ARISTOTLE UNIVERSITY OF THESSALONIKI

(AUTh)

FACULTY OF SCIENCES

SCHOOL OF PHYSICS

SECTION OF ASTROPHYSICS, ASTRONOMY

& MECHANICS

DIPLOMA THESIS

“ANALYSIS OF AND DATA FROM NEAR- EARTH ”

CHRYSOVALANTIS SARAKIS

SUPERVISOR: KLEOMENIS TSIGANIS

THESSALONIKI, GREECE

OCTOBER 2017 2

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Table of Contents

TABLE OF CONTENTS ...... 3

ABSTRACT ...... 7

ΠΕΡΙΛΗΨΗ ...... 7

ACKNOWLEDGEMENTS - INTRODUCTION ...... 9

CHAPTER 1 ...... 11

ASTEROIDS ...... 11

1.1 ORBITAL ELEMENTS ...... 11

1.1.1 Semi-major axis (a) ...... 11

1.1.2 Eccentricity (e) ...... 12

1.1.3 Inclination (풊/Incl) ...... 12

1.1.4 Longitude of the ascending node (Ω/Node) ...... 12

1.1.5 Argument of perihelion (ω/Peri) ...... 13

1.1.6 (M) ...... 13

1.1.7 ...... 13

1.1.8 Perihelion (q) ...... 13

1.1.9 Aphelion (Q) ...... 13

1.2 NEAR-EARTH ASTEROIDS ...... 14

1.3 MAIN BELT ...... 14

1.3.1 Spectral Classification ...... 15

1.3.2 Kirkwood Gaps ...... 16

1.3.3 Resupply of NEOs ...... 17

1.3.4 Asteroid Families ...... 18

1.4 TROJAN ASTEROIDS ...... 19

1.5 CENTAURS ...... 19

1.6 TRANSNEPTUNIAN OBJECTS ...... 19

1.7 WHY DO WE STUDY ASTEROIDS? ...... 20

PHYSICAL PROPERTIES ...... 20

1.7.1 ...... 20 4

1.7.2 Direction of Rotation Axis ...... 21

1.7.3 Size ...... 21

1.7.4 Shape ...... 22

CHEMICAL COMPOSITION ...... 22

1.7.5 Reflectance Spectrum...... 22

CHAPTER 2 ...... 25

PHOTOMETRY ...... 25

2.1 LIGHT-CURVE ANALYSIS ...... 25

2.2 AIR MASS ...... 27

2.3 SEEING ...... 27

2.4 SCINTILLATION ...... 28

2.5 OPPOSITION EFFECT ...... 28

2.6 DIFFERENTIAL PHOTOMETRY ...... 29

2.7 BIAS-FRAME SUBTRACTION ...... 30

2.8 DARK-CURRENT SUBTRACTION ...... 31

2.9 FLAT-FIELD CORRECTION ...... 32

2.9 OBSERVATIONS ...... 33

CHAPTER 3 ...... 35

OCCULTATIONS ...... 35

3.1 WHAT IS A STELLAR BY AN ASTEROID? ...... 35

3.2 WHY DO WE STUDY OCCULTATIONS? ...... 37

3.3 OBSERVATIONS ...... 39

CHAPTER 4 ...... 41

RESULTS ...... 41

4.1 LIGHT-CURVE ANALYSIS ...... 41

4.1.1 Asteroid (100926) 1998 MQ ...... 41

4.1.2 Asteroid (6974) Solti ...... 47

4.1.3 Asteroid (2012) Guo Shou-Jing ...... 51

4.2 ANALYSIS OF STELLAR OCCULTATIONS BY ASTEROIDS DATA ...... 55

4.2.1 Occultation by (45) Eugenia, June 13 2014 ...... 55 5

ASTEROID (45) EUGENIA ...... 55

4.2.2 Occultation by (102) Miriam, March 03 2017 ...... 59

ASTEROID (102) MIRIAM ...... 59

4.2.3 Occultation by (3332) Raksha, March 03 2017 ...... 63

ASTEROID (3332) RAKSHA ...... 63

CHAPTER 5 ...... 67

CONCLUSIONS ...... 67

APPENDIX ...... 69

ASTEROID (100926) 1998 MQ ...... 69

Attempt to track the asteroid ...... 69

Attempt to track the asteroid by implying predictions of its position ...... 71

Attempt to FIND the comparison stars ...... 75

Attempt to TRACK the comparison stars ...... 77

ASTEROID (2012) GUO SHOU-JING ...... 79

Attempt to TRACK the comparison stars ...... 79

ASTEROID (45) EUGENIA ...... 83

Attempt to conduct the differential photometry analysis of asteroid’s (45) Eugenia occultation .. 83

BIBLIOGRAPHY-REFERENCES...... 85 6

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Abstract

Asteroids are small bodies of the Solar System that orbit the Sun and their study is of great importance. They depict the Solar System in its early age and their study can improve its formation model and also contribute to the understanding of how planetary systems beyond ours formed. Furthermore the presence of asteroids in the Earth‟s neighborhood, near-Earth asteroids, and the possibility of a catastrophic impact with her makes their study crucial. Physical properties such as size and rotation period time can be derived from the analysis of photometry data by short-term observations, while shape and direction of the rotation axis by long-term observations. Also, observations of stellar occultations by asteroids is a more direct way of determine their sizes and shapes. The goals of the diploma thesis is the calculation of the rotation period time for asteroids (100926) 1998 MQ, (6974) Solti and (2012) Guo Shou- Jing, by the study of their light-curves using the differential photometry analysis technique and the calculation of the duration and the depth of stellar occultations by asteroids using the same technique.

Περίληψη

Οη αζηεξνεηδείο είλαη κηθξά ζώκαηα ηνπ Ηιηαθνύ Σπζηήκαηνο, ηα νπνία πεξηθέξνληαη γύξω από ηνλ Ήιην θαη ε κειέηε ηνπο είλαη ηδηαίηεξα ζεκαληηθή. Απεηθνλίδνπλ ην Ηιηαθό Σύζηεκα ζηε λεαξή ηνπ ειηθία θαη ε κειέηε ηνπο κπνξεί λα βειηηώζεη ην κνληέιν ηνπ ζρεκαηηζκνύ ηνπ θαη επίζεο λα ζπλεηζθέξεη ζηε θαηαλόεζε πώο πιαλεηηθά ζπζηήκαηα πέξα από ην δηθό καο ζρεκαηίζηεθαλ. Επηπιένλ, ε παξνπζία αζηεξνεηδώλ ζηε γεηηνληά ηεο Γεο, near-Earth asteroids, θαη ε πηζαλόηεηα κίαο θαηαζηξνθηθήο ζύγθξνπζεο καδί ηεο, θαζηζηά ηε κειέηε ηνπο θξίζηκε. Φπζηθά ραξαθηεξηζηηθά, όπωο ην κέγεζνο θαη ε πεξίνδνο πεξηζηξνθήο πξνθύπηνπλ από ηε ηελ αλάιπζε δεδνκέλωλ θωηνκεηξίαο βξαρπρξόληωλ παξαηεξήζεώλ ηνπο, ελώ ην ζρήκα θαη ε δηεύζπλζε ηνπ άμνλα πεξηζηξνθήο κέζω καθξνρξόληωλ παξαηεξήζεωλ. Επίζεο ε παξαηήξεζε ηωλ δηαβάζεωλ ηνπο εκπξόο από αζηέξεο (occultations) απνηειεί έλαλ πην άκεζν ηξόπν θαζνξηζκνύ ηωλ κεγεζώλ θαη ηωλ ζρεκάηωλ ηνπο. Σθνπνί ηεο εξγαζίαο είλαη ν ππνινγηζκόο ηεο πεξηόδνπ πεξηζηξνθήο ηωλ αζηεξνεηδώλ (100926) 1998 MQ, (6974) Solti θαη (2012) Guo Shou-Jing από ηε κειέηε ηωλ θακππιώλ θωηόο ηνπο κέζω ηεο ηερληθήο αλάιπζεο differential photometry θαη ν ππνινγηζκόο ηεο ρξνληθήο δηάξθεηαο θαη ηνπ βάζνπο αζηξηθώλ occultations από αζηεξνεηδείο κέζω ηεο ίδηαο ηερληθήο αλάιπζεο.

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Acknowledgements - Introduction

Asteroids are objects of our Solar System that show great interest. For many years they are the subject of numerous of studies. From 1801 when the first of these objects was discovered by G. Piazzi, (1) Ceres, up to now they still manage to attract the attention of scientists. However in the last few decades great leaps have been done that contributed to their extensive understanding. These are the results of the continuous sophistication of the observational tools, the growth of space missions and the unreduced interest and passion of both professional and amateur astronomers.

With this diploma thesis I tried to place a little stone in the big pile of asteroid studies. In its beginning I will describe what are the asteroids, where can we find them and why do these small bodies attract the interest of astronomers. After that I will focus my discussion to the main belt and near-Earth asteroids. Secondly, I will answer on how we can derive a light-curve from doing asteroid photometry and what are the constraints we should be aware of while observing. In the third part I will discuss about the importance of stellar occultations by asteroids and finally I will appose my results from the analysis that I conducted and state my conclusions. An appendix will include all my simple programming efforts in python codes which I used to my analysis.

