves of (1831) Nicholson, Raw Plot and Phase Plot [Observations by author, plot and analysis made with MPO Canopus.

As is shown in figure 1.11 a typical form of lightcurve in phase plot has two peaks. This is because of the rotation of the typical "potato" form of the asteroid.

With many lightcurves we can determine asteroid characteristics like (Dymock, 2010): its shape. its rotation period. its inner composition (rubble pile or monolithic). the orientation of its spin axis.

24 Chapter 2

Problem Statement

Equipped with his five senses, man explores the universe around him and calls the adventure Science. Edwin Powell Hubble

2.1. BRIEF IMPLICIT PHILOSOPHICAL JUSTIFICATION

The word scientist has its origin from Latin words scientificus, scientia and the suffix -st, that means (to do). A scientist is a person who participates and realizes a regular activity (scientific work) to acquire knowledge. The work of a scientist differs from the work of an engineer. The word engineer has its etymology in the Latin word ingenium, meaning intelligence; engineer applies scientific knowledge to solve human problems or improve comfort, health or any human necessity, even the trivial ones.

Sometimes the engineers need to become scientists, and sometimes the scientists become engineers, their activities are in close relationship. The more notorious difference between these is the purpose. A scientist’s work consists in classifying, measuring, quantifying, and finding correlation between phenomena, or disconnection to know the causes of the effects in nature, to see the invisible in the micro and macrocosms, to find what is beyond current knowledge; and all this is just due to the desire of greater knowledge. What other purpose would have to watch through a telescope and see some luminous points in the sky, or to give a name and category the beaks of the birds? What other purpose could have to know if there are four or ten sub-atomic particles?.

The scientists find what defines the life and death when they watch more deeply and, find correlation between those apparently unrelated phenomena. Scientists find knowledge in those distant luminous dots, or the invisible atom particles. They give us the ability to go or see, where no other human has gone or watched ever before.

25 Ergo, this thesis work must be done because it can be done by the mere fact that it is part of scientific work.

On the other hand, scientific work is a social and historical construction; its development depends on the constant improvement of its models and paradigms.

2.2. WHY DETERMIN ASTEROIDS PROPERTIES?

Asteroids stopped being those "vermins" that used to draw tracks on photographic plates, and they became objects of interest not just for scientific curiosity, but also to maintain humankind alive. The scientist community has discovered the relationship between the thermal emission from the surfaces, the physical and the dynamical properties of the asteroids, as well as relevant information that shows the ranges of physical and orbital parameters. These discoveries give us clues to find the nature of the mechanics that govern a planetary system.

The discovery of trans-Neptunian objects like , Hydra, Cerbero, and Makemake produced a new celestial objects classification. (1)Ceres has officially become a dwarf-planet, according to the IAU in 2006 (Dimock, 2010).

The limited scope and the high cost of space missions to find and characterize the vast number of objects make ground-based observations very important.

Ground-based activities like asteroid astrometry, spectrography, photometry and mathematical techniques like light curve inversion are used to find and characterize asteroids. Amateur astronomers can perform these activities, and in fact, backyard astronomers continue to be a source of data in the field of asteroid surveying.

Despite of all progress that humankind has achieved, more goals have appeared: The and densities determination of known asteroids. The size and shape determination of known asteroids. The rotation properties determination of known asteroids. The taxonomic classification of known asteroids. To find the precise correlation between taxonomy, spin, sizes, shapes, and mineralogy. To determine the orbital parameters of all known asteroids. To have a better understanding of the evolution of the solar system by analyzing statistical distributions of physical and dynamical properties of asteroids.

26 To find how the statistical distributions of characteristics of asteroids differ or agrees for main belt asteroids, near Earth asteroids, and trans-Neptunian asteroids.

2.3. DISCOVERIES MADE BY STATISTICAL ANALYSIS

The discoveries of data and isolated phenomena, tend to break paradigms, such as the life forms in extreme conditions, the first exoplanet or (136199)Eris. Eris, is the second-largest dwarf planet (its orbit is beyond Neptune) and is the 16th most massive in the solar system.(Sicardy, 2011). The discovery of Eris resulted in the change of what is classified as a planet and as a dwarf planet in 2006 by the IAU draft definition of "planet" and "plutons". 1

The study of statistics data leads to discoveries and facts that are not obvious, for example, Kirkwood gaps, main belt families or spin barrier.

2.3.1. RUBBLE PILE SPIN BARRIER In 1979 only 157 asteroid rotation rates were known, in 2005 this known rates were increased to 1686. With this information, various discoveries emerged (Alan et al., 2005): Asteroids larger than 50km have a dispersion of spin rates that are well represented by a single Maxwellian distribution. Smaller asteroids have a more dispersed distribution. Asteroids in the ranges of 1 to 10 kilometers of diameter suggest that even rather small asteroid are “rubble piles”. Among the very slow rotators are some asteroids that are “tumbling” in non- principal axis rotations states. Among the smallest asteroids (less than few hundred metres of diameter) there are some that spin dramatically faster than the “spin barrier”; so it is inferred that these are solid objects. Asteroids smaller than a few tens of kilometers of diameter are affected by radiation pressure torques that tend to either speed up or slow down their spin rates. Both images shown in Figure 2.1 are a clear example of how a large number of data can reveal information which individual or less information can’t prove, the plots of Figure 2.1 graph the rotation period and the diameter of the asteroid in kilometers, the red dashed line marks a rotation spin barrier, below this spin barrier, the asteroids could

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27 be a rubble pile asteroid, this kind of asteroids are slightly gravitationally bounded so a speed above the barrier will take them apart. The spin barrier of asteroids is approximately 2.2 hours period. (Alan, 2005)

Figure 2.1: Comparative between rotations periods data in year 1979 (left) and 2005(right).(Alan et al., 2005)

2.4. EARTH DEFENCE

At the moment, the asteroids in the main belt are far away from Earth and do not represent as much danger as the near-Earth asteroids or even the potentially hazardous asteroids. However, each near-Earth asteroid and potentially hazardous asteroid was once in the main asteroid belt. These asteroids have been expelled from their orbit by processes that includes gravitational resonances, gravitational interactions with planets and experiencing non-gravitational forces such as the YORP effect. Although this orbit change process lasts for thousands or millions of years, it is important to study main belt asteroids to know our potential "enemies".

In order to protect the Earth from an asteroid impact, it is first necessary to find them. These efforts to protect the Earth include both observation and characterization, in particular finding their orbital elements, to determine whether they pose a risk of impacting Earth sometime in the future. Beginning in 204, this task, and others, are carried out bye the International Asteroid Warning Network (IAWN), established under the auspices of the Organization of the United Nations. One of the explicit goals of this program is the characterization of all asteroids, with as much accurate information as possible. Ones IAWN determines that an asteroid has exceeded a threshold of probability of striking the Earth, it would alert the government(s) of the country(ies) that would be impacted. The Space Mission Planning Advisory Group (SMPAG), also established under auspices of the UN, would plan, and if authorized

28 execute space missions to deflect the asteroid to avoid or mitigate the consequences of an impact.

A theoretical example of this danger would be: If an asteroid falls into the 3:1 , its eccentricity will increase and eventually become a Mars-crosser, then Mars could perturb the orbit, or even more, throwing the asteroid into the inner solar system.

The Chicxulub impactor could have had this process. It caused the extinction of many species including dinosaurs in the Cretaceous period. A research made by a team of U.S. and Czech astronomers suggets that Chicxulub asteroid was most likely a part of the Baptistina family. The collision that creates this family broke the original objects into about 140,000 bodies larger than one kilometer in diameter, this collision occurred approximately 160 million years ago. The Chicxulub asteroid after having experimented YORP effect, gravitational resonances in Kirkwood gaps, and gravitational interactions with Mars was propelled to the inner solar system and that eventually causes the impact with Earth.(Dimock, 2010)

2.5. ASTEROID MINING

Commercial asteroids mining will become a fact, it is just matter of time, if humanity continues developing technology (and survives its own annihilation). Planetary Resources, Deep Spaces Industries and Aten Engineering as well as several space agencies are taking this topic very seriously. These organizations will develop ideas that could shape the future of asteroid mining.

Asteroid (436724) 2011 UW158 One example of the potential of this futuristic industry is the asteroid (436724)2011 UW158, figure 2.2, discovered by Arecibo Radio Observatory (Puerto Rico); its observations and data analysis suggest that the value of this object goes from 0.3 to 5,4 billions of dollars. They found that there are platinum and other expensive metals on it. Arecibo Planetary Radar found a shape asteroid much like an unshelled walnut with a diameter of 300 by 600 meters, with rotational period of 37 minutes. The derived period matches the period obtained from visual observations. The asteroid light-curve photometry database catalogs this object as S spectrum class. (Gary, 2016)

Something important to note is its rotational period( 0.5 hours) (figure 2.1) (Gary, 2016). Surely the first targets of mining will be near Earth≈ asteroids and then the entire solar system. To have the most of information of as many asteroids as possible is crucial for the first decisions and the expansion of this future industry.

29 Figure 2.2: Three Arecibo radar images of asteroid 2011UW158 showing a four-minute portion of its 37-minute rotational period. Each pixel is equal to 7.5 metres. The surface of the asteroid looks like a walnut with parallel ridges along the length of the body. (Credit: Arecibo observtory)

2.5.1. GROUND-BASED VISUAL OBSERVATIONS AND ANALYSIS Ground-based observations (like those made in the current thesis work) are a great option to find and characterise asteroids because the coverage of asteroid population is made in a faster and cheaper way; the above due to the low cost of equipment necessary to perform the task. Proof of this is the 2100 2 observatory codes that the has certified.

The caricaturisation of the asteroid 2011UW158 shows that data derived from visual information are as accurate as those made with radar to characterize asteroids, "Ground-based lightcurve observations can help establish limiting parameters before radar observations begin, allowing the astronomers to determine if the radar observations have a reasonable chance of providing useful data." (Brian, 2006).

2.6. FUTURE SPACIAL PROBES TO ASTEROIDS

Considering the future of space missions to asteroids, lightcurve inversion can contribute with the location of landing. The lightcurve inversion model accounts for distance covered by residual bounces as the vehicle comes to rest (considering surface friction coefficient and restitution). A particularly challenging aspect to consider is that some asteroid shapes may have surface locations where a vehicle could get stuck, thus affecting vehicle dynamics on the surface. Conceivably, a hopping rover could be perturbed away from predicted ballistic trajectories by such equilibria. This can affect exploration objectives by constrained the total area that a rover can safely or reliably traverse to on an asteroid surface when stable and unstable equilibrium locations

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30 happen to coincide with surface regions of scientific interest. (Badescu, 2013)

31

Chapter 3

METHODOLOGY

All research in the cultural sciences in an age of specialization, once it is oriented towards a given subject matter through particular settings of problems an has established its methodological principles, will consider the analysis of the data as an end in itself. Max Weber.

3.1. FINDING MORPHOLOGICAL PARAMETERS

It is important to say that in the present work we are not discovering new asteroids, but finding morphological parameters of asteroids with defined orbital elements, in other words, we study asteroids which have its Astrometry well determined. The objectives of this thesis work are: To find the observed asteroids rotation pole orientation. To find the convex-representation of the observed asteroids. To achieve the objectives of this work, we used the Tonantzintla Schmidt Telescope to carry out astronomical observations. We also used other tools like databases to select objects and to get more data necessary to accomplish the requirements of software (and theory) to analyze the information, and get the expected results. The used tools were:

Schmidt Telescope: it was used to make the astronomical observations. CCD SBIG-8300M: it was used as an electronic detector to generate the astronomical pixels. Image Reduction and Analysis Facility (IRAF): it was used for the reduction of RAW images obtained in each observing run, IRAF was used to reduce the noise moreover the bias caused by the electronic of CCD detector. MPO Canopus: it was used to construct the "light curve plot" capturing, the rotation period of the asteroid, and the amplitude of the light curve simultaneously.

33

LOCATION

The telescopes location coordinates on the Earth are of crucial importance; They are needed to determine the position of celestial objects in the sky and to perform their tracking by the telescope control systems.

Table 3.1: Location of INAOE Schmidt Camera, Santa María Tonantzintla, San Andrés Cholula, Puebla, México.(Google Maps)

DATA VALUE Longitude 98◦ 18’ 54.36" West Latitude 19◦ 01’ 53.4" North Altitude 2180 masl Time Zone GMT -6

SCHMIDT TELESCOPE DESIGN

The Schmidt telescope has a catadioptric optic design; the word catadioptric has its root from the Greek words kata (κατα) prefix = because of, dia = through and optomai (oπτoµαι) = to see. The Catadioptric word means that Schmidt telescope is a system that has reflection and refraction elements.

As Figure 3.2 shows, the Schmidt telescope has an aspheric lens (refraction element) and a concave spherical primary mirror (reflection element); the focal length is given by half radius of curvature R, (Valdes et. al., 2016)

The concave spherical primary mirror treats every point in the on-axis and off-axis equal; eliminating all off-axis aberrations such as coma and astigmatism, but creates spherical aberration, see figure 3.3 (Cannon, 1995).

35

3.3. ELECTRONIC DETECTOR

The Tonantzintla Schmidt Telescope has incorporated a CCD detector; This device captures photons and converts them into an electrical charge that, then is translated into "counts". The information can be moved into a computer where it can be visualized and analysed. See table 3.2.

Table 3.2: CCD Characteristics(Santa Barbara Instrument Group, manufacturer of CCD)

Characteristic Symbol Value Description Brand SBIG- SBIG-8300M with 5 Filter Wheel 8300M Physical 17.96mm x Dimensions 13.52mm Pixels 3326mm x 2504 mm Pixel size 5.4µm e− Gain 0.37 ADU Analogue ADC 16 bit Digital Converter Matrix Monochromatic

FILTER WHEEL The filter wheel is an accessory of the CCD that holds filters and allows to change them quickly.

The filters are based in the Johnson-Cousins Standard or Johnson-Cousins Photometric System (this system was extended from UBV to UBVRI in 1973). This system is a well-defined sensitivity to incident radiation. The objective of this is to allow the passage of light through the filter in a defined interval of the electromagnetic spectrum, to which visible light also belongs. (Landolt, 2009), figure 3.6 and table 3.3.

Table 3.3: UBVRI Johnson Standard Characteristics.(Dymock, 2010)

J-C wave Band Color Center frecuency (Å) Band width (Å) U Ultraviolet 350.0 70.0 B Blue 438.0 98.5 V Visual 546.5 870.0 R Red 647.0 151.5 I Infrared 786.5 109.0 C Clear No Filter -

38 Figure 3.6: The Johnson-Cousins UBVRI standard, the plot shows the sensitivity response in 5 colors. (Bessell, 2005)

3.4. SCHMIDT TELESCOPE AND CCD CHARACTERISTICS

Table 3.4: Optic Schmidt Telescope Characteristics

Mirror Characteristics (Valdes et al., 2016) Characteristic Symbol Value Description Diameter D 774mm Physical diameter of the mirror Effective De 762mm Diameter that reflects light and Diameter create the image Radius of Rc 4314.82mm The reflecting surface is a section curvature of a sphere, Rc is the radius of that sphere Rc Focal Distance F 2157.4mm F = 2 Material Pyrex 716 Weight 123.72 Kg Thickness in 134mm Shores Reflecting Deposit of choromium-aluminium surface on the surface in 2015 Aspheric Lens(Corrector Lens) (Valdes et al., 2016) Characteristic Symbol Value Description Diameter Dc 660.4mm Corrector Lens Diameter Material Crystalex Thickness 22.352 Continued on next page

39 Table 3.4 – Continued from previous page Minimal 15 µm Circle of Confusion Dispersion 233.6mm zone distance from center Flattening Field Lens Characteristics (Valdes et al., 2016) Characteristic Symbol Value Description Diameter D 190mm Radius of C1 - Curvature 1 1455.3095mm Radius of C2 1455.3095mm Curvature 2

Table 3.5: Calculed Valules of the Optical System.

Characteristic Symbol Value Description Focal Distance F 2135.24 value with the flattening lens. F latteningLens mm (Valdes, 2016) 1 F 2135.24 Focal Reason F 3.23 Dc = 660.4 arcsec 206265 206265 Plate Scale S 96.6 mm S = F (mm) = 2135.24 Field of View FOV 28.91 x 21.76 arcmin FOVx 28.91arcmin FOVx = 17.96mm 1min × 96.6arcseg/mm( 60sec ) FOVy 21.76 FOVy = 13.52mm 1min × arcmin 96.6arcseg/mm( 60sec ) Image Scale Image 0.52 x 0.52 ImageScalepixel = −3 Scalepixel arcsec ((5.4x10 mm) 96.6arcsec/mm) × Image 0.27 ImageScalepixel = 2 −3 Scalepixel arcsec ((5.4x10 mm) 96.6arcsec/mm)2 × −3 Image Scale 1.04 FOVpix2x2 = ((5.4x10 mm) w/ binning arcsec2 96.6arcsec/mm 2)2 × 2x2 ×

40 3.5. ASTRONOMICAL OBSERVATIONS

Observing the sky through a telescope is an unforgettable and fantastic experience, it is mind-blowing, it makes intense the paradoxical feelings of the greatness and smallness about what a human being means to be.

3.5.1. PLANING AND SELECTING OBJECTS The selection of the asteroids observed for this work had four stages, table 3.6 :

Table 3.6: Observing stages planning.

Stage Database Description and General information Used 1 CALL CALL: collaborative asteroid lightcurve link, ❤♣✿✴✴✇✇✇✳♠✐♥♦♣❧❛♥❡✳✐♥❢♦✴❍✴❝❛❧❧❴❖♣♣▲❈❉❇◗✉❡②✳♣❤♣ It is used to select asteroids that are in opposition in diferent epochs of the year, can filter the , it gets a first approximation of asteroid magnitude and shows the rotational period determined by other observers. 2 MPEph Minor Planet and Comet Ephemeris Service, ❤♣✿✴✴✇✇✇✳♠✐♥♦♣❧❛♥❡❝❡♥❡✳♥❡✴✐❛✉✴▼❊♣❤✴▼❊♣❤✳❤♠❧, It is used to confirm that the selected object can be observed during a given observing run, by requesting their coordinates and magnitudes with a steep of a few minutes. 3 ALCDEF Asteroid Lightcurve Photometry Database ❤♣✿✴✴❛❧❝❞❡❢✳♦❣✴ It includes lightcurves generated by other observers, the phase angle of each light curve, observing/opposition date, includes Lightcurve amplitudes and periods founded in different observations. 4 DAMIT Database of Asteroid Models from Inversion Techniques, this database contain 3D asteroid models that were already computed. It’s operated by The Astronomical Institute of the Charles University in Prague, Czech Republic. ❤♣✿✴✴❛♦✳♦❥❛✳♠❢❢✳❝✉♥✐✳❝③✴♣♦❥❡❝✴❛❡♦✐❞✸❉✴✇❡❜✳♣❤♣

STAGE 1:CALL DATABASE

In this stage, we obtained a first list of candidates of favorable asteroids to observe. Because of three principal variables: magnitude, opposition date, and rotation period. These variables determine which objects can be observed in our location and with the infrastructure that we use. These limitations are:

41 Due to light pollution, Tonantzintla Schmidt telescope can observe objects brighter than magnitude V=17.5 but only at very clear nights. (This empirical data is supported by Arne Henden (MPAPW 1999 meeting) (Warner, 2006) Objects must have a maximum period of 5 hours. If it is longer than that, it will require several nights to complete its rotational period. Declination of the objects was selected grater than -10◦, in order to satisfy the condition that at the beginning of observation (our angle H= 04 hours East) the altitude of the object has to be grater than 30◦. The information entered in the database is: Number; from 0 to 999999; it filters the asteroid by number. Name; Option=Any; It filters asteroid by name. Year and Month of asteroid opposition; To increase the number of objects we should search additionally one month before and one month after the date of observation. Family/Group; We will limit the search for inner, middle and outer main belt asteroids. Favourable Status: option=Ignore; This option shows asteroids which have brightest apparitions in the period between 1995 and 2050. With the option "ignore=on" it includes favourables and non-favourables. Call Status; option=ignore; filters asteroids that have notification to be observed. LCDB Status; Ignore; shows asteroids that have been lightcurved previously, and the rotation period has been determined; not necessarily includes the lightcurve data needed for the lighcurve inversion process. Magnitude Range: From 0 to 18 magnitude. Declination Range: From -10◦ to +90◦. Diameter; 5000; filters the maximum diameter of asteroids in kilometers. In the resulting list, objects with non-determined rotational periods were discarded.

STAGE 2: MINOR PLANET AND COMET EPHEMERIS SERVICE After getting a list of favorable asteroids to observe, we must have ensured that: the object will be in the sky, at least, 1.3 times its rotational period above 30◦ of altitude; the previous considering the nightfall and the sunrise. We achieved this entering our location coordinates, table 3.1, the interval time which was intended to observe the asteroid, with ephemeris interval in minutes according to the precision needed; usually 5 minutes of interval was good enough. The options of "suppress output if sun above the local horizon" and "suppress output if object below the local horizon" must have been checked.

The output of this search shows a list of asteroid coordinates in the sky every several minutes. The coordinates can be switched between J.2000 and current epoch. An empirical analysis was needed to determine if the object was above 30◦ of elevation. This information also shows the , which had to be up to 17.5. The

42 brighter, the better for observation.

We observed objects that allowed an observational night-time less than its rotational period; We also observed asteroids with periods up to 10 hours, but, in this case, we observed these rotation period three consecutive nights, in order to complete the full rotation period of the asteroids.

STAGE 3: ALCDEF DATABASE

The filtering of this stage was not crucial but it was very desirable. Once we had the list of objects that had magnitudes that can be reached with the Tonantzintla Schmidt telescope and favorable rotational periods, it was desirable to have more lightcurves made by another astronomers, to complete the broadest phase angle coverage, and, in the case of main belt asteroids at different opposition dates, at least two different opposition observations are needed to correctly apply the inversion method. The lightcurves contained in ALCDEF database have its own format. Other information was used to validate the data obtained in the lightcurve plotting by MPO Canopus, we will consider this topic later.

STAGE 4: DAMIT DATABASE

The information in this database contains 3D models of asteroids. The models were obtained with observations made by The Astronomical Institute of the Charles University. The lightcurves data contained in this database are in Kaasalainen format. Database contains the information of the spin λ (ecliptic longitude) and β (ecliptic latitude) of the spin axis, rotational period and initial epoch in which was determinated. (Durech, 2010) This last filtering of object selection was not crucial, but it was desirable to have at least one observed object in this database. To have the 3D model of an asteroid as well as its period and its axes of undetermined rotation will help us to confirm and validate the results obtained in this work.

3.5.2. CCD SETTINGS

CCD SETTING

The CCD has incorporated a fan and an electronic cooler. The electronic cooler was set at at -15◦C at the beginning of each night. This temperature never was reached but assured that the CCD was at the lowest possible temperature. Each night the CCD worked at different temperature.

43 CCD BINNING The binning is a merge of pixels. We set a 2 2 binning because the seeing in the region, and a 1 1 raw images made it difficult× to the MPO Canopus process. A 2 2 binning mean× that an array of four pixels, 2 2 pixels is merged into one pixel. The× previous changed the FOV per pixel, but it was× taken into account by the software.

3.5.3. IMAGE REDUCTION BY IRAF Image Reduction and Analysis Facility (IRAF) is a general purpose software system for the reduction and analysis of astronomical data. IRAF is written and supported by the National Optical Astronomy Observatories (NOAO) in Tucson, Arizona.

The CCD is an electronic devise with an inherent noise sources. We used IRAF to make the photometric data reduction, and subtracted these noises. The noises we removed are: Bias: a bias is the dark current generated for a zero seconds exposition, it is temperature dependent. Dark current: This is a noise generated due to the inherent thermal motion of electrons in semiconductor elements of the CCD, it depends of temperature of the CCD, and image integration time. Flat-field noise: is a parameter of the quantum efficiency of each pixel. Each pixel varies 1% or 2% regarding the saturation point. A flat image is taken with a uniformly illuminated surface. Each observation session (called "raw"). Each raw image must be corrected by bias, dark and flat.

BIAS NOISE CORRECTION To correct bias noise we took each night "bias" images: with integration time of 0.1 seconds because is the shortest exposition time that CCD SBIG-8300M allow. with shooter closed to avoid light to come into the CCD. we took ten bias images to find their average (median) value. with temperature similar to the astronomical images. We used the zerocombine and ccdproc tasks in IRAF to find the median value of the ten bias images, and to apply the bias correction, respectively.

DARK NOISE CORRECTION To correct dark noise we took each night "dark" images: with the same integration time as the astronomical images because dark current is time dependent. with closed shooter to avoid light to come into the CCD.

44 we took the images at the end of the night. we took ten dark images to find their average (median) value. We used the imcombine and ccdproc tasks in IRAF to find the median value of the ten dark images, and to apply the dark correction, respectively. We named the result of the combining "dark.fits"

FLATS To correct by flat we took each night "flat" images: dome flats; we illuminated the dome with an halogen lamp and set the correct integration time to get the desirable value of counts, at 70% CCD saturation value; approximately 45000 ADU. we made ten flat images to find their average (median) value. We used the flatcombine and ccdproc and tasks of IRAF to find the median value of the ten dark images, and to apply the flat correction, respectively.

DATA REDUCTION PROCESS The exact process was made according to the next protocol. The protocol was made by Research Faculty, Magdalena Ridge observatory, New Mexico Institute of Mining and Technology. Bill Ryan developed the exact same next protocol( ❤♣✿✴✴✐♥❢♦❤♦✳♥♠✳❡❞✉✴⑦❜②❛♥✴❡❡❛❝❤✴❢❧❛✳❤♠❧, this process was confirmed with the A user’s guide to CCD reductions with IRAF (Massey, 1997): Note: Any page numbers refer to CCD reduction manual by Massey et al, (1997), Change directory to /flattest or whatever directory that you would like to work in.

We decided to only apply dark correction to the images, because bias are included in dark images.

1. Use implot on a flat image to determine the bias overscan section and trim values given in step 2.

2. Use trim on the images.

3. Subtract column dark using naoe. imred. bias.colbias.

4. Create master dark frame with the zerocombine IRAF task.

5. Subtract master dark frame from all flats and data images using the ccdproc package.

6. Create a maser flat field images, using the flatcombine IRAF task.

7. Normalize flat using median and imarith IRAF task.

8. Perform flatfielding correction using ccdproc IRAF task.

45 3.6. MPO CANOPUS

MPO Canopus is a full-featured software for astrometry and photometry. Distributed by Bdw Publishing its algorithm is based on Allan Harris algorithms (Warner, 2006); it can perform the process and measure of images to analyze periods, and times of minimum. It was the software used for taking out the data from the images of the asteroids observed in this thesis work.

To generate the lightcurve, we pointed the telescope to the sky region where the asteroid was located and took a series of images. In addition to this, we assured to have at least three solar analogue comparison stars in the field and to keep the asteroid in the same field during all observation time.

3.6.1. CONFIGURATION SETTINGS The first step was to configure MPO Canopus (figure 3.7) according to the data in table 3.4 and table 3.1:

3.6.2. APERTURES The aperture photometry of stellar like objects is a technique that measures the instrumental magnitude of an object (in this case asteroids), to do this it uses apertures. The apertures in MPO Canopus are three zones which are (Dymock, 2010), figure 3.8 : Sky annulus (the area that measures the brightness of the sky) Dead zone (area between star aperture and inner sky annulus) Inner zone(where the target, comparison star or asteroid is measured) The value of each zone is computed from the histogram of the pixels it contains. The ADU’s contained in the inner zone is a mixture of the signal we pretend to measure and the background(sky brightness). The ADU’s contained in the sky annulus contain information just of the background.

3.6.3. ASTROMETRY The first step is to select an image of the series, at the begging or in the middle of the night, open it and perform astrometry to ensure that the configuration is well working. MPO Canopus computes the position of the stars, from the selected catalogue, and asteroid using its orbital parameters, figure 3.9.

3.6.4. MPO CANOPUS PHOTOMETRY SESSION After taking the images series of an asteroid, we created a session in MPO Canopus following the procedure in the MPO Canopus Users Guide (Bdw Publising, 1993). The concept of a session is "a continuous set of observations that use the same set of

46

Figure 3.8: Aperture (Dymock, 2010)

Figure 3.9: Astrometry of (1318)Nerina in yellow and reference stars in red (By author) comparison stars". (Dimock, 2010) A session includes the number and name of the asteroid, time of integration, temperature of CCD, Earth and Sun distance from the asteroid, phase angle, apparent magnitude, the right ascension and declination of the target, the filter used, the right ascension an declination of comparison stars used, and the photometric band used. This process includes: Up to five comparison stars. The path of the asteroid. Star subtraction, when the path of the asteroid crosses over stars. Functions to make faster the process of image stack. The process is described in the table 3.7.

48 Table 3.7: Procedure for a photometry session in MPO Canopus. /(MPO Canopus user manual, 2003)

No. Step Description and requirements. 1 First image The first image has to be a good quality image. (avoiding clouds or shifted stars provoked for the telescope issues. This image has to be from the beginning or the middle of the observation night because we need another image of at least 40% ahead on the photometric session. 2 Target location Although MPO Canopus can automate the process, the software is "blind" to the objects on the images, so we performed a manual selection of the asteroid and the comparison stars. 3 Comparison stars The comparison stars have to be selected from the "selector tool" because we must select only "solar analogues", this is because the source of light reflected from the asteroid is our sun. The stars are selected from the "MPO Catalog" because it has a more precise selection of invariable stars. To make the selection of the comparison stars on the image must be take into account the SNR and the closeness to the target. (Figure 3.10) 4 Second Image In the second image we located the same comparison stars and target, this second image has to be ahead from the first image at least a 40% of the photometric session. As it is assumed the five stars and target must be on the image but should be in another place. This re- location tells to the software how the target is moving through the image and how the target aperture has to be moved during the image staking. 5 Auto-match We choose two shiny stars as separated as we can. These stars must be in all images. They tell to the program how the images "move" through the observation session so they can be automatched by triangulating the position of these two stars. To be continued on next page

49 Table 3.7 – Continued from previous page 6 Star subtraction If the path of the asteroid moves over stars, we can eliminate them in this step, so their contribution to the ADU’s will be ignored. 7 Image stacking Finally we select all images of the photometric session and perform the photometry.

Figure 3.10: Selection of 4 comparison stars, in yellow one of the reference star. (Dymock, 2010)

3.6.5. VERIFYING COMPARISON STARS Before we proceeded to construct the lightcurve, we verified the behavior of the magnitude of the comparison stars. This is because some times stars are so affected by the atmosphere, or one or more of the selected comparison stars may be variable stars. A bad average behavior of a comparison star looks like figure 3.11 (left) ,a good behavior looks like figure 3.11 (right).

3.6.6. MERGE SESSIONS MPO Canopus can merge photometric sessions. Doing this increases the amount of points in the curve and the result of the analysis is more exact. Is convenient to merge sessions when (Warner, 2006): Sessions are from consecutive nights of observations. If phase angle change less than 5◦. The curves are partial. (does not cover the full rotational period of the asteroid. This can improve the result of the analysis of nights of observations with bad sky as figure 3.12 shows.

50 Figure 3.11: Differential magnitude vs time, (left) Variable comparison star. (right) invariable comparison star (By author)

3.6.7. PERIOD SPECTRUM ANALYZIS

To construct the lightcurves we used the period analyzis tool of MPO Canopus.

The amplitude and the periods are the two fundamental information that can be derived from a lightcurve. To achieve this we must analyze the plot of the period spectrum that is included in the MPO Canopus software.

The period spectrum is based on the Fourier analysis method developed by Alan Harris (Warner, 2006). This tool help to determine which of several possible solution is the most appropriate.

The period spectrum was plotted using six parameters, figure 3.13: The order parameter indicates the number n of a trigonometric polynomial or Fourier series, that is used to find the value of the rotation period; which has the following form:

n 2πnt 2πnt V (t)= H + (a sin ( )+ b cos ( )) R α n n P n n P Xn=1 where Hα is reduced magnitude, n is the order of the series, t is the asteroid rotational phase, P is rotation period, and an and bn are the Fourier coefficients of that series. A Fourier series of order six, has six coefficients a and six coefficients b.

This parameter determine the fit of the Fourier series with the observation data.

In order to restrict the order of the polynomial, we used the relation, suggested by Harris (2014), Between the amplitude of the lightcurve and the order of the fitting

51

Table 3.8: The maximum amplitude for a given primary harmonic.

Primary harmonic Maximum amplitude, magnitude 2 4 0.376∞ 6 0.15 8 0.086 10 0.054 all observation points; a bin value=2 will evaluate the half of observation points because of each two observations are binned into one value.

Max.Dif "enter the maximum interval, in minutes, between any two consecutive data points that are binned to form a single data point. This setting is used to account for large gaps due to clouds, taking reference images for reductions, etc. where binning points well removed from each other in the curve would adversely affect the average value". (MPO Canopus user manual).

The output of this analyzis is shown in figure 3.14.

Figure 3.14: (1831) Nicholson period search results as period spectrum, the minimum RMS match with 3.217 0.001 hours of period.(MPO Canopus) ±

The lightcurve obtained by this analysis is shown in figure 3.12, de dark line is the Fourier fitting function. MPO Canopus gives the corresponding Fourier coefficients. For this particular plot the coefficients are shown in table 3.9.