This diploma thesis is the result of my studies in the school of physics and my specialization in astronomy and astrophysics. It has been worked both in the Aristotle University of Thessaloniki (AUTh) in Greece and in the European Space Research and Technology Centre (ESTEC) and Leiden University (LU) in the Netherlands. I would like to express my gratitude for the completion of this diploma thesis to my supervisors associate professor Kleomenis Tsiganis (AUTh) and Dr. Chrysa Avdellidou ESA researcher in Planetary Science (ESTEC). Also I want to special thank my family and my close friends for their continuous support. Finally I want to thank for providing me with observational data Chrysa Avdellidou, Marco Delbò (Observatoire de la Côte d‟Azur), Detlef Koschny (ESA) and the scientific team of Observatoire de la Côte d‟Azur. For offering me their advices, without an order, my thanks to associate professor Christos Tsagas (AUTh), emeritus professor John-Hugh Seiradakis (AUTh) and Marco Delbò . For the funding and hosting during my participation to the Leiden/ESA Astrophysics Program for Summer Students 2017 I would like to thank the European Space Agency and Leiden University. Special thanks to my home institute the Aristotle University of Thessaloniki.

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Chapter 1

Asteroids

1.1 Orbital Elements

Asteroids are orbiting the Sun in elliptic orbits and they are described by the following orbital elements:

Figure 1.1 : The orbital elements of the asteroid’s elliptic orbit. Credit: Encyclopedia Britannica

1.1.1 Semi-major axis (a)

It defines the half of the distance of the long axis of the asteroid‟s elliptic orbit. This long axis connects the most distant points of the orbit. 12

1.1.2 Eccentricity (e)

It defines the deviation of the asteroid‟s orbit from a circular orbit. The eccentricity of a circular orbit is zero. The eccentricity of an elliptic orbit is greater than zero and less than the unit.

1.1.3 Inclination (풊/Incl)

It defines the angle between the ecliptic plane and the orbital plane of the asteroid. It ranges between 0° - 180 °. If it is greater than zero and less than 90° the asteroid‟s motion is considered prograde. If it is greater than 90° and less than 180° then the motion of the asteroid is considered retrograde.

The line that defines the intersection between the asteroid‟s plane and the ecliptic plane is called “line of nodes”.

“Ascending node” defines the point in the line of nodes where the motion of the asteroid in its orbital plane is in the direction below – above to the ecliptic plane. Respectively the point that the direction of the asteroid‟s motion in the orbital plane is above - below the ecliptic plane, is called “descending node”.

As a reference point in order to define the orbital elements is selected the Vernal Equinox, called “First Point of Aries”.

1.1.4 Longitude of the ascending node (Ω/Node)

The “longitude of the ascending node” defines the direction of the line of nodes in space. It is the angle between the reference point and the line of nodes. It is measured eastwards (increasing Right Ascension) with starting point the First Point of Aries.

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1.1.5 Argument of perihelion (ω/Peri)

It defines the orientation of the semi-major axis in the orbital plane of the asteroid. It is the angle between the ascending mode and the perihelion point. It is measured in the direction of the motion of the asteroid in its orbital plane.

1.1.6 Mean anomaly (M)

It defines the position of the asteroid in a circular orbit with radius equal to the semi- major axis. The frequency of the motion is the same as having a in the real elliptical orbit of the asteroid.

The following are also important to be defined:

1.1.7 Epoch

It defines the date on which the orbital elements of the asteroid are calculated.

1.1.8 Perihelion (q)

It defines the closest point of the asteroid‟s orbit to the Sun. It is calculated by the formula:

푞 = 푎 ∙ (1 − 푒)

1.1.9 Aphelion (Q)

It defines the furthest point of the asteroid‟s orbit to the Sun. It is calculated by:

푄 = 푎 ∙ (1 + 푒)

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According to their heliocentric distances asteroids are categorized in the following groups:

1.2 Near-Earth Asteroids

In small heliocentric distances from the Sun are found the “Near Earth Asteroids” (NEAs). These asteroids are orbiting the Sun in distances that are close to the orbit of the Earth. They are subdivided into four main asteroid groups, “Apollos”, “Atens”, “Amors” and “Atiras” according to their perihelion distance (q), aphelion distance (Q) and semi-major axis (a). They were named after their first discovered member in their group. “Apollos” are named from asteroid (1862) Apollo, they cross Earth‟s orbit and they spend most of their time outside of it. Until now they have been found 9064 “Apollos”. Asteroid (2062) Aten gave its name to the “Atens” group. Also these asteroids are Earth-crossing NEAs and have perihelia between the Earth‟s orbit but they spent most of their time within it. 1227 “Atens” have already been found. “Amors” named from asteroid (1221) Amor have orbits that stay outside Earth‟s orbit. Until today 6386 “Amors” have been discovered. Finally 16 “Atiras” named after asteroid (163693) Atira have orbits that are contained entirely within the Earth‟s orbit. [1]

1.3 Main

In greater heliocentric distances there is the “Main Asteroid Belt”. It lies between 2.1 and 3.3 astronomical units from the Sun and it is between the trajectories of Mars and . It is the biggest reservoir of asteroids in our Solar System [2]. Until now they have been discovered more than 738,271 Main Belt asteroids [3] and it is estimated that it contains between 1.1 and 1.9 million asteroids larger than 1 kilometer in diameter and millions of smaller ones. The first body of this region to be discovered is asteroid (1) Ceres, by Giuseppe Piazzi in 1801. The orbits of the main belt asteroids can be quite elliptical with eccentricities of even 푒 ~ 0.4. Also their inclinations can reach 푖 ~ 30° which indicates that the belt is not a relatively flat disk. 15

Figure 1.2: The Main Asteroid Belt lies from 2.1 to 3.3 au, between the orbits of Mars and Jupiter and it is thought to be the biggest reservoir of asteroids in the Solar System. Credit: NASA/Lunar and Planetary Institute

1.3.1 Spectral Classification

According to the shape of their spectra there are three main spectral classes or types. The first class is the Carbonaceous or C class. Most of the asteroids in the outer Main Asteroid Belt belong to this class. They are relatively blue in color and have the same chemical composition with the Sun. Also they do not include volatiles such as hydrogen or helium. The second class is the S class. Asteroids of this class are stony or rich in silicates and have relatively red color. Finally the third class is the X class which is a mixture of different materials. It is worth noting that until today a large number of discovered asteroids remain to be classified or their classification changes as the observational tools are becoming more sophisticated allowing a better estimation. 16

Figure 1.3: The variations in the geometric visible albedo of asteroids are indicating their diversity in chemical composition and categorize them in C, S or X spectral classes. On the other hand the Kirkwood Gaps can be observed by studying the distributions of the inclination’s sine over the semi-major axis. Credit: Marco Delbò

1.3.2 Kirkwood Gaps

The main asteroid belt is not all dense populated. In 1867 Daniel Kirkwood discovered some regions in the main asteroid belt that they were depleted by asteroids. These regions are named “Kirkwood gaps” and are found at the positions of resonances with Jupiter. The fact that an asteroid is situated at a position of resonance is given by an exact integer ratio of the asteroid‟s mean orbital motion to this of Jupiter‟s. However these gaps are not physical gaps in the belt but orbital periods that are missing. These physical gaps can be filled by the main belt asteroids as they orbit the Sun due to their great eccentricities [4]. 17

Figure1.4: The Yarkovsky effect is a non-gravitational factor that can explain the resupply of NEOs. Credit: Sky & Telescope

1.3.3 Resupply of NEOs

This can explain the resupply of Near Earth Objects. Asteroids that cross the trajectories of the terrestrial have dynamical lifetimes of ~ 107 years, which is relatively short regarding to our Solar System‟s lifetime (~4.6 퐺. 푦. ). However a large number of them are still observed in these trajectories. This is caused because some of the main belt asteroids change their semi-major axis due to collisions between them and are found in the “Kirkwood gaps” or in the 푣6 resonance with Saturn. The gravitational interactions that are affecting them lead either to their ejection from the Solar System or to orbits that cross the terrestrial ‟s trajectories. A possible step is a planetary or a solar impact.

The resupply of Near Earth Objects is caused also by non-gravitational forces. The Yarkovsky effect can lead to the change of their semi-major axis and drift them to orbits resonant with the planets‟ or other larger asteroid‟s orbits. Over time this can lead to the change of the orbit‟s eccentricity and inclination. The amplifying of the eccentricity can lead to planet-crossing or Sun-colliding orbits, which justify the presence of near-Earth objects. 1 The drift of the semi-major axis is size dependent and is changing proportional to 퐷, where 퐷 is the diameter of the asteroid, moving inwards for a retrograde spin or outwards for a retrograde spin. 18

Figure 1.5: Dependence of the inverse size 1/D on the proper semi-major axis ap for the Eos collisional family with the estimated position of the family center (vertical line).

J. Hanuš et al. , 2018 (Icarus)

1.3.4 Asteroid Families

The asteroids can be classified in “asteroid families”. This is the name that Hirayama gave to some significant clumping of asteroids in 1918. They are formed due to either collisional disruption or cratering event excavations of larger parent bodies. According to the Hierarchical Clustering Method asteroids are identified in families by their proper orbital elements, in other words eccentricity, semi-major axis and inclination of the asteroid‟s orbit that are independent of planetary perturbations, the relative velocity between them and a central reference asteroid and selecting all asteroids below a cutoff velocity value.