53 Table 3.9: Fourier fourth order series coefficients of the plot shown on figure. 3.12

Order Sin Cos 1: -0.00293 0.01291 2: 0.09621 -0.09645 3: 0.02478 0.08100 4: 0.00718 0.01253

3.6.8. CSmin/CSmax

Because the change of brightness of an asteroid is produced by a given geometry in rotation; a lightcurve has two peaks (generated by two side-on areas) and two valleys (generated by two point-on). The ratio between minimum and maximum cross sections is correlated with the amplitude by the equation 3.1 (Harris, et al. 2014):

CS A = 2.5log min (3.1) − CSmax

3.7. LIGHTCURVE INVERSION

The lightcurve inversion technique is relatively recent. It was something not achievable and considered impossible, until the year of 2000 when M. Kaasalainen and J. Troppa published a method to determine the shapes, forms, albedo distributions, rotation periods, pole directions of asteroids and scattering parameters from asteroids light curves. (Kaasalainen, 2000)

This method was confirmed inverting synthetic (or simulated) light-curves of objects whose shapes range from almost convex to strongly non-convex, then comparing the results with radar observations of asteroids. The results established the resolution limitations of the shapes of laboratory experiments and real asteroids. (Kaasalainen, 2000)

LIMITATIONS

This method recovers asteroid shape as a convex representation of a non-convex object. This means that craters, even big ones, are not shown in the mathematical representation of the object. Another limitation it is that it assumed a homogeneous albedo distribution in the object, although the shape obtained with sufficient data, can compensate the original albedo distribution, make it almost negligible its contribution with the shape solution.(Kaasalainen, 2000)

54 CONVEX INVERSION The convex inverse problem can be cast in the form (Kassalainen, 2000):

L = Ag

Where: L is the vector of the observed brightnesses, related through the matrix A. g is the vector that contains the parameters to be solved. Vector L describes the Gaussian surface density of the object or curvature function, and determines the product of shape and albedo distribution. After solving the matrix A, the next step is to assign the areas of the facets of a polyhedron, by setting the rotation parameters and scattering properties. The reconstruction of the convex polyhedron corresponding to given facets areas and surface normals can be expressed as a constraint minimization problem; this is done through Minkowski minimisation. As figures 3.15, 3.16, 3.17, and 3.18 show the non-convexities of the objects cannot be shown in the convex representation, but show a near shape approximation. These figures also show that the light curves match for both shapes, original(synthetic) and reconstructed. (Kaasalainen, 2000)

55 Figure 3.15: SHAPE 1(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries ( α=27◦ and α=80◦ (Kassalainen et al., 2001)

56 Figure 3.16: SHAPE 2(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries ( α=53◦ for both; first geometry viewed and illuminated from the equator, the second one from different sides of the equator). (Kassalainen et al., 2001)

57 Figure 3.17: SHAPE 3(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries ( α=27◦ and α=53◦ (Kassalainen et al., 2001)

58 Figure 3.18: SHAPE 3(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries (α=27◦ for both; first geometry viewed and illuminated from the equator, the second one from different sides of the equator).(Kassalainen et al., 2001)

59 PERIOD AND POLE

"Let the pole direction of the asteroid be given by ecliptic polar angle β and longitude λ , and its angular rotations speed by ω. To avoid needless confusion due the historical e notation, we use the following convention: β is measured from the positive z axis and varies between [0,π], and the asteroid always rotates in the positive direction. The e sense of rotations as viewed from the Earth is thus implied by β. The ecliptic latitude β is given by β = 90◦ β. Let r denote a vector in the ecliptic coordinate frame ecl e where the origin is translated− to the asteroid. This vector transform to the vector r e ast in asteorid’s own frame(where z axis is aligned with the rotation axis) by the rotation sequence" (Kaasalainen, 2001). The simulations showed that light curve inversion is possible, but considering that non-linear optimizations methods does not give an unique solution. The convergence toward the correct convex shape can be made by using a global scanning of the parameters. Rotation ecliptic polar angle, longitude and rotation speed of the asteroid, viewed from Earth are different from asteroid’s frame, so this must be considered. Determination of the pole direction is the simplest one, but the period is a crucial initial value to achieve a good shape approximation. The pole must fit considerably with the brightness distribution, so its mathematical distance from the solution should be minimal, then, a ChiSquare test is applied to the period and spin axis search. (Kaasalainen, 2001)

Figure 3.19: (Left panel) Example of Period search with Chisq test. (Kassalainen, 2001), (right panel) and of pole searching value(The bluest the lowest Chisq value)[LCInvert by author]

The next step is to apply the Minkowski shape modeling, to obtain the shape of the asteroid.

60 Figure 3.20: Convex shape model of (1620) Geographos, seen and illuminated from two directions(Kassalainen et al., 2001)

3.7.1. MINKOWSKI REDUCTION The initial step of modeling process starts with a spherical faceted model (rows and columns). The process changes the size and orientation of the facets into a set of areas that duplicates the lightcurve data. The changing process includes the Lomel-Selinguer and the Lambert scattering laws. The set of parameters includes the longitude and latitude of pole as well as the period. The Minkowski reduction puts together all the areas and their directions. This puzzle get together into a form that fits the lightcurve data. The Minkowski file contains the XYZ coordinates of each vertice and an index of the vertices that form each surface, to finally form a the set of polygons (Warner, 2007).

3.8. LIGHT CURVE INVERSION WITH LC INVERT

The general procedure of lightcurve inversion using LC Invert is as indicated below:

1. Data selection from our observations and databases.

2. Retrieve orbital elements of the asteroid.

3. Lightcurves conversion to Kassalainen Format from MPO Canopus or ALDEF database.

4. Import all lightcurves in Kaasalainen format.

5. Search for synodic period.

6. Search for poles.

7. Build the 3D model.

61 3.8.1. DATA SELECTION It is important to say that lightcurve inversion is not a straightforward process. It is easy to found false conclusions. The data requirements for lightcurve inversion are (Warner, 2007): Lightcurves with different geometries. In the case of main belt asteroids at different oppositions; at different oppositions the rotational aspect angle change as shows figure 3.21.

Figure 3.21: Aspect angle and its dependency of H magnitude and geometry variation(Badescu, 2013).

Lightcurves phase angle(α) must cover at least 20◦. This aspect is more important than the density of the data set, but low density of data is an impediment too. The synodic period must be known with a sufficient precision of at least 0.001 hours.

3.8.2. IMPORTING LIGHTCURVE FORMATS There are various lightcurve formats that LC Invert can admit. These are: Kaasalainen Format, the one which is used by the program to perform lightcurve inversion. DAMIT data base use this kind of format on its database. ALCDEF format, the format used in this data base. Canopus format, the format used to export by MPO Canopus. The ALCDEF and Canopus lightcurve formats include raw time-series data and metadata of observations. Metadata is a minimum of critical information so any researcher can use the data correctly in his own investigations. These formats contain for example: the location of the comparison stars, magnitude band of the stars values, filter, lighttime corrections, ecliptic coordinates of Earth, RA and Dec of the asteroid at the time of the observation. The ALCDEF is an lightcurve standard format to share the information internationally, MPO Canopus format was made to be used only by

62 researchers that use MPO Canopus. In the case of Kassalainen lightcurve format is just for realize the lightcurve invert process, so does not includes the metadata of de lightcurves. Figure 3.22 shows an example of Kassalainen lightcurve format. Kassalainen format is a simple ASCII text file. The firs number indicates the quantity of lightcurves include into the file. The first number of the next row indicates the number of points of the first lightcurve. The second number of the first row can be 0 or 1. 0 for relative data (∆ magnitude) and 1 for absolute data. The consecutive rows are the points of the first light curve. It contain the light time corrected Julian date of observation, the intensity of the brightness of the asteroid. The software convert the magnitude into intensity using

I = 10(−0.4×M)

. Columns three, four, and five contain the Sun astrocentric X/Y/Z rectangular ecliptic coordinates; the values consider the asteroid as the center. Columns six, seven, and eight contain the Earth astrocentric X/Y/Z rectangular ecliptic coordinates.

Figure 3.22: An example of the Kassalainen lightcurve format of (22)Kalliope asteroid. (By author).

3.8.3. SYNODIC PERIOD FINDING To find the period is a crucial information to achieve a good 3D model. We considered the periodic reported in the ALCDEF database as a initial parameter of search.

The output of the search is a file which contains:

63

Figure 3.24: Pole of an asteroid (Badescu, 2013).

If there is no previous analyzis of pole orientation in any source, it can be search by LC Invert, first in a coars mode search and then in a medium or fine mode. Software realices a second non-linear optimitation and deliver a list with values of λ and β with chisquare values. The smallest value is the best fit.

3.8.5. MINKOWSKI MODELING The next step is the modelling. That is made with the best fit of the pole search step.

Although the process has been successful, it is usual to obtain erroneous results. One example of this is shown in figure 3.25. The model was generated using the best fit parameters, but the elongated shape on the spin axis is an indicator of bad model.

65 Figure 3.25: (left panel) Bad model of (1831) Nicholson. (right panel) Chisquare results; values from left to right: λ, β, Chisquare value, Period. (LC Invert, By author).

66 Chapter 4

OBSERVATIONS

The eye sees only what the mind is prepared to comprehend Henri Bergson.

4.1. LIGHTCURVES REPORT

The objective of producing light curves is to obtain the amplitude of the light curve and the period of that observation. It should be noted that the parameters of the Fourier analysis was made with the period data reported in the ALCDEF database. We take the period value in the databases to configure the analysis in such a way that our period values were closer to the values already obtained by other observers.

To apply the light curve inversion technique, we added light curves from the DAMIT and ALCDEF databases. Due to this, we organized the information of this chapter by asteroid number, first reporting the observations made in the Schmidt camera of the INAOE and we report the light curves of the databases used in the light curve inversion process.

All asteroids in this thesis work are main belt asteroids. To reduce and analyze the observational data we used the methodology shown in chapter 3.

4.2. OBSERVATIONS REPORT

We organized the information of the observations in three tables: Table 4.1 Observational circumstances for the observed asteroids and magnitude band of comparison stars. Asteroid: number and name of asteroid. • Date: date of the observation in universal time (UT). • RA: right ascension coordinate of the asteroid at the beginning of the • observation.

67 DEC: declination coordinate of the asteroid at the beginning of the • observation. Delta: distance from the asteroid to the Earth in astronomical units. • r: distance from the asteroid to the Sun in astronomical units. • V: visual magnitude of the asteroid at the beginning of the observation. • Filter: filter used to perform the observation. • Mag band: indicates the photometric band used to perform the photometry. Table• 4.2 Selected parameters of the observed asteroids. Asteroid: number and name of asteroid. • a: semi-major axis of the asteroid orbit. • e: eccentricity of the asteroid orbit. • H(mag): H magnitude of the asteroid. • Diameter: diameter derived by equation 1.6 diameter of the asteroid. • Albedo: geometric albedo of the asteroid. • Period: rotational period of the asteroid. • Min amp: minimum amplitude of the asteroid reported on ALCDF • database. Max amp: maximum amplitude of the asteroid reported on ALCDF • database. u: quality designation of the lightcurves reported on ALCDF database. • Commentary: family or general location of the asteroid. Table• 4.3 Rotation period and brightness amplitude of the observed asteroids. Asteroid: number and name of asteroid. • Period: MPO Canopus derived period. • Amplitude: MPO Canopus derived amplitude. • Fourier fit: the Fourier fit order selected on MPO Canopus. • CSmin/CSmax ratio derived by equation 3.1 (Harris, et al. 2014) •

68 Table 4.1: Observational circumstances for the observed asteroids and magnitude band of comparison

Asteroid Date RA DEC ∆ r α (UT) (J2000.0) (J2000.0) (AU) (AU) (degrees) (22) Kalliope 2018 May 29.65 12h35m14.08s +09◦15′21′′.2 2.5344 3.1165 16.97 2018 May 30.56 12h35m12.22s +09◦09′59′′.8 2.5471 3.1172 17.11 (287) Nephtys 2018 May 17.69 12h42m12.90s +10◦06′29′′.6 1.5555 2.3218 20 2018 May 18.57 12h42m06.44s +10◦04′36′′.3 1.5643 2.3216 20.27 2018 May 19.56 12h42m00.42s +10◦02′15′′.9 1.5733 2.3214 20.53 (711) Marmulla 2018 Mar. 12.67 10h19m48.17s +13◦36′59′′.5 1.559 2.518 7.6 2018 Mar. 16.66 10h15m53.52s +13◦47′24′′.7 1.5727 2.5112 9.52 2018 Mar. 24.69 10h09m00.90s +14◦01′28′′.8 1.6102 2.4985 12.95 2018 Apr. 9.63 10h00m32.13s +14◦00′05′′.9 1.4824 2.2052 22.11 (1117) Reginita 2015 Oct. 04.59a 23h04m2.6sa 10◦28′14′′.0a 1.013a 1.956a 13.73a a a a 69 2018 Apr. 8.85 16h34m57.25s −14◦58′38′′.2 1.2511 2.0314 22.7 2018 Apr. 9.63 10h00m32.13s +14− ◦00′05′′.9 1.2435 2.0298 22.51 (1318) Nerina 2018 Mar. 17.59 11h19m14.51s +11◦38′33′′.6 0.8791 1.8634 6.52 2018 Mar. 27.56 11h02m43.16s +09◦13′53′′.3 0.8957 1.8547 12.27 (1346) Gotha 2018 Mar. 29.58 07h12m27.30s +12◦59′32′′.2 2.0593 2.4323 23.88 (1492) Oppolzer 2018 Mar. 18.60 11h26m34.26s +11◦06′23′′.4 0.9991 1.984 5.85 2018 Mar. 26.59 11h19m46.26s +12◦14′25′′.1 1.0114 1.9763 10.27 (3028) Zhangguoxi 2018 Jun. 01.64 15h54m26.91s 08◦34′16′′.1 2.9519 1.9663 5.67 2018 Jun. 02.68 15h53m40.57s −08◦30′59′′.3 2.9521 1.9694 5.95 (4713) Steel 2018 Jun. 05.64 14h11m15.96s −+21◦11′34′′.4 1.7968 1.0464 28.84 (3800) Karayasuf 2018 Mar. 28.79 15h59m16.65s +14◦15′06′′.11 0.6393 1.4596 34.23 (5692) Shirao 2018 Mar. 18.80 12h44m30.81s 03◦20′21′′.6 1.277 2.2558 6.22 2018 Mar. 26.79 12h39m24.54s −01◦46′18′′.8 1.2505 2.2459 2.21 2018 Abr. 7.74 12h31m10.77s +00− ◦37′20′′.9 1.2411 2.2321 4.86 a Vega et al. (2015). Table 4.2: Selected parameters of the observed asteroids

Asteroid aa ea Ha Diame-f Albedo Periodb Min Max u Commentary

(AU) (AU) (mag) ter(km) (Pv) (hours) Amp Amp (22) Kalliope 2.9107 0.0989 6.45 168.245 0.1687 b 4.1483 0.12 0.53 3 Outer (287) Nepthys 2.3531 0.0236 8.3 60.4243 0.238 c 7.605 0.15 0.37 3 Main (711) Marmulla 2.2369 0.1956 11.7 13.1307 0.22 d 2.721 0.03 0.18 3 Flor (1117) Regnita 2.2481 0.1982 11.7 10.2647 0.36 d 2.946 0.1 0.33 3 Inner (1318) Nerina 2.3073 0.2039 12.2 11.7926 0.1721 b 2.528 0.06 0.32 3 Phocaea (1346) Gotha 2.6269 0.1783 11.25 14.3346 0.2794b 2.64067 0.1 0.16 3 Middle 70 (1492) Oppolzer 2.1729 0.1166 12.9 11.8797 0.089 a 3.76945 0.1 0.13 3 Flor (3028) Zhangguoxi 3.0213 0.0316 10.6 25.0868 0.166 e 4.826 0.12 0.25 3 Eos (4713) Steel 1.9265 0.0738 13.2 5.2415 0.3468 b 5.199 0.28 0.44 3 Hung (3800) Karayasuf 1.5779 0.0757 15.1 1.5875 0.657e 2.2319 0.15 0.19 3 Main (5692) Shirao 2.6554 0.1819 12.5 9.0473 0.2218b 2.8878 0.12 0.16 3 Eunomia a From JPL HORIZONS on-line solar system data an ephemeris computation. b (Mainzer et all., 2011). c (Masiero, et. all, 2014) d (Nugent, C.R., et. all, 2016) e (Mainzer, C.R., et. all, 2016) f From equation 1.6 (Muinonen et al., 2010). Table 4.3: Rotation period and brightness amplitude of the observed asteroids.

b Asteroid Period Amplitude Fourier CSmin/CSmax (hours) (mag) fit (Th) (22) Kalliope 4.158 0.017 0.04 8 0.96 (287) Nepthys 7.595±0.002 0.03 6 0.97 (711) Marmulla 2.804±0.001 0.13 8 0.88 2.876±0.074 0.09 6 0.92 2.627±0.041 0.17 6 0.85 (1117) Regnita 2.945±0.002 0.28 6 0.77 2.944±0.012a 0.16a 4 a 0.86a (1318) Nerina 2.586±0.013 0.07 6 0.93 2.463±0.033 0.1 8 0.91 (1346) Gotha 2.563±0.057 0.21 6 0.82 (1492) Oppolzer 3.77 ±0.02 0.11 8 0.90 3.566± 0.075 0.11 8 0.90 3.22 ±0.022 0.31 6 0.75 3.217± 0.001 0.41 8 0.68 (3028) Zhangguoxi 4.835±0.004 0.24 4 0.80 (4713) Steel 5.504±0.278 0.24 4 0.80 (3800) Karayasuf 2.27 ±0.084 0.32 4 0.74 (5692) Shirao 2.957± 0.032 0.13 6 0.88 2.9 0.055± 0.13 4 0.88 2.866± 0.085 0.14 6 0.87 a Vega et al. (2015). ± b From equation 3.1 (Harris, et al. 2014).

To perform the lightcurve inversion our observations are not enough. We used lighcurves of ALCDEF and DAMIT databases to reach at least two oppositions and phase angle the phase angle coverage, both necessary to achieve the objectives of the current thesis work.

4.2.1. DATABASES REPORT DAMIT DATABASE REPORT *Model 1827 is a non-convex model. DAMIT model referencs: (Durechˇ et al. 2010) Model 121, (Kaasalainen et al. 2002a), (Durechˇ et al.2011) Model 1827,(Hanuš et al. 2017b) Model 989,(Hanuš et al. 2016) Model 990, (Hanuš et al. 2016)

71 Table 4.4: Rotation period and brightness amplitude of the observed asteroids.

Asteroid Number of DAMIT λ β Period lightcurves model number (Degrees) (Degrees) (hours)

(22) Kalliope 102 121 196 3 4.1482 1827* 196 2 4.1482 (287) Nepthys 10 989 356 36 7.60411 990 158 39 7.6041

DAMIT lightcurves references: (22) Kalliope: ( Ahmad 1954), ( Gehrels and Owings 1962), (Zappala and van Houten Groeneveld 1979), (Scaltriti et al 1978), (Lupishko et al 1982b), (Weidenschilling et al 1987), (Lupishko et al 1989), (Weidenschilling et al 1987), (Barucci and Dipaoloantonio 1983), ( Zhou et al 1982), (Weidenschilling et al 1987), (Di Martino and Cacciatori 1984b), (Weidenschilling et al 1987) (Ahmad 1954), (Gehrels and Owings 1962), ( Zappala and van Houten Groeneveld 1979), (Surdej et al 1986), ( Weidenschilling et al 1987), ( Dotto et al 1992), (Melillo 1987c), (Michalowski and Velichko 1990), ( Dotto et al 1992), ( Weidenschilling et al 1990), (Hanus et al 2016). (287) Nepthys: (Scaltriti and Zappala 1979), (Harris and Young 1983), (1998/09- 2009/10 USNO), (2003/05-2010/01 Catalina).

ALCDEF DATABASE REPORT

Table 4.5: observational circumstances for the observed asteroids in ALCDEF database(ALCDEF database)

Asteroid Date (UT) α (degrees) Fiterter Mag band Observer (22)Kalliope 2007 Feb. 25.57 20.88 R R B.D. Warner 2007 Feb. 27.60 21.06 V R B. D. Warner 2007 Mar. 02.57 21.28 V R B. D. Warner 2007 Mar. 04.57 21.4 V R B. D. Warner 2007 Mar. 05.58 21.45 V R B. D. Warner 2007 Mar. 08.58 21.58 V R B. D. Warner 2007 Mar. 09.57 21.61 V R B. D. Warner 2007 Mar. 13.58 21.7 V R B. D. Warner 2007 Mar. 14.66 21.72 V R B. D. Warner 2007 Mar. 18.62 21.72 R V V.Reddy 2007 May. 15.66 16.71 R V V.Reddy 2007 May. 17.66 16.42 R V V.Reddy 2007 May 18.68 16.28 R V V.Reddy To be continued on next page

72 Table 4.5 – Continued from previous page 2008 Feb. 23.82 12.47 R V V.Reddy 2012 Feb. 08.48 17.4 C R M. Audejean (287) Nepthys non-lightcurves till 2018 August 07th Asteroid Date (UT) α (degrees) Fiterter Mag band Observer (711)Marmulla 2011 Jan. 15.72 5.9 R SR B. A. Skiff 2011 Jan. 16.64 5.53 R SR B. A. Skiff 2011 Jan. 17.64 5.14 R SR B. A. Skiff (1117)Reginita 2014 Feb.20.65 0.32 R R A. Waszczak 2014 Feb. 21.69 0.35 R R A. Waszczak 2014 Feb. 22.65 0.79 R R A. Waszczak 2014 Feb. 23.65 1.25 R R A. Waszczak (1318)Nerina 2011 Feb. 08.71 20.11 C R R. I. Durkee 2011 Feb. 08.81 20.08 C R R. D. Stephens 2011 Feb. 11.77 18.77 C R R. D. Stephens 2011 Feb. 13.77 17.85 C R R. D. Stephens 2011 Mar. 06.35 5.9 C V Bassano Bresciano Observatory 2011 Mar. 07.33 5.39 C V Bassano Bresciano Observatory 2011 Mar. 08.33 4.92 C V Bassano Bresciano Observatory 2018Mar. 22.35 9.62 C R P.Bacci (1346)Gotha 2012 Jun. 25.90 22.52 R R A. Waszczak 2012 Jun. 26.87 22.45 R R A. Waszczak 2012 Jun. 28.96 22.28 R R A. Waszczak 2012 Jun. 29.85 22.19 R R A. Waszczak 2012 Jun. 30.87 22.09 R R A. Waszczak 2012 Jul. 01.95 21.98 R R A. Waszczak 2012 Jul. 02.87 21.88 R R A. Waszczak 2012 Jul. 11.82 20.63 R R A. Waszczak 2012 Jul. 16.83 19.7 R R A. Waszczak 2012 Jul. 17.87 19.49 R R A. Waszczak 2012 Jul. 18.8 19.29 R R A. Waszczak 2012 Jul. 19.83 19.07 R R A. Waszczak 2012 Jul. 24.81 17.86 R R A. Waszczak 2012 Jul. 25.87 17.58 R R A. Waszczak 2012 Jul. 29.8 16.48 R R A. Waszczak 2012 Jul. 30.88 16.16 R R A. Waszczak 2012 Aug. 08.75 13.12 R R A. Waszczak 2012 Aug. 09.84 12.74 R R A. Waszczak To be continued on next page

73 Table 4.5 – Continued from previous page 2012 Aug. 28.71 4.48 R R A. Waszczak (1492)Oppolzer 2015 Apr. 24.00 6.61 C R J. Oey 2015 Apr. 26.08 6.46 C R J. Oey 2015 May. 10.33 10.71 V V N. Montigiani 2015 May. 17.32 14.05 V V N. Montigiani 2015 May. 18.31 14.52 V V N. Montigiani (3028)Zhangguoxi 2007 Apr. 17.65 5.1 C R R. D. Stephens 2007 Apr. 19.64 5.87 C R R. D. Stephens 2007 Apr. 24.65 7.73 C R R. D. Stephens 2007 Apr. 25.65 8.09 C R R. D. Stephens 2007 May. 05.78 11.49 C R J. W. Brinsfield 2007 May. 06.8 11.8 C R J. W. Brinsfield 2007 May. 07.78 12.1 C R J. W. Brinsfield 2007 May. 08.77 12.4 C R J. W. Brinsfield 2007 May. 14.75 14.06 C R J. W. Brinsfield 2007 May. 19.73 15.29 C R J. W. Brinsfield (4713)Steel 2010 May. 01 22.19 C R B. D. Warner 2010 May. 03 22.23 C R B. D. Warner 2010 May. 04 22.27 C R B. D. Warner 2011 Dec. 12 7.77 R R B. D. Warner 2011 Dec. 14 7.65 C R B. D. Warner 2011 Dec. 15 7.67 C R B. D. Warner 2011 Dec. 16 7.74 C R B. D. Warner 2011 Dec. 17 7.86 C R B. D. Warner 2011 Dec. 27 21 C V B. D. Warner 2011 Dec. 28 20.66 C V B. D. Warner 2011 Dec. 29 20.32 C V B. D. Warner 2011 Dec. 30 19.93 C V B. D. Warner 2018 Apr. 27 22.73 C V B. D. Warner 2018 Apr. 28 22.78 C V B. D. Warner 2018 Apr. 29 22.83 C V B. D. Warner 2018 May. 03 23.15 C V B. D. Warner (3800)Karayusuf 2008 Mar. 25.8 27 C R B. D. Warner 2008 Mar. 26.76 27.15 C R B. D. Warner 2008 Mar. 27.60 27.31 C R B. D. Warner 2008 Apr. 04.64 28.87 C R B. D. Warner 2010 Mar. 17.66 27.37 C R B. D. Warner 2010 Mar. 18.64 27.28 C R B. D. Warner 2010 Mar. 21.63 27.07 C R B. D. Warner 2010 Mar. 22.62 27.03 C R B. D. Warner 2014 Mar. 16.81 32.04 C V B. D. Warner To be continued on next page

74 Table 4.5 – Continued from previous page 2014 Mar. 17.85 31.81 C V B. D. Warner 2018 Jun. 02.71 25.14 C V B. D. Warner 2018 Jun. 03.67 25.47 C V B. D. Warner 2018 Jun. 04.66 25.8 C V B. D. Warner (5692)Shirao 2014 Jun. 10.75 10.9 C V B. D. Warner 2014 Jun. 11.67 11.15 C V B. D. Warner 2014 Jun. 15.78 12.32 C V B. D. Warner 2014 Jun. 16.67 12.62 C V B. D. Warner

4.3. OBSERVATIONS AND DATABASE GRAPHIC REPORTS

In the next figures, we present the results of the analysis of the observations made in MPO Canopus. The observations correspond with the table 4.3. In figures are shown the merge made to obtain each one.

We organize this section in the next structure:

1. MPO Canopus observations organized by number asteroid.

2. DAMIT database observations organized by number asteroid.

3. ALCDEF databse observations organized by number asteroid.

4.3.1. SCHMIDT TELESCOPE OBSERVATIONS LIGHTCURVES

75 Figure 4.1: (22)Kalliope phase plot (MPO Canopus, analysis by author).

Figure 4.2: (22)Kalliope search period analysis (MPO Canopus, analysis by ).

76 Figure 4.3: (287)Nephtys phase plot (MPO Canopus, analysis by author).

Figure 4.4: (287)Nephtys search period analysis (MPO Canopus, analysis by author).

77 Figure 4.5: First (711)Marmulla phase plot (MPO Canopus, analysis by author).

Figure 4.6: First (711)Marmulla search period analysis (MPO Canopus, analysis by author).

78 Figure 4.7: Second (711)Marmulla phase plot (MPO Canopus, analysis by author).

Figure 4.8: Second (711)Marmulla search period analysis (MPO Canopus, analysis by author).

79 Figure 4.9: Third (711)Marmulla phase plot (MPO Canopus, analysis by author).

Figure 4.10: Third (711)Marmulla search period analysis (MPO Canopus, analysis by author).

80 Figure 4.11: First (1117)Reginita phase plot (MPO Canopus, analysis by Vega et al (2017).

Figure 4.12: First (1117)Reginita search period analysis (MPO Canopus, analysis by Vega et al (2017).

81 Figure 4.13: Second (1117)Reginita phase plot (MPO Canopus, analysis by author).

Figure 4.14: Second (1117)Reginita search period analysis (MPO Canopus, analysis by author).

82 Figure 4.15: First (1318)Nerina phase plot (MPO Canopus, analysis by author).

Figure 4.16: First (1318)Nerina search period analysis (MPO Canopus, analysis by author).

83 Figure 4.17: Second (1318)Nerina phase plot (MPO Canopus, analysis by author).

Figure 4.18: Second (1318)Nerina search period analysis (MPO Canopus, analysis by author).

84 Figure 4.19: (1346)Gotha phase plot (MPO Canopus, analysis by author).

Figure 4.20: (1346)Gotha search period analysis (MPO Canopus, analysis by author).

85 Figure 4.21: First (1492)Oppolzer phase plot (MPO Canopus, analysis by author).

Figure 4.22: First (1492)Oppolzer search period analysis (MPO Canopus, analysis by author).

86 Figure 4.23: Second (1492)Oppolzer phase plot (MPO Canopus, analysis by author).

Figure 4.24: Second (1492)Oppolzer search period analysis (MPO Canopus, analysis by author).

87 Figure 4.25: (3028)Zhangguoxi phase plot (MPO Canopus, analysis by author).

Figure 4.26: (3028)Zhangguoxi search period analysis (MPO Canopus, analysis by author).

88 Figure 4.27: (4713)Steel phase plot (MPO Canopus, analysis by author).

Figure 4.28: (4713)Steel search period analysis (MPO Canopus, analysis by author).

89 Figure 4.29: (3800)Karayusuf phase plot (MPO Canopus, analysis by author).

Figure 4.30: (3800)Karayusuf search period analysis (MPO Canopus, analysis by author).

90 Figure 4.31: First (5692)Shirao phase plot (MPO Canopus, analysis by author).

Figure 4.32: First (5692)Shirao search period analysis (MPO Canopus, analysis by author).

91 Figure 4.33: Second (5692)Shirao phase plot (MPO Canopus, analysis by author).

Figure 4.34: Second (5692)Shirao search period analysis (MPO Canopus, analysis by author).

92 Figure 4.35: Third (5692)Shirao phase plot (MPO Canopus, analysis by author).

Figure 4.36: Third (5692)Shirao search period analysis (MPO Canopus, analysis by author).

93

Chapter 5

RESULTS

I’ve learned from experience that if you work harder at it, and apply more energy and time to it, and more consistency, you get a better result. It comes from the work. Louis C. K.

In this chapter we present the models obtained by lightcurve inversion and its analysis. We organized the information by asteroid number.

ABOUT (22) KALLIOPE AND (287) NEPTHYS The cases of (22)Kalliope and (287)Nepthys were special because there are lighcurves from DAMIT database and, as literature suggest, we used these information to obtain the same results from this database and compare them with ours. Then proceeded with all others asteroids.

For (22)Kalliope we did three inversion processes that we report as follows : 1. We get the same results of DAMIT using its lightcurves. Phase angle (α) and phase angle bisector longitude (PABL) plots of lightcurves used in the inversion from DAMIT database. Lightcurve inversion using the λ, β and period parameters posted on the DAMIT database. Generate the 3D shape.

2. Implementation of lightcurve inversion with LC Invert using the same lightcurves from DAMIT database Find Period Find Poles and Shape Generate the 3D shape.

95 3. Implementation of lighcurve inversion with LC Invert using only ALCDEF databes and INAOE lightcurves. Phase angle (α) and phase angle bisector longitude (PABL) plots of lightcurves used in the inversion from ALCDEF database and INAOE observations. Find Period Find Poles and Shape Generate the 3D shape.

For (287) Nephtys we did two processes that we report as follows :

1. We get the same results of DAMIT using its lightcurves. Phase angle (α) and phase angle bisector longitude (PABL) plots of lightcurves used in the inversion from DAMIT database. Lightcurve inversion using the λ, β and period parameters posted on the DAMIT database. Generate the 3D shape.

2. Because of the lack of lighcurves from ALCDEF database we only merged the lighcurves from DAMIT database and INAOE observations. Phase angle (α) and phase angle bisector longitude (PABL) plots of lightcurves used in the inversion from DAMIT database and INAOE observations. Find Period Find Poles and Shape Generate the 3D shape.

ASTEROIDS WITHOUT ENTRY IN DAMIT DATABASE As for the rest of asteroids results, there are lightcurves from ALCDEF database and INAOE observations. These asteroids are: (711) Marmulla (1117) Regnita (1318) Nerina (1346) Gotha (1492) Oppolzer (3028) Zhangguoxi (3800) Karayasuf (4713) Steel (5692) Shirao We used the lightcurves from ALCDEF database and INAOE observations to perform the period, pole axis orientation and model search process. The process is detailed below:

96 1. We merged the two data sources and plot their distribution of phase angle (α) and phase angle bisector longitude (PABL).

2. We set the initial period search interval parameters using the periods reported in ALCDEF database and INAOE observations. We procure that all periods values were included in the interval of search.

3. From the period search results, we chose the period with the lowest chi-square value to enter it as an initial parameter of the first pole axis orientation and shape search.

4. We set β = -1 and λ=0 to indicate to LC Invert that we do not know the pole axis orientations. This first pole axis and shape search was performed in medium mode.

5. From the first pole axis orientation and shape search we obtained a value of period, β, and λ that later we entered into the second search, but this one was performed in fine mode.

6. Using the lightcurves and parameters of the period, β, and λ found by the second search we generate the 3D shape. This shape was developed using the Minkowski optimization.

97 5.1. (22) KALLIOPE

(22) Kalliope was discovered in 1852 by J. R. Hind.It is a low density M-type asteroid gr with bulk density of 2.37 0.4 cm3 . It has a satellite called I Linus (Schmadel, 2012).For this asteroid there± are light curves in DAMIT and ALCDEF databases in addition to our own observations.

In the case of this asteroid, we made three different inversions. We obtained three shapes varying the lightcurves sources.

1. In the first process, we used the exact parameters (λ,β, and period) of DAMIT database to made the lightcurve inversion process of (22)Kalliope, and we used 100 lightcurves (the maximum lightcurves allowed by LC invert). This process consisted in loading the lightcurves file and then enter the three parameters posted on the database. The output was the first shape of (22) Kalliope.