Due to the size dependency of the semi-major axis drift caused by the Yarkovsky 1 effect asteroid families acquire a characteristic V-shape [5] in the plane 푎 vs 퐷 on 푀. 푦. time scales. All asteroids are drifting away from the center of the family and the determination of whether an asteroid has a prograde or retrograde spin results in its determination if it is a member of the family or not and put constraints to the limits of the family. V-shapes have been used to measure the age of families [5] and to determine which bodies are “interlopers” and they do not belong to the despite the fact that according to the HCM model they do. [5] 19

1.4 Trojan Asteroids

After the main asteroid belt there are the “Jupiter Trojans” asteroids. They are thought to be the second biggest reservoir of asteroids in our Solar System [2] and form the most significant population of Trojan asteroids. These asteroids are sharing the same orbit and of Jupiter but they are situated around the Lagrangian points 퐿4 and 퐿5 of its trajectory, respectively leading and trailing by 60° in the planet„s orbit. In these points the pull from the Sun and the planet are balanced with the tendency of the Trojan asteroid to fly out from its orbit [6]. However until today around 6,456 asteroids have been discovered and studied and there also been discovered 17 “Neptune Trojans” 1 “Uranus Trojan”, 4 “Martian Trojans” and 1 “Earth Trojan” [6].

1.5 Centaurs

In heliocentric distances of 5.5 to 29 astronomical units are found the “Centaurs”. The semi-major axis of their elliptic orbits is situated within the orbits of Jupiter and Neptune. 685 Centaurs and Scattered Disk Objects have been discovered until today. Due to their position in the Solar System and to the high possibility of being perturbated by the giant planets they pose a threat to Earth. [7]

1.6 TransNeptunian Objects

Even further there are the “transNeptunian Objects” (TNOs).

“Classical Edgeworth- Objects” have orbits between 42 to 48 astronomical units. Their orbit‟s eccentricities are low and they characterized as “cold” if they have low inclinations or “hot” if they have high inclinations. [7]

“Scattered Disk Objects” have orbits greater than 48 astronomical units. They have also high eccentricities meaning that their aphelion distance can reach great heliocentric distances, even hundreds of astronomical units. 20

The first of the “Detached Objects” to be discovered was (90377) Sedna. These bodies have orbits with high eccentricities. They are not gravitationally perturbated by either Neptune or the Edgeworth-Kuiper Belt Objects. [7]

1.7 Why do we study asteroids?

Today asteroids can depict the picture of the Solar System as it was around 4.6 퐺. 푦. ago. Most of them are situated in the region between the terrestrial and the giant planets where Jupiter‟s perturbations are the main cause for the failure of a planet formation. [8] Asteroids are the “most direct remnants of the original building blocks that formed the planets”. [2] Despite the fact that their collisional evolution is enormous, their geological, thermal and orbital evolutions are of little progress.

Physical Properties

1.7.1 Rotation Period

Asteroids can be distinguished in solid bodies and gravitational aggregates (commonly referred to in the literature as “rubble piles”). An asteroid with a diameter less than 100 or 150 meters is considered to be a solid body [7] and was formatted due to collisions with the other asteroids. Larger bodies with greater diameters are consisted of a big number of smaller fragments loosely bounded together. The main reason of their formation is also the collisions between larger asteroids. However in this case after the collision the smaller fragments came together, re-forming these rubble piles. 21

The rotation periods that characterize asteroids that are thought to be gravitational aggregates are greater than two and a half or a little less than this. This is because if the asteroid is rotating faster than this its smaller fragments may come in loose. In this case the centrifugal force overpowers the asteroid‟s self gravity. As a result every asteroid that is found to have a smaller period time of rotation around its spin axis is considered to be a monolithic rock, in other words a solid body. So, the computation of the rotation period time by the analysis of photometry data can distinguish asteroids either in solid bodies or gravitational aggregates.

Studying slow rotators are more difficult due to the great amount of time needed to be spent for their observations and to the difficulty of choosing the right comparisons stars. Because of luck of gathered data their study is considered necessary in order suggestions to be made of where and how these bodies formed.

1.7.2 Direction of Rotation Axis

Another characteristic of asteroids is the direction of the rotation axis. Its calculation can lead to the determination if the asteroid has a prograde or retrograde spin. This can help to calculate if it is drifting outwards to greater heliocentric distances or inwards to smaller heliocentric distances. Thus scientists can determine if this asteroid is a member of an asteroid family of interest and put constraints to the limits of its V-shape.

1.7.3 Size

The largest asteroid discovered is (1) Ceres, with a diameter of 939.4 kilometers. [9] The determination of the size can result to the computation of the asteroid‟s volume. If its mass has been also determined then scientists can calculate its density. This can lead to general knowledge of the asteroids‟ composition as well as meteorites‟ origin. The mass can be determined from the gravitational effects in the orbits either of Mars for the largest asteroids or of other asteroids, asteroids‟ satellites or spacecrafts that orbit them or fly by them.

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1.7.4 Shape

Except from the largest asteroids, most of them have irregular shapes. The determination of the asteroid‟s shape can lead to the computation of its volume. Finally its determination can lead to numerical solutions about the calculation of the direction of the rotation axis.

Chemical Composition

1.7.5 Reflectance Spectrum

Asteroids are bodies that do not emit their own light. Instead they reflect the Sun‟s light. Thus their spectra are referred to as “reflectance” spectra. The reflectance spectrum of an asteroid has many similar features with the Sun‟s spectrum. However it differs from this mostly due to the composition of the asteroid‟s surface. Specifically minerals in the asteroid‟s surface reflect the sunlight differently. Some other reasons that can change the shape of the spectra are the increased phase angle which causes the reddening of the spectrum, space weathering that darkens or reddens asteroid‟s surface, the size of regolith‟s particles in asteroid‟s surface and the surface temperature. [7]

Geometric albedo and wavelength are used to give the shape of the reflectance spectrum. The former is the ratio between the reflected light from the asteroid‟s surface and the reflected light from a perfectly white sphere of the same size and at the same distance from the Sun. In the last case the incident light is all reflected and the geometric albedo equals to 1. Typical values for asteroids vary from 0.05 to 0.25. [7] Finally for the latter mostly are used visible wavelengths (390-700 nm).

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The information that a researcher gets from the reflectance spectrum varies. The reflectance spectrum can give an estimation of the asteroid‟s surface mineralogy. It can show if specific crystalline and amorphous compounds are present, including rocks, metals and ices. [10] Also major chemical abundances can be determined. As different minerals may be formed from the same chemicals scientists can then make suggestions about the processes of asteroid‟s origin and evolution. This is due to the temperature and the pressure under the minerals were formed and on the following processes of alternation [10]. Finally the interiors of differianted bodies can be studied as a result of their collisional disruption.

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Chapter 2

Photometry

2.1 Light-curve Analysis

Figure 2.1: The observable variations in the asteroid’s flux can result in the determination of its light-curve. Light-curve for asteroid (206) Kleopatra . Credit : Database of Asteroid Models from Inversion Techniques (DAMIT)

Asteroids are celestial objects that are spinning around their rotational axis. Due to their irregular shapes an observer can detect little changes in the measured flux. Light-curve photometry is the measurement of these variations in brightness over time [11]. The most common measured quantities in the horizontal axis are or flux and in the vertical axis is time or phase angle (the angle Sun-asteroid-Earth). In the light-curve analysis of an asteroid the observer can derive the asteroid‟s rotation period time due to its spinning and the amplitude of the curve. 26

However a researcher can keep on with further analysis. By gathering data for a relatively short period of time and plotting them, unexpected “dips” (usually a dimming of 0.01-0.03 magnitudes) in the general light-curve [11] may appear, supporting the claim that the asteroid is a binary system. Also gathering data for a longer time (several months or years) may lead to the deriving of the shape [11] taking also into account the variations of its light- curve‟s amplitude. Furthermore scientists can determine the direction of its spin axis. [11] This can lead to the determination if the asteroid has a prograde or retrograde spin and belongs or not to an asteroid family of interest setting the limits in the V-shape of the family.

Figure 2.2: Long-term observations can result in the determination of the asteroid’s shape taking also into account the amplitude’s variations of its light-curve over time. Shape of asteroid (206) Kleopatra. Credit: Database of Asteroid Models from Inversion Techniques (DAMIT)

It is essential though that the observations include several different apparitions of the asteroid-target. This is because with every apparition the asteroid‟s spin axis has a different angle from the line of sight to the observer [11]. As a result the amplitude of the curve is different for each apparition. The researcher can measure these changes over time and result in the determination of the pole orientation (spin axis) with only five to seven light-curves. [11]

Radar observations can be used to measure the frequency shift of the signal sent to the asteroid and its distance from the researcher. A further step to the analysis is the deriving of the shape of the asteroid and whether if it has a companion body. Light-curve photometry can be used to pinpoint if a radar observation of an asteroid-target will be valuable. If the optical and radar observations are taking place at the same time then the light-curve photometry can put constraints to the results of the radar observations. [11]

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2.2 Air Mass

Ground-based observations take into account “air mass”. The amount of air is not the same in every direction the observer looks to observe a celestial object. In the zenith the amount of air (air mass) is the least, while nearer to the horizon it increases and affects the measured flux. If the light travels a long distance inside the air before it is detected this means the measured flux would be low. Also the blue light is scattered more than the red light and the objects seem redder.

The air mass values are increasing significantly below 30° from the horizon and it is affected by changes in humidity, barometric pressure, clouds, haze and pollution. [11] Also these small altitudes must be avoided due to differential extinction (first and second order) across the frame, increased scintillation and differential refraction, which can make objects become miniature spectra and blue stars change positions with respect to red stars. [11]

First-order extinction is applied when there is a dimming in the light of a celestial object due to effect the earth‟s atmosphere has on it. Closer to the horizon more the extinction to the blue light in compare to the red light, due to the light scattering. It is measured in units of magnitudes or air mass. Second-order extinction is taking into account the colour of the target observed and the air mass in which it is measured. It is expressed in units of magnitudes, air mass or colour index.