2. In the second process, we used the same 100 lightcurves, but this time using the LC Invert tool of searching the period and the pole axis orientation. For the pole search first we made a medium mode pole search, then we made a fine mode pole search to finally find the shape.

3. Finally, for the last process, we only used the lightcurves of ALCDEF and INAOE following the same process of searching period at medium, fine mode to then we obtained a shape.

5.1.1. DAMIT LIGHTCURVES OF (22) KALLIOPE

The lightcurves used to perform the following inversion are reported on table 4.4. The phase angle (α) and phase angle bisector longitude (PABL) distributions are shown on figure 5.1.

98 (a)

(b) Figure 5.1: Coverages of lightcurves of (22) Kalliope from DAMIT database. (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) 99 5.1.2. PERIODS REPORTED FOR (22) KALLIOPE

To determine the initial interval values for searching the period we used the periods reported on ALCDEF and DAMIT databases and the period found by INAOE observations.

Table 5.1: Pole axis and periods values reported for (22) Kalliope.

Source Period(hours) λ β Reference ALCDEF 4.1476 - - Hamanowa 2011web ALCDEF 4.1478 - - Behrend 2006web ALCDEF 4.14794 - - Behrend 2012web ALCDEF 4.148 - - Scaltriti 1978a ALCDEF 4.148 - - Lupishko 1982a ALCDEF 4.148199 194 -8 Drummond 1991 DAMIT 4.148199 196 3 Kaasalainen 2002a ALCDEF 4.148199 196 3 Durech 2011a DAMIT 4.148199 196 2 Hanuš et al. (2017b) ALCDEF 4.14827 - - Behrend 2007web ALCDEF 4.14828 - - Warner 2007e ALCDEF 4.14828 - - Alton 2007 ALCDEF 4.148343 - - Garlitz 2012web ALCDEF 4.1484 - - Behrend 2004web INAOE 4.158∗ - - Table 4.3 * Period found merging all observations made at INAOE Periods are arranged from lowest to highest value.

5.1.3. DAMIT SHAPE RECREATION FOR (22)KALLIOPE USING DAMIT PARAMETERS

We get the shape of (22) Kalliope using directly the lightcurves of DAMIT and its parameters. There were 102 curves uploaded in DAMIT database; but LC Invert just can make calculus with maximum 100 lightcurves, so we deleted the two first lightcurves. To achieve the shape we entered the parameters show in the table 5.2 into the "Lightcurve Import and Inversion" step of LC Invert. The parameters to achieve the shape are the same values of pole axis orientations and period from DAMIT databese of Kaasalainen 2002a (see table 5.1) , the objective of this was to recreate the same result of DAMIT and compare DAMIT and LC Invert shapes.

100 Table 5.2: Parameters used by DAMIT database of (22) Kalliope (Kaasalainen 2002a) to generate the 3D shape.

Parameter Value Pole λ 196 Pole β 3 Period 4.148199 NOTE: Shape searching and modelling made in fine mode

Figure 5.2: Shape of (22) Kalliope obtained by LC Invert using the parameters in table 5.2

5.1.4. DAMIT SHAPE RECREATION FOR (22) KALLIOPE FOLLOWING THE PROCESS OF SEARCH PERIOD AND POLES WITH LC INVERT

To perform this process we used the same 100 lightcurves from DAMIT database and followed the process to determine period and poles, the process is detailed below:

SEARCHING PERIOD FOR (22)KALLIOPE USING DAMIT LIGHTCURVES.

We defined the interval of search in: Period low=4.14800 Period high=4.1483 We determined this interval using the period used by Kaassalainen (2002a) showed in the table 5.1. The results of this search are shown on table 5.3 arranged by chi-square value.

101 Table 5.3: Period search results for (22)Kalliope using 100 lightcurves from DAMIT database

Period(hours) Chisquare Value 4.1482018 2.69425 4.14819944 3.1613 4.14821234 5.54521 4.1481808 10.12197 4.14799935 12.56755 4.14839221 14.65809 4.14800799 14.71297 4.14838032 14.78229 4.14822411 16.30710 NOTE: Arranged from the lowest to highest chi-square value.

SEARCHING POLES AND SHAPE FOR (22)KALLIOPE USING DAMIT LIGHTCURVES

To search the poles we used the parameters showed in table 5.4. The parameter λ=-1 is to test all possible pole axis orientations. We used the period with the lowest chi-square value found in previous step.

Table 5.4: Initial parameters of sear for (22) Kalliope (Kaasalainen 2002a)

Parameter Value Pole λ -1 Pole β 0 Period 4.1482018 NOTE: Pole searching and modelling was made in medium mode

The output of the search is another chi-square optimization. The results are shown in figure5.3.

102

Figure 5.4: 3D Minkowsky representation of (22)Kalliope using parameters shown in table 5.5 [LC Invert By Author]

The last step is to search the pole axis orientation and inversion shape into fine mode. The initial search parameters were found previously in the medium search mode, table 5.5.

Figure 5.5: Plot of chi-square values in fine mode, darkest blue is the lowest logarithm of chi- square value. (LC Invert by author)

The parameters found for (22) Kalliope in fine search mode are shown in table 5.6.

Table 5.6: Parameters to 3D conversion of (22)Kalliope in fine mode

Parameter Value Period 4.14820084 Pole λ 196.2 Pole β 3.9 NOTE: Pole searching and modelling was made in fine mode

104 The 3D shape produced with parameters in table 5.6 is shown in figure 5.6.

Figure 5.6: 3D Minkowsky representation of (22)Kalliope using the parameters shown in table 5.6 [LC Invert By Author]

5.1.5. LIGHT CURVE INVERSION FOR (22)KALLIOPE USING ALCDEF AND INAOE OBSERVATIONS

ALCDEF AND INAOE LIGHTCURVES OF (22)KALLIOPE

The lightcurves used to perform the following inversion are reported on table 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figures 5.8, 5.7 and 5.9.

105 (a)

(b) Figure 5.7: Coverages of lightcurves of (22) Kalliope from INAOE observations. (a) coverage of phase angle (α), and (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author)

106 (a)

(b) Figure 5.8: Coverages of lightcurves of (22) Kalliope from ALCDEF database. (a) coverage of phase angle (α), and (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author)

107 (a)

(b) Figure 5.9: Merged coverages of lightcurves of (22) Kalliope from ALCDEF (green) and INAOE (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author)

108 Table 5.8: Parameters used to pole and shape search for (22) Kalliope in medium mode by using the lightcurves from ALCDEF database and INAOE.

Parameter Value Pole λ -1 Pole β 0 Period 4.14819955 NOTE: Pole searching and modelling was made in medium mode

PERIOD SEARCH INTERVAL

We used the same initial search interval used previously: Period low=4.14800 Period high=4.1483 The output of the search is shown on table 5.7:

Table 5.7: Period search results for (22)Kalliope using lightcurves from ALCDEF and INAOE observations.

Period(hours) Chi-square Value 4.14819955 2.40523 4.14819972 2.42537 4.14819987 2.43987 4.14798802 2.67651 4.14838629 2.77321 NOTE: Data arranged from lowest chi-square to highest.

SHAPE AND POLES SEARCH

We used as initial parameters of search those shown in table 5.8. The search was performed as we did not know the pole orientation. The output of this first search is shown on table 5.9 and figure 5.10.

Table 5.9: Pole search parameters of (22) Kalliope by using the ALCDEF and INAOE lightcurves.

Parameter Value Period 4.14819831 Pole λ 180 Pole β 0 NOTE: Pole searching and modelling was made in medium mode.

109 Figure 5.10: Plot of chi-square values in medium mode, darkest blue is the lowest logarithm of chi-square value. (LC Invert by author)

After this we made a second search of shape and pole orientation using the fine mode. The results are shown on table 5.10 and figure 5.11.

Table 5.10: Pole search final parameters of (22) Kalliope using the ALCDEF and INAOE lightcurves.

Parameter Value Period 4.1481922 Pole λ 176.6 Pole β 11.7 NOTE: Pole searching and modelling was made in fine mode

110 Figure 5.11: Plot of chi-square values in fine mode, darkest blue is the lowest logarithm of chi-square value. (LC Invert by author)

MINKOWSKI 3D OPTIMIZATION The shape found through this process which only includes ligthcurves from ALCDEF database and INAOE observations is shown on figure 5.12

Figure 5.12: 3D Minkowsky representation of (22)Kalliope using parameters shown in table 5.10 This shape were developed using the lightcurves from ALCDEF and INAOE observations and was performed at fine mode. [LC Invert By Author]

111 5.2. (287) NEPTHYS

(287) Nepthys was discovered in 1889 by C. H. F. Peters. Nepthys is the name of Egyptian goddess of the dead, sister of Isis and wife of Seth. (Schmadel, 2012).For this asteroid there are light curves in DAMIT database in addition to our own observations.

We used these to perform two process. The process that we implement was:

1. In the first process we applied the same initial search parameters (λ,β, and period) that DAMIT database used to find the lightcurve inversion of (22)Kalliope, so we obtained directly the 3D shape.

2. In the second process we used the lightcurves of DAMIT merged with INAOE observations, then we followed the process of searching period, poles and finally obtained the 3D shape.

5.2.1. DAMIT LIGHTCURVES OF (287) NEPTHYS

The lightcurves used to perform the following inversion are reported on table 4.4. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.13.

112 (a)

(b) Figure 5.13: Coverages of lightcurves of (287) Nepthys from DAMIT database: (a) coverage of phase angle (α), and (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) 113 5.2.2. PERIODS REPORTED FOR (287) NEPTHYS

To determine the interval of period search we used the periods reported on DAMIT database and INAOE observations.

Table 5.11: Pole axis orientation and periods reported for (287) Nepthys.

Source Period(hours) λ β Reference ALCDEF 7.58 279 -54 Blanco 1996b INAOE 7.595 ∗ - - Table 4.3 ALCDEF 7.60409 - - Alton 2011 DAMIT 7.6041 158 39 Hanuš et al. (2016) DAMIT 7.60411 356 36 Hanuš et al. (2016) ALCDEF 7.605 - - Fauerbach 2008 * Period found merging all the observations made at INAOE Periods are arranged from the lowest to highest.

5.2.3. DAMIT SHAPE RECREATION FOR (287) NEPHTYS USING DAMIT PARAMETERS

We get the shape of (287) Nepthys directly the lightcurves of DAMIT and as initial pole search parameters those reported by Hanuš (2016). To achieve the shape we entered the parameters show in the table 5.12 into the "Lightcurve Import and Inversion" step of LC Invert. The parameters to achieve the shape match exactly the values of poles and period from DAMIT databese of Hanuš (2016) (see table 5.11) , the objective of this was to recreate the same result of DAMIT and to compare it with the shape obtained by LC Invert. There are two shapes from DAMIT database so we applied the lighcurve inversion to both set of parameters.

FIRST SET OF PARAMETERS AND SHAPE POSTED ON DAMIT DATABASE

Table 5.12: First set of parameters used by DAMIT database for (287) Nepthys (Hanuš 2016)

Parameter Value Pole λ 356 Pole β 36 Period 7.60411 NOTE:The shape and modelling searching was made in fine mode.

114 Figure 5.14: 3D shape of (287) Nephtys using the parameters in table 5.12

SECOND SET OF PARAMETERS AND SHAPE POSTED ON DAMIT DATABASE

Table 5.13: Second set of parameters used by DAMIT database for (287) Nepthys (Hanuš 2016)

Parameter Value Pole λ 158 Pole β 39 Period 7.60410 NOTE:The shape searching and modelling was made in fine mode

115 Figure 5.15: 3D shape of (287) Nepthys using the parameters in table 5.13

5.2.4. SHAPE FOR (287) NEPHTYS USING DAMIT AND INAOE LIGHTCURVES

The lightcurves used to perform the following inversion are reported on table 4.3 and 4.4. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figures 5.13, and 5.16.

116 (a)

(b) Figure 5.16: Coverages of lightcurves of (287) Nephtys from INAOE observations. (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author)

117 SEARCHING PERIOD FOR (287) NEPHTYS USING DAMIT AND INAOE LIGHTCURVES.

We defined the initial interval of search as: Period low=7.56 Period high=7.613 We defined the interval using the period found by Hanuš (2016) shown in the table 5.11. The results of this search are shown on table 5.14.

Table 5.14: Period search results for (287)Nephtys the lightcurves from DAMIT database.

Period(hours) Chisquare Value 7.60410096 5.32053 7.60482387 5.54541 7.6044428 5.57366 7.56115597 5.58154 7.59928087 5.58345 7.60957663 5.60052 7.60593962 5.6041 7.6029999 5.64176 7.59382055 5.64661 7.60407749 5.65932 7.59712319 5.67663 7.56045646 5.69908 7.60183843 5.69919 7.59450068 5.7077 7.59573774 5.70924 7.60830047 5.7111 7.59379171 5.71395 7.56570604 5.71786 7.60189118 5.72163 7.59640053 5.72633 7.59643197 5.72714 7.59455633 5.72806 7.60777669 5.73384 7.60228898 5.73773 7.60883802 5.7465 7.59932978 5.7529 7.58443107 5.75368 7.60117001 5.7593 NOTE: Data arranged from the lowest to highest chi-square.

118 SEARCHING POLE AND SHAPE FOR (287) NEPTHYS USING DAMIT AND INAOE LIGHTCURVES To search the pole orientation we used the parameters shown in table 5.15. The parameter λ=-1 is to test all possible pole axis orientations. We used the period with the lowest chi-square value found on previous step.

Table 5.15: Parameters used to search pole axis orientation and shape of (287) Nepthys.

Parameter Value Pole λ -1 Pole β 0 Period 7.60410096 NOTE: Pole axis orientation searching and modelling was made in medium mode.

The output of the search was another chi-square optimization, it is shown in table 5.16 and figure5.17.

Table 5.16: Parameters used to search poles and shape in medium mode of (22) Kalliope

Parameter Value Pole λ 0.0 Pole β 45.0 Period 7.60410447 NOTE: Pole searching and modelling was made in medium mode.

Figure 5.17: Pole and inversion search for (287) Nepthys. The plot of chi-square values was made in medium mode, darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

119 After this search we performed another in fine mode. The resulsts are shown in table 5.17 and figure5.18.

Table 5.17: Parameters used to search poles and shape in fine mode of (22) Kalliope

Parameter Value Pole λ 352.3 Pole β 34.2 Period 7.60410240 NOTE: Pole searching and modeling was made in fine mode

Figure 5.18: Pole and inversion search for (287) Nepthys, Plot of chi-square values in fine mode, darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

We created the 3D representation by selecting the iteration with the lowest chi-square value. The parameters to make the final search are shown in table 5.17 and the 3D shape is shown in figure 5.19.

120 Figure 5.19: 3D Minkowski representation of (287) Nephtys using parameters shown in table 5.17. This shape was developed using lightcurves from DAMIT and INAOE observations and it was performed at fine mode. [LC Invert By Author]

121 5.3. (711) MARMULLA

(711) Marmulla was discovered in 1911 by J. Palisa. Marmulla is the medieval German word for marble (Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations.

We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.3.1. LIGHTCURVES OF (711) MARMULLA The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.20.

5.3.2. PERIOD SEARCH INTERVAL To determine the search period interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.18: Periods reported for (711) Marmulla

Source Period (hours) Reference ALCDEF 2.721 Skiff 2011web INAOE 2.723∗ Table 4.3 * Period found merging all observations made at INAOE Periods are arranged from the lowest to highest value.

5.3.3. SEARCHING PERIOD FOR (711) MARMULLA USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=2.715 Period high=2.725 The results of this search are shown on table 5.19 arranged by chi-square value.

122 (a)

(b) Figure 5.20: Coverages of merged lightcurves of (711) Marmulla from INAOE observations (green) and ALCDEF database (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by123 author) Table 5.19: Period search results for (711) Marmulla using lightcurves from INAOE and ALCDEF database.

Period(hours) Chisquare Value 2.72151354 0.26972 2.7210428 0.2701 2.72436076 0.2706 2.72127858 0.2713 2.72080612 0.27145 2.72459693 0.27208 2.72057374 0.27221 2.72412539 0.27354 2.72016789 0.27724 2.720978 0.27891 NOTE: Periods are arranged from the lowest to highest chi-square value.

5.3.4. SEARCH POLES AND MODEL FOR (711) MARMULLA

To search the poles we used the parameters showed in table 5.20. The parameter λ=-1 is to test all possible pole axis orientations. We used the period from the previous search with the lowest chi-square value.

Table 5.20: Search pole parameters of (711) Marmulla

Parameter Value Pole λ -1 Pole β 0 Period 2.72151354 NOTE: Pole searching and shape was made in medium mode

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.21.

124 Figure 5.21: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.26

Table 5.21: Search pole parameters of (711) Marmulla

Parameter Value Pole λ 165 Pole β -15 Period 2.72151721 NOTE: Pole searching and modelling was made in medium mode.

From the fine pole search chi-square optimization we obtained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.27

Table 5.22: Final fine search pole parameters of (711) Marmulla

Parameter Value Pole λ 165.0 Pole β -17.9 Period 2.72151939 NOTE: Pole searching and modelling was made in fine mode

125 Figure 5.22: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.3.5. GENERATING 3D SHAPE FOR (711) MARMULLA We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.23

Figure 5.23: 3D shape of (711) Marmulla (LC Invert by author)

126 5.4. (1117) REGINITA

(1117) Reginita was discovered in 1927 by J. Comas Solá, named in honor of the niece of the discoverer.(Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations. We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.4.1. LIGHTCURVES OF (1117) REGINITA The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.24.

5.4.2. PERIOD SEARCH INTERVAL To determine the search period interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.23: Periods reported for (1117) Reginita

Source Period Reference INAOE 2.945∗ Table 4.3 ALCDEF 2.9458 Behrend 2007web ALCDEF 2.946 Tan 2017 ALCDEF 2.9463 Wisniewski 1997 * Period found merging all observations made at INAOE Periods are arranged from lowest to highest value.

5.4.3. SEARCHING PERIOD FOR (1117) REGINITA USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=2.94 Period high=2.957 The results of this search are shown on table 5.24 arranged by chisquare value.

127 (a)

(b) Figure 5.24: Coverages of merged lightcurves of (711) Marmulla from INAOE observations (green) and ALCDEF database (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by128 author) Table 5.24: Period search results for (1117) Reginita using lightcurves from INAOE and ALCDEF database.

Period(hours) Chisquare Value 2.94600696 0.10703 2.94621042 0.10803 2.94351562 0.10813 2.94505217 0.10876 2.9447913 0.10885 2.94372407 0.10915 2.94190157 0.10927 2.94517851 0.10985 2.94334118 0.10999 2.94438934 0.11002 2.94526 0.1101 2.9471786 0.11035 2.94391835 0.1104 2.9461137 0.1106 NOTE: Periods are arranged from lowest to highest chi-square value.

5.4.4. SEARCH POLES AND MODEL FOR (1117) REGINITA

To search the poles we used the parameters showed in table 5.25. The parameter λ=-1 is to test all possibles poles axis orientation. We used the period from the previous search with the lowest chi-square value.

Table 5.25: Search pole parameters of (1117) Reginita

Parameter Value Pole λ -1 Pole β 0 Period 2.94600696 NOTE: Pole and shape searching was made in medium mode

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.21.

129 Figure 5.25: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.26

Table 5.26: Search pole parameters of (1117) Reginita

Parameter Value Pole λ 165 Pole β -15 Period 2.72151721 NOTE: Pole and modelling searching was made in medium mode

From the fine pole search chi-square optimization we implemented we obtained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.27

Table 5.27: Final fine search pole parameters of (711) Marmulla

Parameter Value Pole λ 165.0 Pole β -17.9 Period 2.72151939 NOTE: Pole searching and modelling made in fine mode

130 Figure 5.26: Plot of chi-square values in fine mode, darkest blue the lowest logarithm of chi- square value. (LC Invert by author)

5.4.5. GENERATING 3D SHAPE FOR (1117) REGINITA We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.27

Figure 5.27: 3D shape of (1117) Reginita (LC Invert by author)

131 5.5. (1318) NERINA

(1318) Nerina was discovered in 1934 by C. Jackson.(Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations. We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.5.1. LIGHTCURVES OF (1318) NERINA The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.28.

5.5.2. PERIOD SEARCH INTERVAL To determine the period search interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.28: Periods reported for (1318) Nerina

Source Period(hours) Reference ALCDEF 2.5275 Behrend 2018web ALCDEF 2.5277 Franco 2018c INAOE 2.528 Table 4.3 ALCDEF 2.528 Stephens 2011g * Period found merging all observations made at INAOE Periods are arranged from lowest to highest value.

5.5.3. SEARCHING PERIOD FOR (1318) NERINA USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=2.514 Period high=2.535 The results of this search are shown on table 5.29 arranged by chisquare value.

132 (a)

(b) Figure 5.28: Coverages of merged lightcurves of (1318) Nerina from INAOE observations (green) and ALCDEF database (orange). (a)133 coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) Table 5.29: Period search results for (1318) Nerina using lightcurves from INAOE and ALCDEF database.

Period(hours) Chisquare Value 2.52908548 0.27587 2.52919469 0.27717 2.52929188 0.27809 2.52898583 0.27858 2.5286739 0.27858 2.52898574 0.27868 2.52919313 0.27895 2.52888213 0.27911 2.52752877 0.27939 2.52940041 0.27939 2.52940198 0.2794 NOTE: Periods are arranged from lowest to highest chi-square value.

5.5.4. SEARCH POLES AND MODEL FOR (1318) NERINA

To search the poles we used the parameters showed in table 5.30. The parameter λ=-1 is to test all possibles pole axis orientations. We used the period from the previous search with the lowest chi-square value.

Table 5.30: Search pole parameters of (1318) Nerina

Parameter Value Pole λ -1 Pole β 0 Period 2.52908548 NOTE: Pole and modelling searching was made in medium mode.

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.29.

134 Figure 5.29: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.31

Table 5.31: Search pole parameters of (1318) Nerina

Parameter Value Pole λ 105.0 Pole β 45.0 Period 2.52898331 NOTE: Pole and modelling searching was made in medium mode.

From the fine pole search chi-square optimization we implemented we obtain the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.32

Table 5.32: Final fine search pole parameters of (1318) Nerina

Parameter Value Pole λ 101.5 Pole β 49.3 Period 2.52898152 NOTE: Pole and modelling searching was made in fine mode.

135 Figure 5.30: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.5.5. GENERATING 3D SHAPE FOR (1318) NERINA We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.31

Figure 5.31: 3D shape of (1318) Nerina (LC Invert by author)

136 5.6. (1346) GOTHA

(1346) Gotha was discovered in 1929 by K. Rinmuth, named for the city of Germany.(Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations.We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.6.1. LIGHTCURVES OF (1346) GOTHA

The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.32.

5.6.2. PERIOD SEARCH INTERVAL

To determine the search period interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.33: Periods reported for (1346) Gotha

Source Period Reference INAOE 2.563∗ Table 4.3 ALCDEF 2.64067 Behrend 2011web ALCDEF 2.642 Aznar 2017b * Period found merging all observations made at INAOE. Periods are arranged from lowest to highest value.

5.6.3. SEARCHING PERIOD FOR (1346) GOTHA USING ALCDEF AND INAOE LIGHTCURVES.

According the reported periods we used the interval of search: Period low=2.54 Period high=2.65 The results of this search are shown on table 5.34 arranged by chisquare value.

137 (a)

(b) Figure 5.32: Coverages of merged lightcurves of (1346) Gotha from INAOE observations (orange) and ALCDEF database (green). (a)138 coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) Table 5.34: Period search results for (1346) Gotha using lightcurves from INAOE and ALCDEF database.

Period (hours) Chisquare Value 2.54066408 0.08304 2.54056079 0.08483 2.54004298 0.0849 2.5401061 0.08602 2.54020394 0.08612 2.54030217 0.08636 2.54040343 0.08648 2.54026059 0.08688 2.54133004 0.0876 2.54035507 0.0876 2.54173665 0.08776 2.54046586 0.08776 2.54015493 0.08823 NOTE: Periods are arranged from lowest to highest chi-square value.

5.6.4. SEARCH POLES AND MODEL FOR (1346) GOTHA

To search the poles we used the parameters showed in table 5.35. The parameter λ=-1 is to test all possibles pole axis orientations. We used the period from the previous search with the lowest chi-square value.

Table 5.35: Search pole parameters of (1346) Gotha

Parameter Value Pole λ -1 Pole β 0 Period 2.54066408 NOTE: Pole and modelling searching was made in medium mode.

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.33.

139 Figure 5.33: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.36

Table 5.36: Search pole parameters of (1346) Gotha

Parameter Value Pole λ 270 Pole β -45 Period 2.5406617 NOTE:Pole and modelling searching was made in medium mode.

From the fine pole search chi-square optimization we implemented we obtained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.37

Table 5.37: Final fine search pole parameters of (1346) Gotha

Parameter Value Pole λ 249.5 Pole β -33.5 Period 2.54066893 NOTE: Pole and modelling searching was made in fine mode.

140 Figure 5.34: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.6.5. GENERATING 3D SHAPE FOR (1346) GOTHA We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.35

Figure 5.35: 3D shape of (1346) Gotha (LC Invert by author)

141 5.7. (1492) OPPOLZER

(1492) Oppolzer was discovered in 1938 by Y. Väisälä. It was named in honour of Hofrath professor Theodor Ritter von Oppolzer, professor of astronomy in Vienna. The name was suggested by Jean Meeus.(Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations. We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.7.1. LIGHTCURVES OF (1492) OPPOLZER The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.36.

5.7.2. PERIOD SEARCH INTERVAL To determine the search period interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.38: Periods reported for (1492) Oppolzer

Source Period(hours) Reference INAOE 3.769∗ Table 4.3 ALCDEF 3.76945 Pravec 2018web ALCDEF 3.77 Salvaggio 2015c ALCDEF 3.7702 Pravec 2015web ALCDEF 3.789 Behrend 2018web * Period found merging all observations made at INAOE. Periods are arranged from lowest to highest value.

5.7.3. SEARCHING PERIOD FOR (1492) OPPOLZER USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=3.765 Period high=3.782

142 (a)

(b) Figure 5.36: Coverages of merged lightcurves of (1492) Oppolzer from INAOE observations (orange) and ALCDEF database (green). (a) coverage143 of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) The results of this search are shown on table 5.39 arranged by chisquare value.

Table 5.39: Period search results for (1492) Oppolzer using lightcurves from INAOE and ALCDEF database.

Period (hours) Chisquare Value 3.76900786 0.16575 3.76920896 0.16661 3.76876318 0.16802 3.76809262 0.16931 3.76966831 0.16936 3.77012961 0.16954 3.76831693 0.16955 NOTE:Periods are arranged from lowest to highest chi-square value.

5.7.4. SEARCH POLES AND MODEL FOR (1492) OPPOLZER

To search the poles we used the parameters showed in table 5.40. The parameter λ=-1 is to test all possible pole axis orientation. We used the period from the previous search with the lowest chi-square value.

Table 5.40: Search pole parameters of (1492) Oppolzer

Parameter Value Pole λ -1 Pole β 0 Period 3.76900786 NOTE: Pole and shape searching was made in medium mode

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.37.

144 Figure 5.37: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.41

Table 5.41: Search pole parameters of (1492) Oppolzer

Parameter Value Pole λ 75 Pole β 30 Period 3.76902000 NOTE: Pole and modelling searching was made in medium mode

From the fine pole search chi-square optimization we implemented we obtain the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.42

Table 5.42: Final fine search pole parameters of (1492) Oppolzer

Parameter Value Pole λ 70.7 Pole β 40.9 Period 3.76901463 NOTE: Pole and modelling searching was made in fine mode

145 Figure 5.38: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.7.5. GENERATING 3D SHAPE FOR (1492) OPPOLZER We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.39

Figure 5.39: 3D shape of (1492) Oppolzer (LC Invert by author)

146 5.8. (3028) ZHANGGUOXI

(3028) Zhangguoxi was discovered in 1978 at the Purple Mountain Observatory at Nanjing. (Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations. We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.8.1. LIGHTCURVES OF (3028) ZHANGGUOXI The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.40.

5.8.2. PERIOD SEARCH INTERVAL To determine the interval of searching the period we used the periods reported on INAOE observations and ALCDEF database.

Table 5.43: Periods reported for (3028) Zhangguoxi

Source Period(hours) Reference ALCDEF 4.826 Behrend 2007web ALCDEF 4.826 Stephens 2007f ALCDEF 4.827 Brinsfield 2007b INAOE 4.835∗ Table 4.3 * Period found merging all observations made at INAOE. Periods are arranged from lowest to highest value.

5.8.3. SEARCHING PERIOD FOR (3028) ZHANGGUOXI USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=4.82 Period high=4.938 The results of this search are shown on table 5.44 arranged by chisquare value.

147 (a)

(b) Figure 5.40: Coverages of merged lightcurves of (3028) Zhangguoxi from INAOE observations (orange) and ALCDEF database (green). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) 148 Table 5.44: Period search results for (3028) Zhangguoxi using lightcurves from INAOE and ALCDEF database.

Period (hours) Chisquare Value 4.82663762 0.72224 4.8270213 0.72625 4.82719166 0.73487 4.82671743 0.73653 4.8276743 0.73669 4.82749955 0.74151 4.82654702 0.74251 4.82689087 0.74644 4.82623906 0.74878 4.82615962 0.75105 4.82641304 0.75133 4.82738138 0.75175 4.8256806 0.75459 4.82785942 0.75561 4.82792696 0.75869 NOTE: Periods are arranged from lowest to highest chi-square value.

5.8.4. SEARCH POLES AND MODEL FOR (3028) ZHANGGUOXI

To search the poles we used the parameters showed in table 5.45. The parameter λ=-1 is to test all possible pole axis orientations. We used the period from the previous search with the lowest chi-square value.

Table 5.45: Search pole parameters of (3028) Zhangguoxi

Parameter Value Pole λ -1 Pole β 0 Period 4.82663762 NOTE: Pole and shape searching was made in medium mode

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.41.

149 Figure 5.41: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.46

Table 5.46: Search pole parameters of (3028) Zhangguoxi

Parameter Value Pole λ 180 Pole β 30 Period 4.82664836 NOTE: Pole and shape searching was made in medium mode

From the fine pole search chi-square optimization we implemented we obtrained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.47

Table 5.47: Final fine search pole parameters of (3028) Zhangguoxi

Parameter Value Pole λ 169.8 Pole β 38.7 Period 4.82664673 NOTE: Pole and shape searching was made in fine mode

150 Figure 5.42: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.8.5. GENERATING 3D SHAPE FOR (3028) ZHANGGUOXI We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.43

Figure 5.43: 3D shape of (3028) Zhangguoxi (LC Invert by author)

151 5.9. (3800) KARAYUSUF

(3800) Karayusuf was discovered in 1984 by E. F. Helin. It was named in honor of Alford S. Karayusuf. (Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations.We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.9.1. LIGHTCURVES OF (3800) KARAYUSUF The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.44.

5.9.2. PERIOD SEARCH INTERVAL To determine the period search interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.48: Periods reported for (3800) Karayusuf

Source Period(hours) Reference ALCDEF 2.221 Warner 2014j ALCDEF 2.225 Skiff 2018web ALCDEF 2.2319 Warner 2008m ALCDEF 2.232 Warner 2010p ALCDEF 2.2328 Warner 2018web INAOE 2.27∗ Table 4.3 * Period found merging all observations made at INAOE. Periods are arranged from lowest to highest value.

5.9.3. SEARCHING PERIOD FOR (3800) KARAYUSUF USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=2.23 Period high=2.25

152 (a)

(b) Figure 5.44: Coverages of merged lightcurves of (3800) Karayusuf from INAOE observations (orange) and ALCDEF database (green). (a) coverage153 of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) The results of this search are shown on table 5.49 arranged by chisquare value.

Table 5.49: Period search results for (3800) Karayusuf using lightcurves from INAOE and ALCDEF database.

Period (hours) Chisquare Value 2.23204956 0.69752 2.23206575 0.7061 2.23176399 0.71635 2.2323446 0.72959 2.23203085 0.74678 2.23148474 0.74699 2.23120152 0.75229 2.23232198 0.75423 2.23091671 0.76545 NOTE: Periods are arranged from lowest to highest chi-square value.

5.9.4. SEARCH POLES AND MODEL FOR (3800) KARAYUSUF

To search the poles we used the parameters showed in table 5.50. The parameter λ=-1 is to test all possible pole axis orientations. We used the period from the previous search with the lowest chi-square value.

Table 5.50: Search pole parameters of (3800) Karayusuf

Parameter Value Pole λ -1 Pole β 0 Period 2.23204956 NOTE: Pole and shape searching was made in medium mode

The output of the search is another chi-square optimizzation. The output of medium search is showed in figure5.45.

154 Figure 5.45: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimizzation we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.51

Table 5.51: Search pole parameters of (3800) Karayusuf

Parameter Value Pole λ 270.0 Pole β -45.0 Period 2.23204909 NOTE: Pole and shape searching was made in medium mode

From the fine pole search chi-square optimization we implemented we obtained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.52

Table 5.52: Final fine search pole parameters of (3800) Karayusuf

Parameter Value Pole λ 294.5 Pole β -23.0 Period 2.23205023 NOTE: Pole and shape searching was made in fine mode

155 Figure 5.46: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.9.5. GENERATING 3D SHAPE FOR (3800) KARAYUSUF We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.47

Figure 5.47: 3D shape of (3800) Karayusuf (LC Invert by author)

156 5.10. (4713) Steel

(4713) Steel was discovered in 1989 by R. H. McNaught. It was named in honor of Duncan Steel. (Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations. We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.10.1. LIGHTCURVES OF (4713) STEEL The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.48.

5.10.2. PERIOD SEARCH INTERVAL To determine the period search interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.53: Periods reported for (4713) Steel

Source Period (hours) Reference ALCDEF 5.186 Behrend 2002web ALCDEF 5.193 Warner 2012c ALCDEF 5.1963 Warner 2018web ALCDEF 5.199 Warner 2010p ALCDEF 5.203 Warner 2015j INAOE 5.504∗ Table 4.3 * Period found merging all observations made at INAOE. Periods are arranged from lowest to highest value.