2.3 Seeing

Another factor that has an effect to the accuracy of the analysis is the circumstances within the observational data were taken in. In an ideal situation a celestial object should be observed like a point-light source. However for ground-based observations this is not the case. “Seeing” is the factor that measures the atmospheric turbulences. When the different layers of air in earth‟s atmosphere have different values of temperature, the light does not follow a parallel path. In fact the light reaches the observer with a non-zero angle from the asteroid-observer line due to refraction. Each air layer has a different refraction index, resulting in an enlarged image of the celestial object, which occupies a larger number of pixels. Seeing is measured in arc-seconds and represents the size of the object‟s image at full 28 width at half maximum (FWHM). It usually ranges from 0.5 arc-seconds up to 3 arc-seconds and even greater values while precision decreases. [9]

2.4 Scintillation

“Scintillation” is causing the sparkling of light observed at the night sky. It is due to the fact that the light is reaching the observer not in a simple packet but in several. Long – time exposures eliminate this factor.

2.5 Opposition Effect

The “opposition effect” can help to determine asteroid‟s surface composition. As the asteroid is approaching opposition, that means low phase angles (the angle Sun-asteroid- Earth), a non-linear rise of about 0.3 to 0.5 magnitudes occurs [7]. This happens due to the way the particles in the asteroid‟s surface scatter the sunlight and it is independent of rotational variations. [8] Important factors are the roughness of the surface, the size, the shape and the porosity of the particles. Thus low albedo results in little “opposition effect”. However the brightness of an asteroid can be different from opposition to opposition due to different obliquity. This means that the angle between the asteroid‟s spin axis and the ecliptic plane is not always the same from one opposition to another. The larger the angle means more area of the asteroid‟s surface is lighted and the asteroid is brighter at that time.

29

Figure 3.3: Upper side – 1st data image, Lower side – 29th data image from the analysis that we conducted to asteroid (100926) 1998 MQ. The only difference in these two images is the position of a moving object, the asteroid of interest. Observational data were provided by Chrysa Avdellidou and Marco Delbò

2.6 Differential Photometry

For this diploma thesis our goal is to get an automatic way to track the asteroid-target and to find and track the comparisons stars in the star field of interest. In the APPENDIX we include all our PYTHON scripts that we used. For the analysis we used the “differential photometry” technique. In this method the magnitude of the asteroid – target is compared to the magnitude of one comparison star, or to the average value of many comparisons. Differential photometry is often used when very small variations in the magnitude of the asteroid-target are observed and the amplitude of the curve is less than 0.1 magnitudes. [11] Furthermore differential photometry allows using only the raw instrumental values and working with the differential values for the data analysis. 30

The field of view of a Charge- Coupled Device (CCD) camera is so small enough that the same comparison stars can be used for the same observation night. As a result the difference of the air-mass between the asteroid and the comparison stars is not taken into account. The same happens with the extinction values as long as the researcher selects comparison stars having the same colour with the asteroid-target. Thus it is helpful to avoid observing targets below 30° due to the higher values of extinction. [11]

The selection of the comparison star must be done with care. It must be almost of the same flux value as the asteroid or brighter. Also the researcher has to be reassured that the star selected is not a variable one. Plotting the flux or magnitude over time can eliminate this possibility. It is also useful to select as many comparison stars as possible for eliminating small errors done during the measurement of the flux. [11] This will smooth the average value resulting to better analysis. For the subtraction or correction of the following calibration frames we used the SOURCE EXTRACTOR (SE) program and SE and PYTHON routines by Chrysa Avdellidou (ESA).

Figure 3.4: Bias frame by the light-curve analysis that we conducted to asteroid (100926) 1998 MQ. Observational data were provided by Chrysa Avdellidou and Marco Delbò

2.7 Bias-Frame Subtraction

Bias frames are taken with a zero-second exposure time and with the dark shutter closed. This way the noise in the image due to the CCD‟s electronics is eliminated. More specifically it is necessary to eliminate the low spatial frequency variation in the amplifiers across the chip. Also a number of bias fames can be taken and then average or median combine them to form a master bias. In this way the readout noise that can be produced from the subtraction of a single bias frame is eliminated. [12] 31

Figure 3.5: Dark Frame by the photometry analysis that we conducted of (45) Eugenia’s stellar occultation on June 13 2014. Observational data provided by the scientific team of Observatoire de la Côte d’Azur.

2.8 Dark-Current Subtraction

Dark frames are taken with the same exposure time and temperature as the data image, with the shutter closed. It is also necessary to have the same binning. Their purpose is to eliminate the thermal noise. Dark currents may add a uniform and noisy background level to the data image. A number of dark frames can be taken and then average or median combine them to form a master dark. The dark frames include the bias frames. However due to the very low dark-count levels of modern CCDs the subtraction of dark frames is not always necessary. This is why their subtraction adds noise and cosmic rays that are present in them can also present in the data images. [12]

Figure 3.6: Flat Frame by the light-curve light-curve analysis that we conducted to asteroid (100926) 1998 MQ. Observational data were provided by Chrysa Avdellidou and Marco Delbò 32

2.9 Flat-Field Correction

Flat fields are taken mainly for two reasons. The first is to eliminate the differences between different pixels. These variations exist because each pixel is manufactured with different sensitivity from the average pixel. The second reason regards the response of both the telescope and the CCD camera. Flat fields can eliminate any dimming in light due to dust grains either in the chip or in the telescope. It is essential that the telescope is never pointing the same stars for two flat fields in the row. In this way the stars can be eliminated in the final image. Also the telescope must have the same focus as the data image, usually this is infinity. A number of flat fields can be taken and then average them or median combine them to form a master flat. Once done with the master flat it is essential not to make changes in the overall system, like changing the position of the camera. Otherwise the process must be repeated.

There are mainly two ways that a researcher can take the flat fields. The first is during the twilight pointing at a uniformly lighted part of the sky. The sky must be enough dark to avoid saturation and not having too many stars in the final image. The second way is pointing at an evenly illuminated portion of the inside of the dome.

Figure 3.7 : Left side – Non-calibrated data image from the analysis that we conducted of asteroid (100926) 1998 MQ. Right side: The same data image after the calibration. The noise has been eliminated and the uniformities both in the telescope and the CCD camera. Observational data were provided by Chrysa Avdellidou and Marco Delbò

33

2.9 Observations

The observational data for the analysis of asteroids (100926) 1998 MQ, (6974) Solti and (2012) Guo Shou-Jing are provided by Chrysa Avdellidou (ESA) and Marco Delbò (Observatoire de la Côte d‟Azur). Asteroid (100926) 1998 MQ was observed in the Observatoire de Haute Provence in France with the 1.2 m Newton telescope. On the other hand asteroids (6974) Solti and (2012) Guo Shou-Jing were observed in the Calar Alto Observatory in Spain with the 0.8 m Schmidt telescope.

Figure 3.8: Left side - The 1.2m Newton telescope in the Observatoire de Haute Provence in France. Credit: Ministère de la Culture, France.

Right side – The 0.8m Schmidt telescope in the Calar Alto Observatory in Spain. Credit: Calar Alto Observatory

34

35

Chapter 3

Occultations

3.1 What is a stellar occultation by an asteroid?

Figure 3.1: A stellar occultation by an asteroid is occurring when the asteroid passes in front of the observer’s-star line causing a drop in the star’s flux. Stellar occultations by asteroids can be used to determine the diameter of the asteroid. Models made by Chrysovalantis Sarakis.

The determination of asteroid‟s size and shape is based on observation of occultations of stars by asteroids. During a stellar occultation by an asteroid, the small body, meaning the asteroid, passes through the hypothetic line of the observer and the distant star. This basically make the asteroid seems like it approaches the star from one direction until it blocks its light and then continue its way from the other direction. [13] The blocking of the star‟s light can be partially or even total depending on the horizontal parallax. Thus many different observations are needed by different observers at different locations. 36

Figure 3.2: Prediction of the asteroid’s shadow path on Earth. (225) Henrietta occults

TYC 0035-00981-1 Credits : IOTA Asteroidal Occultation Results for North America

Figure 3.3: The positions of the observers in the asteroid’s shadow path. . (225) Henrietta occults

TYC 0035-00981-1 Credit: IOTA Asteroidal Occultation Results for North America

In more detail, by measuring the time interval of the stellar occultation, the observer can have the length of one chord across the asteroid. [13] Many chords can set limits to the dimensions of the asteroid. Thus a big number of observers equals to a better estimation of the asteroid‟s size. The observers must be situated in a perpendicular line to the asteroid‟s shadow path on Earth and in relatively large distances between them to avoid chords that are too close spaced. However of great importance are also the negative reports of observation, meaning that no drop in the star‟s flux is noted. These observations can work again as constraints to the asteroid‟s dimensions. It is worth noting that the calculation of the length of one chord is in linear units and not in angular. 37

Figure 3.4: The chords determined by a number of stellar occultations corresponded to the derived shape of the asteroid. Asteroid (206) Kleopatra. Credit: Database of Asteroid Models from Inversion Techniques

3.2 Why do we study Occultations?

For my diploma thesis our goals are the determination of the time interval that the occultation occurs and the determination of the flux‟s drop depth. Thus for the analysis we used the “differential photometry technique, but without moving to the calibration of the raw data in our part. However occultations of stars by asteroids are observed for various reasons. By measuring the asteroid‟s dimensions with such a direct method the scientists can derive the asteroid‟s diameter with a relatively accurate way and only using modest equipment. [13] This can also helps to better calibrate the observations of the same asteroid in radio wavelengths. [13] Another reason is that instead of observations made in different wavelengths, occultation observations provide a method with a very high accuracy on the determination of the asteroid‟s shape. [13] Occultation data can also reveal the roughness of an asteroid‟s surface. [13]

Furthermore occultation data can result in the computation of the asteroid‟s density. In many cases the mass of an asteroid can be calculated from the perturbations from the action of another body to them. If so with the derived diameters from occultation data the scientists can result to the density of this asteroid. Density, on the other hand can be used to make further suggestions about the composition and the origin of the asteroids [13] and the meteorites. 38

Occultation observations even gave the proof of the existence of asteroid‟s satellites. There have been already reported occultation observations in which observers that were situated outside the asteroid‟s shadow path have noted a drop in the star‟s flux indicating that the star has been occulted by another object and not the asteroid itself. [13] In this case occultation observations have the positive side that can be used to detect asteroid‟s satellites that are either too faint or too close to the asteroid by other methods. Another positive side is that those observations that are considered to be “negative observations”, in other words no flux drop was noticed can be used as constrains to the asteroid‟s size.

An approximate determination of the asteroid‟s diameter can be done with the following way:

The revolution speed of Earth can be calculated by the formula:

2휋푎퐸 푉퐸 = 푇퐸 where 푎퐸 is its semi-major axis and 푇퐸 its revolution period.

A correction by the addition of the Earth‟s rotational speed in its revolution speed can be done by applying the formula:

푟 2휋푟퐸 푉퐸 = 푟 푇퐸

푟 where 푟퐸 is the Earth‟s radius and 푇퐸 its rotation period.

The revolution speed of the asteroid can be calculated by the formula:

2 1 푉퐴 = 퐺푀 − 푟퐴 푎퐴

where 푟퐴 is the distance from the Sun and 푎퐴 its semi-major axis.

From the geometry of the models above that depicts a stellar occultation the law of cosines can be applied to measure the angle 휑:

푟2 + 푟2 − 푟2 푐표푠휑 = 퐸푆 퐴푆 퐴퐸 2푟퐸푆푟퐴푆 where 푟퐸푆 is the Earth‟s-Sun distance, 푟퐴푆 is the asteroid‟s-Sun distance and 푟퐴퐸 is the asteroid‟s-Earth distance. 39

A correction by deriving the dot product can be done for the 3D space by applying the formula:

푉 ∙ 푉 푐표푠휑 = 퐸 퐴 푉퐸 푉퐴

The asteroid‟s relative projected speed to Earth can be calculated by the formula:

푟푒푙 푉퐴 = 푉퐸 − 푉퐴푐표푠휑

Finally the approximate asteroid‟s dimension can be calculated by the formula:

푟푒푙 푙퐴 = 푉퐴 푡표푐푐 where 푡표푐푐 is the time interval of the stellar occultation by the asteroid.

3.3 Observations

The observational data for the analysis of stellar occultations by asteroid (45) Eugenia, (102) Miriam and (3332) Raksha are provided by the scientific team of Observatoire de la Côte d‟Azur in France and by Detlef Koschny (ESA) respectively. The stellar occultation by asteroid (45) Eugenia was observed in the Observatoire de la Côte d‟ Azur in Calern (France). On the other hand stellar occultations by asteroids (102) Miriam and (3332) Raksha were observed in the Kryoneri Observatory in Greece with the 1.2m Cassegrain telescope.

Figure 3.5: The 1.2m Cassegrain telescope in the Kryoneri Observatory in Greece. Credit: Theofanis Matsopoulos 40

41

Chapter 4

Results

4.1 Light-curve Analysis

4.1.1 Asteroid (100926) 1998 MQ

According to the JPL Small-Body Database Browser [9] :

Classification: Amor [NEO]

SPK-ID: 2100926

Epoch: 2458000.5 (Sep-Sep-04.0)

Table 4. 1: The orbital elements of asteroid (100926) 1998 MQ

Element Value Uncertainty (1-sigma) Units e 0.4076394040984813 3.3977e-08 -

a 1.782854028134288 4.5332e-09 au i 24.24416633151929 8.4548e-06 Deg peri 138.8003503283143 1.0751e-05 Deg node 221.1141608758828 9.9613e-06 Deg M 333.1207676200772 5.1461e-06 Deg q 1.05609247451105 6.0752e-08 au Q 2.509615581757526 6.3811e-09 au Table 4. 2: The already found values of asteroid’s (100926) 1998 MQ physical properties

Parameter Value Uncertainty (1-sigma) Units Diameter 1.174 0.135 km Rotation Period 2.328 n/a h NOTE: Result based on less than full coverage, so that the period may be wrong by 30 percent or so

42

According to the Asteroid Lightcurve Photometry Database [14] :

Table 4. 3: The already found range of the light-curve’s amplitude for asteroid (100926) 1998 MQ

Parameter Minimum Value Maximum Value Amplitude 0.11 0.12

The photometry data were divided in four observational nights:

Table 4. 4: The observational photometry data of asteroid (100926) 1998 MQ

Date of Observation Number of Images Exposure Time (sec) (/Month/Year) 22/05/2017 57 90 23/05/2017 55 120 24/05/2017 45 150 26/05/2017 13 180 10 200

For the bias-frame subtraction and the flat field correction were used:

Table 4. 5: The calibration frames of asteroid’s (100926) 1998 MQ observational photometry data analysis

Date of Observation Number of Bias-Frames Number of Flat-Fields (Day/Month/Year) 22/05/2017 11 11 23/05/2017 21 11 24/05/2017 11 11 26/05/2017 11 11

A number of comparison stars were selected for each observation night:

Table 4. 6: The number of comparison stars of asteroid’s (100926) 1998 MQ differential photometry data analysis

Date of Observation Number of Comparison Stars (Day/Month/Year) 22/05/2017 7 23/05/2017 3 24/05/2017 3 26/05/2017 4 5 43

Figure 4.1: Light-curve of asteroid (100926) 1998 MQ we derived from its differential photometry data analysis

Table 4. 7: The rotation period and the light-curve’s amplitude for asteroid (100926) 1998 MQ that we derived from its differential photometry data analysis

Parameter Value Uncertainty (1-sigma) Rotation Period 2 h 17 min 57.1 s 1.7 s Amplitude 0.229 mag 0.017 mag 44

Figure 4.2: Light-curve of asteroid (100926) 1998 MQ on May 22 2017

Figure 4.3: Light-curve of asteroid (100926) 1998 MQ on May 23 2017

45

Figure 4.4: Light-curve of asteroid (100926) 1998 MQ on May 24 2017

Figure 4.5: Light-curve of asteroid (100926) 1998 MQ on May 26 2017

46

47

4.1.2 Asteroid (6974) Solti

According to the JPL Small-Body Database Browser [9] :

Classification: Main-belt Asteroid

SPK-ID: 2006974

Epoch: 2458000.5 (2017-Sep-04.0)

Alternate Designations: 1992 MC = 1941 HE = 1959 RF = 1980 RZ4 = 1995 DO

Table 4. 8: The orbital elements of asteroid (6974) Solti

Element Value Uncertainty (1-sigma) Units e 0.1412126970696198 4.4568e-08 - a 2.599378136344619 1.3321e-08 au i 15.79437334992312 4.6773e-06 Deg peri 202.7066453837908 2.5274e-05 Deg node 166.0112833225569 1.8575e-05 Deg M 281.5730049103065 1.8854e-05 Deg q 2.232312939007594 1.1362e-07 au Q 2.966443333681644 1.5203e-08 au

Table 4. 9: The already found values of asteroid’s (6974) Solti physical properties

Parameter Value Uncertainty (1-sigma) Units Diameter 9.818 0.607 km Rotation Period 2.423 n/a h

According to the Asteroid Lightcurve Photometry Database [14] :

Table 4. 10: The already found range of the light-curve’s amplitude for asteroid (6974) Solti

Parameter Minimum Value Maximum Value Amplitude - 0.16 The photometry data were divided in two observational nights: 48

Table 4. 11: The observational photometry data of asteroid (6974) Solti

Date of Observation Number of Images Exposure Time (sec) (Day/Month/Year) 29/07/2017 39 120 30/07/2017 60 120

For the bias-frame subtraction and the flat field correction were used:

Table 4. 12: The calibration frames of asteroid’s (6974) Solti observational photometry data analysis

Date of Observation Number of Bias-Frames Number of Flat-Fields (Day/Month/Year) 29/07/2017 9 3 30/07/2017 9 3

A number of comparison stars were selected for each observation night:

Table 4. 13: The number of comparison stars of asteroid’s (6974) Solti differential photometry data analysis

Date of Observation Number of Comparison Stars (Day/Month/Year) 29/07/2017 3 30/07/2017 3

49

Figure 4.6: Light-curve of asteroid (6974) Solti we derived from its differential photometry data analysis

Table 4. 14: The rotation period and the light-curve’s amplitude for asteroid (6974) Solti that we derived from its differential photometry data analysis

Parameter Value Uncertainty (1-sigma) Rotation Period 2 h 18 min 1.8 s 5.4 s Amplitude 0.186 mag 0.008 mag

50

Figure 4.7: Light-curve of asteroid (6974) Solti on July 29 2017

Figure 4.8: Light-curve of asteroid (6974) Solti on July 30 2017 51

4.1.3 Asteroid (2012) Guo Shou-Jing

According to the JPL Small-Body Database Browser [9] :

Classification: Main-belt Asteroid

SPK-ID: 2002012

Epoch: 2458000.5 (2017-Sep-04.0)