5.10.3. SEARCHING PERIOD FOR (4713) STEEL USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=5.184 Period high=5.22

157 (a)

(b) Figure 5.48: Coverages of merged lightcurves of (4713) Steel from INAOE observations (orange) and ALCDEF database (green). (a)158 coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) The results of this search are shown on table 5.54 arranged by chi-square value.

Table 5.54: Period search results for (4713) Steel using lightcurves from INAOE and ALCDEF database.

Period (hours) Chisquare Value 5.19936924 0.6056 5.19975037 0.77785 5.19821352 0.80273 5.19898443 0.81991 5.20052202 0.83468 5.20205885 0.8367 5.20090481 0.88496 5.20013362 0.90064 5.20167158 0.91056 5.20514912 0.97366 5.1970553 1.01884 5.20128701 1.02122 5.19397167 1.02764 5.19744456 1.03927 5.19744848 1.04039 5.20361227 1.06706 5.20283062 1.08406 5.1997446 1.14797 5.19513269 1.1889 5.19859605 1.2249 5.18858152 1.26322 5.2024488 1.26728 5.198213 1.27577 5.19783176 1.28755 NOTE: Periods are arranged from lowest to highest chi-square value.

5.10.4. SEARCH POLES AND MODEL FOR (4713) STEEL

To search the poles we used the parameters showed in table 5.55. The parameter λ=-1 is to test all possible pole axis orientations. We used the period from the previous search with the lowest chi-square value.

159 Table 5.55: Search pole parameters of (4713) Steel

Parameter Value Pole λ -1 Pole β 0 Period 5.19936924 NOTE: Pole and shape searching was made in medium mode

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.49.

Figure 5.49: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.56

Table 5.56: Search pole parameters of (4713) Steel

Parameter Value Pole λ 45 Pole β 45 Period 5.19936963 NOTE: Pole and shape searching was made in medium mode

From the fine pole search chi-square optimization we implemented we obtained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.57

160 Table 5.57: Final fine search pole parameters of (4713) Steel

Parameter Value Pole λ 19.7 Pole β 54.2 Period 5.19936872 NOTE: Pole and shape searching was made in fine mode

Figure 5.50: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.10.5. GENERATING 3D SHAPE FOR (4713) STEEL We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.51

Figure 5.51: 3D shape of (4713) Steel (LC Invert by author)

161 5.11. (5692) SHIRAO

(5692) Shirao was discovered in 1992 by K. Endate. It was named in honor of Motomaro Shirao, a geologist and photographer. (Schmadel, 2012). For this asteroid, there are lightcurves from ALCDEF database and INAOE observations. We present the next information: Plot of phase angle (α) and phase angle bisector longitude (PABL) coverage for the available lightcurves. Periods reported for the asteroid. Period interval of search. Results of first search of pole axis orientations and period. Results of second search of pole axis orientations and period. 3D shape of asteroid.

5.11.1. LIGHTCURVES OF (5692) SHIRAO The lightcurves used to perform the following inversion are reported on tables 4.3 and 4.5. The distribution of phase angle (α) and phase angle bisector longitude (PABL) are shown on figure 5.52.

5.11.2. PERIOD SEARCH INTERVAL To determine the period search interval we used the periods reported on INAOE observations and ALCDEF database.

Table 5.58: Periods reported for (5692) Shirao

Source Period (hours) Reference ALCDEF 2.886 Pray 2005b INAOE 2.887∗ Table 4.3 ALCDEF 2.8878 Warner 2014l ALCDEF 2.9 Behrend 2001web ALCDEF 2.9 Behrend 2006web * Period found merging all observations made at INAOE. Periods are arranged from lowest to highest value.

5.11.3. SEARCHING PERIOD FOR (5692) SHIRAO USING ALCDEF AND INAOE LIGHTCURVES. According the reported periods we used the interval of search: Period low=2.85 Period high=2.93

162

The results of this search are shown on table 5.59 arranged by chisquare value.

Table 5.59: Period search results for (5692) Shirao using lightcurves from INAOE and ALCDEF database.

Period (hours) Chisquare Value 2.88690235 0.36439 2.88700826 0.36685 2.88789607 0.36741 2.88740357 0.36788 2.8872075 0.3688 2.88671106 0.36944 2.88709949 0.36969 2.8871549 0.37082 2.88679317 0.37153 NOTE: Periods are arranged from lowest to highest chi-square value.

5.11.4. SEARCH POLES AND MODEL FOR (5692) SHIRAO

To search the poles we used the parameters showed in table 5.60. The parameter λ=-1 is to test all possible pole axis orientations. We used the period from the previous search with the lowest chi-square value.

Table 5.60: Search pole parameters of (5692) Shirao

Parameter Value Pole λ -1 Pole β 0 Period 2.88690235 NOTE: Pole and shape searching was made in medium mode.

The output of the search is another chi-square optimization. The output of medium search is showed in figure5.53.

164 Figure 5.53: Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

From the medium pole search chi-square optimization we implemented a fine search using the output parameters with the lowest chi-square test. The parameters we used are shown on table 5.61

Table 5.61: Search pole parameters of (5692) Shirao

Parameter Value Pole λ 105.0 Pole β 45.0 Period 2.88689998 NOTE: Pole and shape searching was made in medium mode

From the fine pole search chi-square optimization we implemented we obtained the final parameters to create the 3D shape. The results of this fine pole search are shown in the table 5.62

Table 5.62: Final fine search pole parameters of (5692) Shirao

Parameter Value Pole λ 113.1 Pole β 19.5 Period 2.88679688 NOTE: Pole and shape searching was made in fine mode

165 Figure 5.54: Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author)

5.11.5. GENERATING 3D SHAPE FOR (5692) SHIRAO We again selected the lowest chi-square value to generate the 3D shape. The output of this step is shown in figure 5.55

Figure 5.55: 3D shape of (5692) Shirao (LC Invert by author)

166 Chapter 6

CONCLUSIONS AND FUTURE WORK.

6.1. GENERAL CONCLUSIONS

The INAOE observational infrastructure is appropriate to make differential photometry to solar system minor objects, despite the light pollution at the zone.

The results using MPO LC INVERT are similar for the control objects obtained from the DAMIT database. DAMIT uses different software to derive the shape, pole and period, so we have confidence in our results.

The light curves obtained can be uploaded to databases like ALCDEF to apply the inversion lightcurve method and obtaining physical parameters of asteroids.

We derived period, pole axis orientation and shape of seven main belt asteroids. The data contribute to the objectives of International Asteroid Warning Network (IWAN), which INAOE is founding member.

6.2. RESULTS CONCLUSIONS

The results obtained in the present thesis work can be divided into two groups:

1. The results of asteroids with parameters determined by DAMIT.

2. The results of asteroids without parameters determined by DAMIT.

It is important to note that DAMIT uses the same method but with different software to

167 obtain its results and we have obtained very similar values using MPO LC Invert. Thus, this gives us the confidence that the results obtained in the present work are correct.

6.2.1. ASTEROIDS WITH PARAMETERS DETERMINED BY DAMIT DATABASE. (22) KALLIOPE AND (287) NEPTHYS This category includes (22) Kalliope and (287) Nepthys, which have parameters determined by DAMIT.

(22) KALLIOPE We can observe that the lightcurve inversions using LC Invert and the lightcurves from DAMIT, figures 5.2, 5.6 and 5.12, present consistency and repeatability with the data obtained by other researchers. We can see on figure 6.1 that both the data reported by DAMIT and the found using LC Invert with the same lightcurves are very similar.

The table 6.1 shows a comparative of the results.

(287) NEPTHYS The results of (287) Nepthys using DAMIT lightcurves and LC Invert show values very similar with the parameters found by other researchers.

The tables 6.2 and 6.3 show a comparison between the data reported in DAMIT database and the found applying the lighcurve inversion using the LC Invert process.

168 Table 6.1: Results of period and pole search comparative of (22) Kalliope

Period Period Pole λ Pole β Period by LCI Pole λ Pole β by LCI Period by LCI Pole λ β by by by by using DAMIT by LCI using DAMIT using ALCDEF using ALCDEF DAMIT DAMIT DAMIT parameters and using DAMIT parameters and and INAOE and (hours) (hours) (◦)(◦) lighcurves parameters and lighcurves (◦) lightcurves lightcurv ◦ ◦ (hours) lighcurves (◦) (hours) 4.1483 4.1482 196 3 4.14820084 196.2 3.9 4.1481922 176.6 - 4.1482 196 2 - - - - - 169

Table 6.2: Results of period and pole search comparative of (287) Nepthys

Period reported Period reported Pole λ Pole β Period by LCI Pole λ by LCI β by ALCDEF by DAMIT by DAMIT by DAMIT using DAMIT using DAMIT (hours) (hours) (◦)(◦) parameters and parameters and lighcurves (hours) lighcurves (◦) ◦ 7.605 7.60411 356 36 7.6041024 352.3 - 7.6041 158 39 - - NOTE: The comparison is between data reported in DAMIT database and the process of LC Invert using DAMIT lightcurves only. Table 6.3: Results of period and pole search comparative of (287) Nepthys

Period reported Period reported Pole λ Pole β Period by LCI Pole λ by LCI Pole β by ALCDEF by DAMIT by DAMIT by DAMIT using DAMIT using DAMIT using (hours) (hours) (◦)(◦) and INAOE and INAOE and 170

7.605 7.60411 356 36 - - - - 7.6041 158 39 7.6041 158 39 NOTE: The comparison is between data reported in DAMIT database and the process of LC Invert using DAMIT and INAOE lightcurves. 6.3. ASTEROIDS WITH OUT ENTRIES IN DAMIT.

After analyzing the results of (22) Kalliope and (287) Nepthys and the information in the chapter of metodology, we selected the asteroids with a wider (α) and PABL coverage. We included in this category asteroids with at least two oppositions with at least 45◦ angular distance of PABL. These asteroids does not have parameters in DAMIT database.

Table 6.4: Results of period and pole search of asteroids with at least two oppositions.

Asteroid Period by LCI Pole λ by LCI Pole β by LCI using ALCDEF using ALCDEF using ALCDEF and INAOE and INAOE and INAOE lighcurves (hours) lighcurves (◦) lighcurves (◦) (1117) Regnita 2.72151939 165 -17.9 (1346) Gotha 2.54066893 249.5 -33.5 (1492) Oppolzer 3.76901463 70.7 40.9 (3028) Zhangguoxi 4.82664673 169.8 38.7 (3800) Karayasuf 2.23205023 294.5 -23 (4713) Steel 5.19936872 19.7 54.2 (5692) Shirao 2.88679688 113.1 19.5 NOTE: We included in this category asteroids with at least two oppositions with at least 45◦ of angular distance of PABL.

6.4. ASTEROIDS WITH NOT ENOUGH COVERAGE OF α or PABL

Table 6.5: Results of period and pole search of asteroids with not enough α coverage.

Asteroid Period by LCI Pole λ by LCI Pole β by LCI using ALCDEF using ALCDEF using ALCDEF and INAOE and INAOE and INAOE lighcurves (hours) lighcurves (◦) lighcurves (◦) (711) Marmulla 2.72151939 165 -17.9 (1318) Nerina 2.52898152 101.5 49.3 These asteroids have a low α coverage.

171 6.5. FUTURE WORK

Our results are truthful and it is necessary to continue the observations of main belt asteroids to contribute databases like ALCDEF and DAMIT, likewise contributing with the International Warning Asteroid Network (IWAN).

Use different software to find the shape and pole axis orientation, like the posted on DAMIT web site.

Continue using the INAOE observational infrastructure to perform lightcurves and to obtain shape and pole axis orientation of more asteroids.

6.5.1. RECOMMENDATIONS TO MAKE THE FUTURE OBSERVATIONS One whole night should be devoted to the observation of each asteroid. The number of points (images) per lightcurve should be limited to approximately 60 exposures per . This could be achieved by configuring the pause parameter in the CCD control software.

RECOMMENDED PERIOD OF NEXT OBSERVATIONS These recommendations were developed by a orbit simulation made at JPL Small-Body Database Browser. The recommended period of observation for asteroids contained in this work with lack of data are shown on table 6.6.

Table 6.6: Recommended observational period for next apparition of asteroids with not enough data.

Asteroid Initial date of observation Final date of observation (711) Marmulla August 15th, 2019 November 15th, 2019 (1318) Nerina July 15th, 2019 October 15th, 2019

172 List of Figures

1.1. Dwarf Planet Cross-section Model (Dymock, 2010) ...... 5 1.2. Breakup of comet 73/P. The original fragments were created during a splitting event in 1995. Figure displays a wider field of view, showing several of the original fragments. (Dymock, 2010) ...... 5 1.3. Rubble pile asteroid reforming after a collision, (Dimock, 2010). . . . . 6 1.4. Asteroid Spectra, The horizontal axis indicates the wavelength in microns. The vertical axis shows the asteroid flux light at specific wavelengths. The the left x-axis shows the albedo normalized at 0.56µm (Dimock, 2010) ...... 7 1.5. Schematic definition of orbital elements of asteroids. (Badescu, 2013) . 9 1.6. Histogram showing the four most prominent Kirwood gaps and a possible division into inner, middle and outer main belt asteroids: blue-inner main-belt(2 < a <2.5 AU); yellow- intermediate main-belt (2.5 < a <2.82 AU); green-outer main-belt (2.82 < a < 3.5 AU) [based on plot by Alan Chamberlain, JPL/Caltech] ...... 11 1.7. How the Yarkovsky effect changes the orbit of an asteroid. (Dimock, 2010) ...... 12 1.8. (a) Clustering algorithm applied to he asteroid belt separates dynamical families (yellow) from the background (red). (b) Variation in reflectance properties of main belt asteroids. Here we plot 25,000 asteroids that were observed by both SDSS and WISE, The color code was chosen to highlight the albedo/color contrast of different families. (Nesvorny et al., 2015) ...... 14 1.9. (a) Phase angles vs magnitude of asteroid (21)Lutetia (Bucheim et al., 2010). (b). Geometry of orbits: Earth distance, Sun distance, and solar phase angle. (Bucheim et al., 2010) ...... 20 1.10. Views of the first four comets (lower right) and nine asteroid systems that were imaged close-up by interplanetary spacecraft, shown at the same scale. The object name and dimensions, as well as the name of the imaging spacecraft and the year of the encounter, are listed below each figure. Note the wide range of sizes. Dactyl is a moon of Ida. (Lissauer, 2013) ...... 23

173 1.11. Light curves of (1831) Nicholson, Raw Plot and Phase Plot [Observations by author, plot and analysis made with MPO Canopus. . 24

2.1. Comparative between rotations periods data in year 1979 (left) and 2005(right).(Alan et al., 2005) ...... 28

2.2. Three Arecibo radar images of asteroid 2011UW158 showing a four- minute portion of its 37-minute rotational period. Each pixel is equal to 7.5 metres. The surface of the asteroid looks like a walnut with parallel ridges along the length of the body. (Credit: Arecibo observtory) . . . . 30

3.1. Schmidt Telescope property of INAOE [By Author] ...... 34 3.2. Optical design of Schmidt Camera. (Credits:❤♣✿✴✴✇✇✇✳✈✐❦❞❤✐❧❧♦♥✳❛❢❢✳❤❡❢✳❛❝✳✉❦✮ ...... 36 3.3. Spheric mirror aberration.(Credits: ❤♣✿✴✴✇✇✇✳✈✐❦❞❤✐❧❧♦♥✳❛❢❢✳❤❡❢✳❛❝✳✉❦✴✮ ...... 36 3.4. The focal plane consists of an array of 42 charge coupled devices (CCDs). Each CCD is 2.8 x 3.0 cm with 1024 x 1100 pixels. The entire focal plane contains 95 mega pixels. (Credit: NASA Ames and Ball Aerospace)...... 37 3.5. Flattening Lens mounted on CCD compartment. [By Author] ...... 37 3.6. The Johnson-Cousins UBVRI standard, the plot shows the sensitivity response in 5 colors. (Bessell, 2005) ...... 39 3.7. Configuration MPO Canopus. (By Author) ...... 47 3.8. Aperture (Dymock, 2010) ...... 48 3.9. Astrometry of (1318)Nerina in yellow and reference stars in red (By author) ...... 48 3.10. Selection of 4 comparison stars, in yellow one of the reference star. (Dymock, 2010) ...... 50 3.11. Differential magnitude vs time, (left) Variable comparison star. (right) invariable comparison star (By author) ...... 51 3.12. Sessions of (1831) Nicholson 20, 22 and 23 merged. Despite two nights were cloudy, the period are close enough to the 3.228 hours period reported in CALL database. (MPO Canopus, data by author ...... 52 3.13. Period spectrum plotting parameters.(MPO Canopus) ...... 52 3.14. (1831) Nicholson period search results as period spectrum, the minimum RMS match with 3.217 0.001 hours of period.(MPO Canopus) ...... ± 53 3.15. SHAPE 1(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries ( α=27◦ and α=80◦ (Kassalainen et al., 2001) . . 56

174 3.16. SHAPE 2(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries ( α=53◦ for both; first geometry viewed and illuminated from the equator, the second one from different sides of the equator). (Kassalainen et al., 2001) ...... 57 3.17. SHAPE 3(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries ( α=27◦ and α=53◦ (Kassalainen et al., 2001) . . 58 3.18. SHAPE 3(a) The original shape shown from two directions, (b) corresponding view of the convex hull, (c) the convex shape solution obtained by polyhedron inversion, and (d) lightcurves produced by the shpaes (a) (solid line), (b) (dotted line, and (c) (dashed line) in two observing geometries (α=27◦ for both; first geometry viewed and illuminated from the equator, the second one from different sides of the equator).(Kassalainen et al., 2001) ...... 59 3.19. (Left panel) Example of Period search with Chisq test. (Kassalainen, 2001), (right panel) and of pole searching value(The bluest the lowest Chisq value)[LCInvert by author] ...... 60 3.20. Convex shape model of (1620) Geographos, seen and illuminated from two directions(Kassalainen et al., 2001) ...... 61 3.21. Aspect angle and its dependency of H magnitude and geometry variation(Badescu, 2013)...... 62 3.22. An example of the Kassalainen lightcurve format of (22)Kalliope asteroid. (By author)...... 63 3.23. An example of finding period results for (22)Kalliope asteroid. (By author)...... 64 3.24. Pole of an asteroid (Badescu, 2013)...... 65 3.25. (left panel) Bad model of (1831) Nicholson. (right panel) Chisquare results; values from left to right: λ, β, Chisquare value, Period. (LC Invert, By author)...... 66

4.1. (22)Kalliope phase plot (MPO Canopus, analysis by author)...... 76 4.2. (22)Kalliope search period analysis (MPO Canopus, analysis by ). . . . 76 4.3. (287)Nephtys phase plot (MPO Canopus, analysis by author)...... 77 4.4. (287)Nephtys search period analysis (MPO Canopus, analysis by author). 77 4.5. First (711)Marmulla phase plot (MPO Canopus, analysis by author). . . 78 4.6. First (711)Marmulla search period analysis (MPO Canopus, analysis by author)...... 78 4.7. Second (711)Marmulla phase plot (MPO Canopus, analysis by author). 79

175 4.8. Second (711)Marmulla search period analysis (MPO Canopus, analysis by author)...... 79 4.9. Third (711)Marmulla phase plot (MPO Canopus, analysis by author). . 80 4.10. Third (711)Marmulla search period analysis (MPO Canopus, analysis by author)...... 80 4.11. First (1117)Reginita phase plot (MPO Canopus, analysis by Vega et al (2017)...... 81 4.12. First (1117)Reginita search period analysis (MPO Canopus, analysis by Vega et al (2017)...... 81 4.13. Second (1117)Reginita phase plot (MPO Canopus, analysis by author). . 82 4.14. Second (1117)Reginita search period analysis (MPO Canopus, analysis by author)...... 82 4.15. First (1318)Nerina phase plot (MPO Canopus, analysis by author). . . . 83 4.16. First (1318)Nerina search period analysis (MPO Canopus, analysis by author)...... 83 4.17. Second (1318)Nerina phase plot (MPO Canopus, analysis by author). . 84 4.18. Second (1318)Nerina search period analysis (MPO Canopus, analysis by author)...... 84 4.19. (1346)Gotha phase plot (MPO Canopus, analysis by author)...... 85 4.20. (1346)Gotha search period analysis (MPO Canopus, analysis by author). 85 4.21. First (1492)Oppolzer phase plot (MPO Canopus, analysis by author). . . 86 4.22. First (1492)Oppolzer search period analysis (MPO Canopus, analysis by author)...... 86 4.23. Second (1492)Oppolzer phase plot (MPO Canopus, analysis by author). 87 4.24. Second (1492)Oppolzer search period analysis (MPO Canopus, analysis by author)...... 87 4.25. (3028)Zhangguoxi phase plot (MPO Canopus, analysis by author). . . . 88 4.26. (3028)Zhangguoxi search period analysis (MPO Canopus, analysis by author)...... 88 4.27. (4713)Steel phase plot (MPO Canopus, analysis by author)...... 89 4.28. (4713)Steel search period analysis (MPO Canopus, analysis by author). 89 4.29. (3800)Karayusuf phase plot (MPO Canopus, analysis by author). . . . . 90 4.30. (3800)Karayusuf search period analysis (MPO Canopus, analysis by author)...... 90 4.31. First (5692)Shirao phase plot (MPO Canopus, analysis by author). . . . 91 4.32. First (5692)Shirao search period analysis (MPO Canopus, analysis by author)...... 91 4.33. Second (5692)Shirao phase plot (MPO Canopus, analysis by author). . . 92 4.34. Second (5692)Shirao search period analysis (MPO Canopus, analysis by author)...... 92 4.35. Third (5692)Shirao phase plot (MPO Canopus, analysis by author). . . . 93 4.36. Third (5692)Shirao search period analysis (MPO Canopus, analysis by author)...... 93

176 5.1. Coverages of lightcurves of (22) Kalliope from DAMIT database. (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 99 5.2. Shape of (22) Kalliope obtained by LC Invert using the parameters in table 5.2 ...... 101 5.3. Pole and inversion search for (22)Kalliope, (a) Output of search, values from left to right are: Chi-square value, λ, β, number of iteration, (b) Plot of chi-square values in medium mode, darkest blue is the lowest logarithm of chi-square value. (LC Invert by author) ...... 103 5.4. 3D Minkowsky representation of (22)Kalliope using parameters shown in table 5.5 [LC Invert By Author] ...... 104 5.5. Plot of chi-square values in fine mode, darkest blue is the lowest logarithm of chi-square value. (LC Invert by author) ...... 104 5.6. 3D Minkowsky representation of (22)Kalliope using the parameters shown in table 5.6 [LC Invert By Author] ...... 105 5.7. Coverages of lightcurves of (22) Kalliope from INAOE observations. (a) coverage of phase angle (α), and (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 106 5.8. Coverages of lightcurves of (22) Kalliope from ALCDEF database. (a) coverage of phase angle (α), and (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 107 5.9. Merged coverages of lightcurves of (22) Kalliope from ALCDEF (green) and INAOE (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author)108 5.10. Plot of chi-square values in medium mode, darkest blue is the lowest logarithm of chi-square value. (LC Invert by author) ...... 110 5.11. Plot of chi-square values in fine mode, darkest blue is the lowest logarithm of chi-square value. (LC Invert by author) ...... 111 5.12. 3D Minkowsky representation of (22)Kalliope using parameters shown in table 5.10 This shape were developed using the lightcurves from ALCDEF and INAOE observations and was performed at fine mode. [LC Invert By Author] ...... 111 5.13. Coverages of lightcurves of (287) Nepthys from DAMIT database: (a) coverage of phase angle (α), and (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 113 5.14. 3D shape of (287) Nephtys using the parameters in table 5.12 . . . . . 115 5.15. 3D shape of (287) Nepthys using the parameters in table 5.13 . . . . . 116 5.16. Coverages of lightcurves of (287) Nephtys from INAOE observations. (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 117 5.17. Pole and inversion search for (287) Nepthys. The plot of chi-square values was made in medium mode, darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 119

177 5.18. Pole and inversion search for (287) Nepthys, Plot of chi-square values in fine mode, darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 120 5.19. 3D Minkowski representation of (287) Nephtys using parameters shown in table 5.17. This shape was developed using lightcurves from DAMIT and INAOE observations and it was performed at fine mode. [LC Invert By Author] ...... 121 5.20. Coverages of merged lightcurves of (711) Marmulla from INAOE observations (green) and ALCDEF database (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 123 5.21. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 125 5.22. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 126 5.23. 3D shape of (711) Marmulla (LC Invert by author) ...... 126 5.24. Coverages of merged lightcurves of (711) Marmulla from INAOE observations (green) and ALCDEF database (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 128 5.25. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 130 5.26. Plot of chi-square values in fine mode, darkest blue the lowest logarithm of chi-square value. (LC Invert by author) ...... 131 5.27. 3D shape of (1117) Reginita (LC Invert by author) ...... 131 5.28. Coverages of merged lightcurves of (1318) Nerina from INAOE observations (green) and ALCDEF database (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 133 5.29. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 135 5.30. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 136 5.31. 3D shape of (1318) Nerina (LC Invert by author) ...... 136 5.32. Coverages of merged lightcurves of (1346) Gotha from INAOE observations (orange) and ALCDEF database (green). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 138 5.33. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 140 5.34. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 141 5.35. 3D shape of (1346) Gotha (LC Invert by author) ...... 141

178 5.36. Coverages of merged lightcurves of (1492) Oppolzer from INAOE observations (orange) and ALCDEF database (green). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 143 5.37. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 145 5.38. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 146 5.39. 3D shape of (1492) Oppolzer (LC Invert by author) ...... 146 5.40. Coverages of merged lightcurves of (3028) Zhangguoxi from INAOE observations (orange) and ALCDEF database (green). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 148 5.41. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 150 5.42. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 151 5.43. 3D shape of (3028) Zhangguoxi (LC Invert by author) ...... 151 5.44. Coverages of merged lightcurves of (3800) Karayusuf from INAOE observations (orange) and ALCDEF database (green). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 153 5.45. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 155 5.46. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 156 5.47. 3D shape of (3800) Karayusuf (LC Invert by author) ...... 156 5.48. Coverages of merged lightcurves of (4713) Steel from INAOE observations (orange) and ALCDEF database (green). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 158 5.49. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 160 5.50. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 161 5.51. 3D shape of (4713) Steel (LC Invert by author) ...... 161 5.52. Coverages of merged lightcurves of (5692) Shirao from INAOE observations (green) and ALCDEF database (orange). (a) coverage of phase angle (α), (b) coverage of phase angle bisector longitude (PABL) (LC Invert by author) ...... 163 5.53. Plot of chi-square values in medium mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 165

179 5.54. Plot of chi-square values in fine mode, the darkest blue shows the lowest logarithm of chi-square value. (LC Invert by author) ...... 166 5.55. 3D shape of (5692) Shirao (LC Invert by author) ...... 166

180 List of Tables

1.1. Bower asteroid discovery periods.(Peebles, 2000) ...... 8 1.2. Orbital elements ...... 10 1.3. Main Belt Asteroid Familes(Nesvorny et al., 2015) ...... 15 1.4. Variation in values of H and G for (1)Ceres and (8) Flora. (Dimock, 2010)...... 19

3.1. Location of INAOE Schmidt Camera, Santa María Tonantzintla, San Andrés Cholula, Puebla, México.(Google Maps) ...... 35 3.2. CCD Characteristics(Santa Barbara Instrument Group, manufacturer of CCD) ...... 38 3.3. UBVRI Johnson Standard Characteristics.(Dymock, 2010) ...... 38 3.4. Optic Schmidt Telescope Characteristics ...... 39 3.5. Calculed Valules of the Optical System...... 40 3.6. Observing stages planning...... 41 3.7. Procedure for a photometry session in MPO Canopus. /(MPO Canopus user manual, 2003) ...... 49 3.8. The maximum amplitude for a given primary harmonic...... 53 3.9. Fourier fourth order series coefficients of the plot shown on figure. 3.12 54

4.1. Observational circumstances for the observed asteroids and magnitude band of comparison stars...... 69 4.2. Selected parameters of the observed asteroids ...... 70 4.3. Rotation period and brightness amplitude of the observed asteroids. . . 71 4.4. Rotation period and brightness amplitude of the observed asteroids. . . 72 4.5. observational circumstances for the observed asteroids in ALCDEF database(ALCDEF database) ...... 72

5.1. Pole axis and periods values reported for (22) Kalliope...... 100 5.2. Parameters used by DAMIT database of (22) Kalliope (Kaasalainen 2002a) to generate the 3D shape...... 101 5.3. Period search results for (22)Kalliope using 100 lightcurves from DAMIT database ...... 102 5.4. Initial parameters of sear for (22) Kalliope (Kaasalainen 2002a) . . . . 102 5.5. Parameters to 3D generation of (22)Kalliope in medium mode . . . . . 103

181 5.6. Parameters to 3D conversion of (22)Kalliope in fine mode ...... 104 5.8. Parameters used to pole and shape search for (22) Kalliope in medium mode by using the lightcurves from ALCDEF database and INAOE. . . 109 5.7. Period search results for (22)Kalliope using lightcurves from ALCDEF and INAOE observations...... 109 5.9. Pole search parameters of (22) Kalliope by using the ALCDEF and INAOE lightcurves...... 109 5.10. Pole search final parameters of (22) Kalliope using the ALCDEF and INAOE lightcurves...... 110 5.11. Pole axis orientation and periods reported for (287) Nepthys...... 114 5.12. First set of parameters used by DAMIT database for (287) Nepthys (Hanuš 2016) ...... 114 5.13. Second set of parameters used by DAMIT database for (287) Nepthys (Hanuš 2016) ...... 115 5.14. Period search results for (287)Nephtys the lightcurves from DAMIT database...... 118 5.15. Parameters used to search pole axis orientation and shape of (287) Nepthys...... 119 5.16. Parameters used to search poles and shape in medium mode of (22) Kalliope ...... 119 5.17. Parameters used to search poles and shape in fine mode of (22) Kalliope 120 5.18. Periods reported for (711) Marmulla ...... 122 5.19. Period search results for (711) Marmulla using lightcurves from INAOE and ALCDEF database...... 124 5.20. Search pole parameters of (711) Marmulla ...... 124 5.21. Search pole parameters of (711) Marmulla ...... 125 5.22. Final fine search pole parameters of (711) Marmulla ...... 125 5.23. Periods reported for (1117) Reginita ...... 127 5.24. Period search results for (1117) Reginita using lightcurves from INAOE and ALCDEF database...... 129 5.25. Search pole parameters of (1117) Reginita ...... 129 5.26. Search pole parameters of (1117) Reginita ...... 130 5.27. Final fine search pole parameters of (711) Marmulla ...... 130 5.28. Periods reported for (1318) Nerina ...... 132 5.29. Period search results for (1318) Nerina using lightcurves from INAOE and ALCDEF database...... 134 5.30. Search pole parameters of (1318) Nerina ...... 134 5.31. Search pole parameters of (1318) Nerina ...... 135 5.32. Final fine search pole parameters of (1318) Nerina ...... 135 5.33. Periods reported for (1346) Gotha ...... 137 5.34. Period search results for (1346) Gotha using lightcurves from INAOE and ALCDEF database...... 139 5.35. Search pole parameters of (1346) Gotha ...... 139

182 5.36. Search pole parameters of (1346) Gotha ...... 140 5.37. Final fine search pole parameters of (1346) Gotha ...... 140 5.38. Periods reported for (1492) Oppolzer ...... 142 5.39. Period search results for (1492) Oppolzer using lightcurves from INAOE and ALCDEF database...... 144 5.40. Search pole parameters of (1492) Oppolzer ...... 144 5.41. Search pole parameters of (1492) Oppolzer ...... 145 5.42. Final fine search pole parameters of (1492) Oppolzer ...... 145 5.43. Periods reported for (3028) Zhangguoxi ...... 147 5.44. Period search results for (3028) Zhangguoxi using lightcurves from INAOE and ALCDEF database...... 149 5.45. Search pole parameters of (3028) Zhangguoxi ...... 149 5.46. Search pole parameters of (3028) Zhangguoxi ...... 150 5.47. Final fine search pole parameters of (3028) Zhangguoxi ...... 150 5.48. Periods reported for (3800) Karayusuf ...... 152 5.49. Period search results for (3800) Karayusuf using lightcurves from INAOE and ALCDEF database...... 154 5.50. Search pole parameters of (3800) Karayusuf ...... 154 5.51. Search pole parameters of (3800) Karayusuf ...... 155 5.52. Final fine search pole parameters of (3800) Karayusuf ...... 155 5.53. Periods reported for (4713) Steel ...... 157 5.54. Period search results for (4713) Steel using lightcurves from INAOE and ALCDEF database...... 159 5.55. Search pole parameters of (4713) Steel ...... 160 5.56. Search pole parameters of (4713) Steel ...... 160 5.57. Final fine search pole parameters of (4713) Steel ...... 161 5.58. Periods reported for (5692) Shirao ...... 162 5.59. Period search results for (5692) Shirao using lightcurves from INAOE and ALCDEF database...... 164 5.60. Search pole parameters of (5692) Shirao ...... 164 5.61. Search pole parameters of (5692) Shirao ...... 165 5.62. Final fine search pole parameters of (5692) Shirao ...... 165

6.1. Results of period and pole search comparative of (22) Kalliope . . . . . 169 6.2. Results of period and pole search comparative of (287) Nepthys . . . . 169 6.3. Results of period and pole search comparative of (287) Nepthys . . . . 170 6.4. Results of period and pole search of asteroids with at least two oppositions...... 171 6.5. Results of period and pole search of asteroids with not enough α coverage.171 6.6. Recommended observational period for next apparition of asteroids with not enough data...... 172

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