Alternate Designations: 1964 TE2 = 1971 SF1 = 1974 MS

Table 4. 15: The orbital elements of asteroid (2012) Guo Shou-Jing

Element Value Uncertainty (1-sigma) Units e 0. 1781821218594249 3.9226e-08 -

a 2.328646680900046 8.6013e-09 au i 2.906634549089457 4.1522e-06 Deg peri 36.69555237146083 8.0852e-05 Deg node 277.1072254579385 8.0091e-05 Deg M 6.725234977065448 1.3189e-05 Deg

q 1.913723474236368 9.1988e-08 au Q 2.743569887563723 1.0134e-08 au

Table 4. 16: The already found values of asteroid’s (2012) Guo Shou-Jing physical properties

Parameter Value Uncertainty (1-sigma) Units Diameter 12.248 0.035 km Rotation Period 12.000 n/a h NOTE: lower-limit; Result based on fragmentary light-curve(s), may be completely wrong

52

According to the Asteroid Lightcurve Photometry Database [14] :

Table 4. 17: The already found range of the light-curve’s amplitude for asteroid (2012) Guo Shou-Jing

Parameter Minimum Value Maximum Value Amplitude - 0.05

The photometry data were divided in two observational nights:

Table 4. 18: The observational photometry data of asteroid (2012) Guo Shou-Jing

Date of Observation Number of Images Exposure Time (sec) (Day/Month/Year) 21/07/2017 113 90 29/07/2017 88 120

For the bias-frame subtraction and the flat field correction were used:

Table 4. 19: The calibration frames of asteroid’s (2012) Guo Shou-Jing observational photometry data analysis

Date of Observation Number of Bias-Frames Number of Flat-Fields (Day/Month/Year) 21/07/2017 9 3 29/07/2017 9 3

A number of comparison stars were selected for each observation night:

Table 4. 20: The number of comparison stars of asteroid’s (2012) Guo Shou-Jing differential photometry data analysis

Date of Observation Number of Comparison Stars (Day/Month/Year)

21/07/2017 3

29/07/2017 4 53

Figure 4.1: Light-curve of asteroid (2012) Guo Shou-Jing on July 21 2017 54

Figure 4.2: Light-curve of asteroid (2012) Guo Shou-Jing on July 29 2017

Table 4. 21: The rotation period and the light-curve’s amplitude for asteroid (2012) Guo Shou- Jing that we derived from its differential photometry data analysis

Parameter Value Rotation Period Undefined Amplitude 0.07 mag

55

4.2 Analysis of Stellar Occultations by Asteroids Data

4.2.1 Occultation by (45) Eugenia, June 13 2014

Asteroid (45) Eugenia

According to the JPL Small-Body Database Browser [9] :

Classification: Main-belt Asteroid

SPK-ID: 2000045

Epoch: 2458000.5 (2017-Sep-04.0)

Alternate Designations: 1941 BN

Table 4. 22: The orbital elements of asteroid (45) Eugenia

Element Value Uncertainty (1-sigma) Units e 0.8399506498648018 3.5893e-08 - a 2.719553001321756 6.5125e-09 au i 6.60400093601017 3.5543e-06 Deg peri 88.75662002024241 4.0604e-05 Deg node 147.6713443442752 3.4598e-05 Deg M 265.975038609184 2.1887e-05 Deg q 2.491123970241558 9.851e-08 au

Q 2.947982032401954 7.0595e-09 au

Table 4. 23: The already found values of asteroid’s (45) Eugenia physical properties

Parameter Value Uncertainty (1-sigma) Units Diameter 202.327 2.168 km Rotation Period 5.699 n/a h

56

According to the Centre Pedagogique Planete et Univers [15] :

Table 4. 24: The already found values of asteroid’s (45) Eugenia occultation’s duration and depth

Parameter Value Uncertainty (1-sigma) Units Duration 17.71 0.12 s Depth 0.40 n/a mag

Table 4. 25: The observation photometry data of asteroid’s (45) Eugenia occultation

Number of Images Exposure Time (sec) Cycle Time (sec) 16380 0.02000 0.02175

Table 4. 26: : The number of comparison stars of asteroid’s (45) Eugenia occultation differential photometry data analysis

Date of Observation (Day/Month/Year) Number of Comparison Stars 13/06/2014 1 57

Figure 4.3: Occultation of 3UCAC163-139142 by asteroid (45) Eugenia on June 13 2017 light- curve we derived from its differential photometry data analysis

Table 4. 27: The duration and the depth for asteroid’s (45) Eugenia occultation that we derived from its differential photometry data analysis

Parameter Value Uncertainty (1-sigma) Units Duration 17.661 0.392 s Depth 0.39 n/a mag 58

59

4.2.2 Occultation by (102) Miriam, March 03 2017

Asteroid (102) Miriam

According to the JPL Small-Body Database Browser [9] :

Classification: Main-belt Asteroid

SPK-ID: 2000102

Epoch: 2458000.5 (2017-Sep-04.0)

Alternate Designations: 1944 FC = 1972 PC

Table 4. 28: The orbital elements of asteroid (102) Miriam

Element Value Uncertainty (1-sigma) Units e 0.252890397471242 3.1598e-08 -

a 2.660303102419047 7.8584e-09 au i 5.178530482611002 3.3105e-06 Deg peri 147.1918649213123 4.0929e-05 Deg node 210.8570244562909 4.0035e-05 Deg M 114.8153533744478 1.0285e-05 Deg

q 1.987537993454316 8.1264e-08 au

Q 3.333068211383778 9.8457e-09 au

Table 4. 29: The already found values of asteroid’s (102) Miriam physical properties

Parameter Value Uncertainty (1-sigma) Units Diameter 82.595 0.400 km Rotation Period 23.613 n/a h

60

Table 4. 30: The observation photometry data of asteroid’s (102) Miriam occultation

Number of Images Exposure Time (sec) Cycle Time (sec) Filter 1384 0.300005 0.300106 I

1431 0.300005 0.300106 R

Table 4. 31: The number of comparison stars of asteroid’s (102) Miriam occultation differential photometry data analysis

Date of Observation (Day/Month/Year) Number of Comparison Stars 03/03/2017 3

Figure 4.4: Occultation by asteroid (102) Miriam on March 03 2017 light-curve on the Infrared wavelength band we derived from its differential photometry data analysis. The analysis did not result in its observation. 61

Figure 4.5: Occultation by asteroid (102) Miriam on March 03 2017 light-curve on the Red wavelength band we derived from its differential photometry data analysis. The analysis did not result in its observation.

Table 4. 32: The duration and the depth for asteroid’s (102) Miriam occultation that we derived from its differential photometry data analysis

Parameter Value Duration Undefined Depth Undefined

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63

4.2.3 Occultation by (3332) Raksha, March 03 2017

Asteroid (3332) Raksha

According to the JPL Small-Body Database Browser [9] :

Classification: Main-belt Asteroid

SPK-ID: 2003332

Epoch: 2458000.5 (2017-Sep-04.0)

Alternate Designations: 1978 NT1 = 1936 FT = 1950 TC4 = 1952 CB = 1962 TH = 1970 PP = 1974 OR = 1978 RF1

Table 4. 33: The orbital elements of asteroid (3332) Raksha

Element Value Uncertainty (1-sigma) Units e 0.08443583636362131 4.0475e-08 -

a 2.545864036324445 1.3448e-08 au i 14.84568745931939 5.188e-06 Deg peri 277.2753145783832 3.5444e-05 Deg node 138.7638104239246 1.6168e-05 Deg M 128.9995774079368 3.2695e-05 Deg q 2.330901877149326 1.0009e-07 au Q 2.760826195499564 1.4584e-08 au

Table 4. 34: The already found values of asteroid’s (3332) Raksha physical properties

Parameter Value Uncertainty (1-sigma) Units Diameter 14.581 0.202 km Rotation Period 4.8065 n/a h

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Table 4. 35: The observation photometry data of asteroid’s (3332) Raksha occultation

Number of Images Exposure Time (sec) Cycle Time (sec) Filter 1179 0.400005 0.400105 I

1193 0.400005 0.400105 R

Table 4. 36: The number of comparison stars of asteroid’s (3332) Raksha occultation differential photometry data analysis

Date of Observation (Day/Month/Year) Number of Comparison Stars 03/03/2017 3

Figure 4.6: Occultation by asteroid (3332) Raksha on March 03 2017 light-curve on the Infrared wavelength band we derived from its differential photometry data analysis. The analysis did not result in its observation. 65

Figure 4.7: Occultation by asteroid (3332) Raksha on March 03 2017 light-curve on the Red wavelength band we derived from its differential photometry data analysis. The analysis did not result in its observation.

Table 4. 37: The duration and the depth for asteroid’s (3332) Raksha occultation that we derived from its differential photometry data analysis

Parameter Value Duration Undefined Depth Undefined

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67

Chapter 5

Conclusions

We succeed to get an automatic way to track the asteroid and to find and track the comparison stars within the star field of interest.

Also our attempt to conduct the differential photometry analysis proved correct as it seems below.

After the light-curve study of Near-Earth asteroid (100926) 1998 MQ the value of its rotation period that we computed has almost the same value with the already found. This can be an indication that the differential photometry analysis technique was correctly conducted. However a greater value of the light-curve‟s amplitude of about 0.10 magnitudes was also found for the same asteroid.

The analysis of the light-curve of the main belt asteroid (6974) Solti resulted in a smaller value of about 7 min 20 sec of its rotation period and in a greater value of 0.026 mag of its amplitude in compare to the already found values.

The study of the light-curve of the main belt asteroid (2012) Guo Shou-Jing included only two observation nights of about 3 observation hours each. However this was not enough for the computation of its rotation period. The second day analysis resulted to a value of the light-curve‟s amplitude of about 0.07 mag, which however is greater than the already found value.

The analysis of the stellar occultation by main belt asteroid (45) Eugenia on June 13th 2014 resulted to values of its duration and its relative flux depth very similar to these that have already been found. This is an indication that the analysis technique of differential photometry was correctly conducted.

The analysis of the occultation by main belt asteroid (102) Miriam on March 3rd 2017 did not result in the observation of a significant drop of flux that could be considered as an occultation. The drop of flux should be present at the same time both in the Infrared and Red wavelength bands which did not occur. 68

The analysis of the occultation by main belt asteroid (3332) Raksha on March 3rd 2017 did not also result in the observation of a significant drop of flux that could be considered as an occultation. The drop of flux should be present at the same time both in the Infrared and Red wavelength bands which did not occur.

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Appendix

Below there are the Python scripts that we used for the differential photometry data analysis.

Asteroid (100926) 1998 MQ

Attempt to track the asteroid

##Asteroid (100926) 1998 MQ analysis 22/05/2017 ##Attempt to track the asteroid by CHRYSOVALANTIS SARAKIS ##Take in notice that the telescope is re-pointed during the observation night ##The star field is changing dividing the images in sets ##During each set the star field is relatively stable ##It is necessary to see manually how the coordinates of the asteroid are changing and then with the script apply it to all ##Notice in which images the coordinates are taking their maximum and minimum value for each axis ##For these images note manually from the txt files the value of x or y and the value of the flux ##The script searches within a rectangle ##In the two axes the rectangle is limited between the minimum and maximum values ##For the flux set a range that can include the asteroid in all the images but not other stars ##It tracks the asteroid ##lowest x: fr15 ,highest x: fr14 ##lowest y: fr60 ,highest y: fr15 ##from the analysis are excluded : fr 20, 21, 22 ,23 ,24 (remember) import pyfits as pf import numpy as np from numpy import * import glob from math import sqrt

print '~~~~~~~Asteroid (100926) 1998 MQ analysis 22052017~~~~~~' print 'Attempt to track the asteroid by CHRYSOVALANTIS SARAKIS'

##set the x coordinates x1 = 384.6123 x1f = 508.0984

##set the y coordinates y1 = 450.3108 y1f = 559.7350

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##set the range of the flux f1 = 16500. f2 = 31000.

##get the lists list_f1 = []

##get the txt files txtlist = glob.glob('r_100926*.txt') txtlist = sort(txtlist) for txtname in (txtlist) : txtfile = open(txtname) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] f = txtdata[j,5] ##the tracking starts if x1<=x<=x1f and y1<=y<=y1f and f1<=f<=f2 : list_f1.append(txtdata[j,0:7]) array_f1 = array(list_f1) np.savetxt('asteroid_100926_22052017.txt', array_f1, fmt='%f') print '~~~~~~~~~~~~~~~~~~~~~eof~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'

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Attempt to track the asteroid by implying predictions of its position

##Asteroid (100926) 1998 MQ analysis 22/05/2017 ##Attempt to track the asteroid by CHRYSOVALANTIS SARAKIS ##The script is taking from the headers the exposure time and the Julian Date of the images ##It converts the exposure time to units of julian date ##Set a maximum value of "readout time" that will be added to the exposure time ##It also computes the time difference between two images in a row in units of julian date ##Take in notice that the telescope is re-pointed during the observation night ##The star field is changing dividing the images in sets ##During each set the star field is relatively stable ##If the time difference is greater than the exposure time plus the readout time then it means that the telescope was repointed and the star field has changed, thus it is not stable anymore ##It prints the sets of images where the star field has changed ##It is neccessary to find the asteroid in these images manually and note its coordinates from the txt files ##It asks the coordinates of the first and last image of each set ##It divides the two point distance in each axis with the number of images to predict the position of the asteroid ##It makes a small circle around the predicted position ##Set the radius of the small circle ##It tracks the asteroid ##from the analysis are excluded : fr 20, 21, 22 ,23 ,24 (remember) import math from math import sqrt import numpy as np import pyfits as pf import glob from numpy import *

print '~~~~~~Asteroid (100926) 1998 MQ analysis 22052017~~~~~~~' print 'Attempt to track the asteroid by CHRYSOVALANTIS SARAKIS'

##get the lists astx = [] asty = [] list_xi = [] list_yi = [] ##get the txt files open files = glob.glob('r_*_R.txt') files = sort(files) #print 'files=', files for fl in files : filename = open(fl) data = np.loadtxt(filename) ##get the fit images open fits = glob.glob('r_*_R.fit') 72 fits = sort(fits) list_n = [] ##get the exposure time and julian date from headers for ft in fits : fitname = pf.open(ft) z = fitname[0].header['EXPTIME'] w = fitname[0].header['JD'] ##convert the exposure time to julian date units if z == 60.0 : z = 0.000694 elif z == 90.0 : z = 0.001042 elif z == 120.0 : z = 0.001389 elif z == 180.0 : z = 0.002083 else : print 'warning : Problem with the exposure time' list_n.append(z) list_n.append(w) n = array(list_n) n = reshape(n,(60,2)) ##get the lists list_t = [] list_m = [] m = 0 for m in range(0,len(n)-1) : t1 = n[m,1] ##add to the exposure time a "readout time" in jd units z_dt = n[m,0] + 0.00005 m+= 1 t2 = n[m,1] ##get the difference in julian date of two images dt = t2 - t1 ##compare the difference with the the exposure+readout if dt <= z_dt : print 'ok' 'for m=', fits[m] , 'dt=', dt else : print 'warning : The star field changed in fit:', fits[m] , 'dt=', dt list_m.append(m) ##get the first and the last image in the list list_m.insert(0,0) list_m.append(len(fits)) ##get the ste of images where the star field has changed print 'list_m=', list_m ##set the coordinates for each set of images for k in range(0, len(list_m)-1) : ##get the first image of the set print 'Give coords for fit :' , fits[list_m[k]] xi = float(raw_input('x initial:')) yi = float(raw_input('y initial:')) k+= 1 ##get the last image of the set print 'Give coords for fit :' , fits[list_m[k]-1] xf = float(raw_input('x final:')) 73

#print 'xf=', xf yf = float(raw_input('y final:')) dx = (xf - xi) dy = (yf - yi) a = (yi - yf) / (xi - xf) b = (yf*xi - yi*xf) / (xi - xf) for fl in range(list_m[k-1],list_m[k]) : fl = files[fl] filename = open(fl) data = np.loadtxt(filename) print 'fl =' , fl ##divide the distance in each axis with the number of images of the set lx = dx/(list_m[k] - (list_m[k-1]+1)) ly = dy/(list_m[k] - (list_m[k-1]+1)) ##get the coordinates of the predicted position xi+= lx #sqrt(lx**2 + ly**2) yi+= ly print 'xi=', xi, 'yi=', yi list_xi.append(xi) list_yi.append(yi) finalx = [] finaly = [] for q in range(0,len(data)) : x = data[q,1] y = data[q,2] ##get a small circle around the predicted position rad = sqrt((x - xi)**2 + (y - yi)**2) ##set the radius of the small circle ##the tracking starts if rad <= 7. : finalx.append(x) astx.append(x) finaly.append(y) asty.append(y) predx = array(list_xi) predy = array(list_yi) asterx = array(astx) astery = array(asty) np.savetxt('asteroid_22052017_pred.txt', c_[predx,predy], fmt='%f') np.savetxt('asteroid_22052017_aster.txt', c_[asterx,astery], fmt='%f') print '~~~~~~~~~~~~~~~~~~~~~eof~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'

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75

Attempt to FIND the comparison stars

##Asteroid (100926) 1998 MQ analysis 22/05/2017 ##Attempt to find the comparison stars by CHRYSOVALANTIS SARAKIS ##The script divides the first image of the observation in 4 even quarters ##It is necessary to note the flux of the asteroid manually from the txt of the first image ##Set a flux range for the search to be done ##It finds the comparison stars ##Iake in notice that the telescope is re-pointed during the observation night ##The star field is changing dividing the images in sets ##During each set the star field is relatively stable ##It is necessary to exclude manually in the end from the found comparison stars those that are not going to be in all the images ##lenght of axes : 1024 x 1024 ##first image of the observation night : fr04 ##asteroid's flux in the first image (counts): 23077.05

import pyfits as pf import numpy as np from numpy import * import glob from math import sqrt print '~~~~~~~~Asteroid (100926) 1998 MQ analysis 22052017~~~~~~~~~~' print 'Attempt to find the comparison stars by CHRYSOVALANTIS SARAKIS'

##set the 4 quarters x2 = 512. x3 = 512. x4 = 1024. x5 = 1024. x2f = 0. x3f = 0. x4f = 512. x5f = 512. y2 = 0. y3 = 512. y4 = 0. y5 = 512. y2f = 512. y3f = 1024. y4f = 512. y5f = 1024.

##get the lists list_f2 = [] list_f3 = [] list_f4 = [] 76 list_f5 = []

##get the flux range f1 = 20000.0 f2 = 40000.0

##get the txt files txtlist = glob.glob('r_100926-004_R.txt') txtlist = sort(txtlist) for txtname in (txtlist) : txtfile = open(txtname) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] f = txtdata[j,5] ##the searching starts if x>=x2f and x<=x2 and y>=y2 and y<=y2f and f1<=f<=f2: list_f2.append(txtdata[j,0:7]) elif x>=x3f and x<=x3 and y>=y3 and y<=y3f and f1<=f<=f2 : list_f3.append(txtdata[j,0:7]) elif x>=x4f and x<=x4 and y>=y4 and y<=y4f and f1<=f<=f2 : list_f4.append(txtdata[j,0:7]) elif x>=x5f and x<=x5 and y>=y5 and y<=y5f and f1<=f<=f2 : list_f5.append(txtdata[j,0:7]) array_f2 = array(list_f2) array_f3 = array(list_f3) array_f4 = array(list_f4) array_f5 = array(list_f5) np.savetxt('quarter1_100926_22052017.txt', array_f2, fmt='%f') np.savetxt('quarter2_100926_22052017.txt', array_f3, fmt='%f') np.savetxt('quarter3_100926_22052017.txt', array_f4, fmt='%f') np.savetxt('quarter4_100926_22052017.txt', array_f5, fmt='%f') print '~~~~~~~~~~~~~~~~~~~~~~~~~~eof~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'

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Attempt to TRACK the comparison stars

##Asteroid (100926) 1998 MQ analysis 22/05/2017 ##Attempt to track the comparison stars by CHRYSOVALANTIS SARAKIS ##Take in notice that the telescope is re-pointed during the observation night ##The star field is changing dividing the images in sets ##During each set the star field is relatively stable ##It is necessary to see manually how the coordinates of a star are changing and then with the script apply it to all ##Note when the coordinates are significantly changing in each axis (not always in each set) ##The script is taking these values (must be noted manually from the txt files) ##It seaches within a small rectangle from the given coordinate ##The rectangles can be adjusted nanually in order to include the comparison star in every image and not other stars ##It tracks the comparison stars ##1st x: fr04 ,2nd x: fr15 ,3rd x: fr43 ##1st y: fr04, 2nd y: fr15 or fr43 ##from the analysis are excluded : fr 20, 21, 22 ,23 ,24 (remember) import pyfits as pf import numpy as np from numpy import * import glob from math import sqrt print '~~~~~~~~~Asteroid (100926) 1998 MQ analysis 22052017~~~~~~~~~~' print 'Attempt to track the comparison stars by CHRYSOVALANTIS SARAKIS' ##get the txt files txtlist = glob.glob('r_100926*.txt') txtlist = sort(txtlist) ##get the nunber of comparison stars for i in range(1,10) : list_f1 = [] print'Asteras%s'%i ##get the different values for the axis x1 = float(raw_input('1st x:')) x2 = float(raw_input('2nd x:')) x3 = float(raw_input('3rd x:')) ##get the different values for the y axis y1 = float(raw_input('1st y:')) y2 = float(raw_input('2nd y:')) ##get the small rectangles a1=x1-1.5 b1=x1+1.5 a2=x2-1.5 b2=x2+1.5 a3=x3-1.5 b3=x3+1.5 c1=y1-1.5 d1=y1+1.5 c2=y2-2.7 78

d2=y2+2.7 for txtname in (txtlist) : txtfile = open(txtname) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] ##the tracking starts if a1<=x<=b1 and c1<=y<=d1 : list_f1.append(txtdata[j,0:7]) elif a2<=x<=b2 and c2<=y<=d2 : list_f1.append(txtdata[j,0:7]) elif a3<=x<=b3 and c2<=y<=d2 : list_f1.append(txtdata[j,0:7]) array_f1 = array(list_f1) np.savetxt('asteras%s_100926_22052017.txt'%i, array_f1, fmt='%f') print '~~~~~~~~~~~~~~~~~~~~~~~~~eof~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'

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Asteroid (2012) Guo Shou-Jing

Attempt to TRACK the comparison stars

##Asteroid (2012) Guo Shou-Jing analysis 21/07/2017 ##Attempt to track the comparison stars by CHRYSOVALANTIS SARAKIS ##Take in notice that the telescope is re-pointed during the observation night ##The star field is changing dividing the images in sets ##During each set the star field is relatively unstable meaning that the stars are not in fixed positions ##It is necessary to see manually how the coordinates of a star are changing and then with the script apply it to all ##If the change of the coordinates between some sets is not significant then combine them in one set ##After that note where the coordinates are taking their minimum and maximum values for each axis ##The script is searching the comparison star within a rectangle ##The rectangle is limited between the minimum and maximum values of x and y and of the flux ##There is the possibility that one of the coordinates does not change significantly within a set but the other does ##Then it searches within a smaller rectangle for the specific axis ##The smaller rectangle can be adjusted manually in order to include the comparison star in all the images ##The flux range can be adjusted manually in order to include the comparison star in all the images but not other stars ##It tracks the comparison stars ##first set - smaller rectangle: 1st x : fr1.00, 1st y : fr1.00, 2nd y : fr1.13 ##second set - rectangle : 2nd x (lowest) : fr2.99, 3rd x (highest) : fr2.00, 3rd y (lowest) : fr2.00, 4th y (highest) : fr2.56 ##1st f (lowest): fr2_26 ,2nd f (highest) : fr1_13 ##in the flux range add 25000.0 to the highest value

import pyfits as pf import numpy as np from numpy import * import glob from math import sqrt

print '~~~~~~~~Asteroid (2012) Guo Shou-Jing analysis 21072017~~~~~~~~~' print 'Attempt to track the comparison stars by CHRYSOVALANTIS SARAKIS'

##get the number of the comparison stars for k in range(1,6) : list_f1 = [] print'Asteras%s'%k ##get the x coordinates 80 x1 = float(raw_input('1st x:')) x2 = float(raw_input('2nd x:')) x3 = float(raw_input('3rd x:')) ##get the y coordinates y1 = float(raw_input('1st y:')) y2 = float(raw_input('2nd y:')) y3 = float(raw_input('3rd y:')) y4 = float(raw_input('4th y:')) ##get the flux values f1 = float(raw_input('1st f:')) f2 = float(raw_input('2nd f:')) ##set the smaller rectangle a1=x1-2.5 b1=x1+2.5 ##get the first ten images of the first set for i in range(0,10) : print i txtfile = open('r_a2012_1_0%s.txt'%i) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] f = txtdata[j,5] if a1<=x<=b1 and y1<=y<=y2 and f1<=f<=f2 : list_f1.append(txtdata[j,0:7]) ##get the rest of the images of the first set for i in range(10,14) : print i txtfile = open('r_a2012_1_%s.txt'%i) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] f = txtdata[j,5] if a1<=x<=b1 and y1<=y<=y2 and f1<=f<=f2 : list_f1.append(txtdata[j,0:7]) ##get the first ten images of the second set for i in range(0,10) : print i txtfile = open('r_a2012_2_0%s.txt'%i) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] f = txtdata[j,5] if x2<=x<=x3 and y3<=y<=y4 and f1<=f<=f2 : list_f1.append(txtdata[j,0:7]) ##get the rest of the images of the second set for i in range(10,100) : print i txtfile = open('r_a2012_2_%s.txt'%i) txtdata = np.loadtxt(txtfile) for j in range(len(txtdata)) : x = txtdata[j,1] y = txtdata[j,2] 81 f = txtdata[j,5] if x2<=x<=x3 and y3<=y<=y4 and f1<=f<=f2 : list_f1.append(txtdata[j,0:7]) array_f1 = array(list_f1) np.savetxt('asteras%s_2012_21072017.txt'%k, array_f1, fmt='%f') print '~~~~~~~~~~~~~~~~~~~~~eof~~~~~~~~~~~~~~~~~~~~~~'

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83

Asteroid (45) Eugenia

Attempt to conduct the differential photometry analysis of asteroid’s (45) Eugenia occultation

#LC relativeflux(mag)-time(sec) (45)Eugenia occultation analysis 13/06/2014 X4 #Convert flux to magnitudes #exposure time: 0.02 s #cycle time: 0.02175 s # import math from math import log10 from math import log from math import sqrt import numpy as np import pylab as pl from numpy import *

#get the txt files opened f1st13 = open('comparisonsasteras1_occultation45Eugenia_13062014_X4.txt') focst13 = open('occultedstar_occultation45Eugenia_13062014_X4.txt') ft13 = open('time_occultation45Eugenia_13062014_X4.txt')

#get the txt data loaded d1st13 = np.loadtxt(f1st13) docst13 = np.loadtxt(focst13) dt13 = np.loadtxt(ft13)

#get the lists dmlist13 = [] dmerrlist13 = [] tlist13 = [] #get the loop for each row of the txt data for i in range(len(d1st13)) : #get the time (hours) t = dt13[i,0] #get them in list tlist13.append(t) print 't=', t #get the flux (exclude the exposure time) f = docst13[i,5] f1 = d1st13[i,5] ferr = docst13[i,6] ferr1 = d1st13[i,6] #get the exposure time (sec) expt = 0.02 #get the magnitude (mag=-2.5*log10(flux/exp_time_sec)) 84

m = -2.5 * log10(f/expt) m1 = -2.5 * log10(f1/expt) #get the magnitude error (merr=-(2.5/ln10)*(flux_error/flux)) merr = -2.5 * ferr/(log(10.) * f) merr1 = -2.5 * ferr1/(log(10.) * f1) #get them in one list mlist = [m1] #get the average av = sum(mlist)/len(mlist) #get the average eror averr = (sqrt(merr1**2.))/1. #get the difference in magnitudes dm = abs(m - av) #get them in list dmlist13.append(dm) #get the difference error dmerr = sqrt(merr**2.+averr**2.) #get them in list dmerrlist13.append(dmerr) #get them in array dm13 = array(dmlist13) dmerr13 = array(dmerrlist13)

#save them in txt np.savetxt('data_occultation45Eugenia_13062014_X4.txt', c_[dm13,dmerr13], fmt='%f') print '~~~~~~~~~~eof_13062014~~~~~~~~~~~~~~'

#get the txt files closed f1st13.close() focst13.close() ft13.close()

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