A SURVEY STRATEGY FOR LIGHT ECHOES
FROM HISTORICAL SUPERNOVAE IN THE MILKY WAY A SURVEY STRATEGY FOR LIGHT ECHOES FROM
HISTORICAL SUPERNOVAE IN THE MILKY WAY
By LINDSAY E. OASTER, B.Sc.
A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree
Master of Science
August 2008
McMaster University
© 2008 Lindsay Oaster I grant McMaster University the non-exclusive right to use this work for the University's own purposes and to make single copies of the work available to the public on a not-for-profit basis if copies are not otherwise available.
Lindsay Oaster The thesis of Lindsay Oaster was reviewed and approved* by the following:
Dr. Douglas Welch Professor, McMaster University Thesis Advisor, Chair of Committee
Dr. Christine Wilson Professor, McMaster University Committee member
Dr. Laura Parker Professor, McMaster University Committee member
*Signatures are on file in the Graduate School. MASTER OF SCIENCE (2008) McMaster University
(Department of Physics and Astronomy) Hamilton, Ontario
TITLE: A Survey Strategy for Light Echoes from Historical Supernovae in the Milky Way AUTHOR: Lindsay Oaster, B.Sc. SUPERVISOR: Douglas Welch NUMBER OF PAGES: viii, 77
ii L..--_____ABSTRAC~
Hundreds of years after exploding, the original light from a supernova can still be observed in the form of light echoes. This light scatters off interstellar dust and is re-directed back toward Earth; due to the extra travel time, we observe the echo after the initial outburst. At some time t after observing the outburst, the surface of equal travel paths defines an ellipsoid with Earth and the supernova at the foci. If dust intersects this ellipsoid it is possible to scatter the light and produce an echo. In this thesis, I develop a relative probability model for the detection of supernova light echoes based on the physical characteristics of interstellar dust and absorption near the Galactic plane. This model includes a dust scat tering function, distribution (scale height) of dust in the Galaxy, the dilution of echo flux with distance, and absorption along the supernova-dust-Earth travel paths. I have tested the model's predictions against observations and compared it with a prior survey strategy based on IRIS (re-processed IRAS) maps. Currently the IRIS-based strategy is more effective at selecting good paintings but its detection rate is only around 5%, highlighting the elusiveness of echo appearances. This work considers six historical supernovae in the Milky Way, all of which exploded in the pre-telescopic era (with the possible exception of Cas A) and were recorded as "guest stars" in astronomy records from Asia, Europe, and the Middle East. Their light echoes could give us information on these historically significant events and an opportunity to simultaneously study a supernova in outburst and several hundred years later. Early investigations suggest that the distribution of CO in the Galaxy may anti-correlate with the best paintings for light echoes; if a CO-echo link can be established, this would be useful in future light echo surveys.
iii ~----ACKNOWLEDGMENT~
During my experiences here, I have been supported by more people than I could possibly name within these pages. I would like to give my sincere thanks to the following people.
To Doug Welch: I got to come to McMaster and study astronomy with a fantastic supervisor - everyone should be so lucky! Thank you for sharing your wisdom and experience (and many stories!), and for all of your guidance during my time here.
To Christine Wilson and Laura Parker: thank you for being on my thesis com mittee and for your valued input on my research.
To the incredible staff in the department office: Mara Esposto, Cheryl John ston, Daphne Kilgour, Rosemary McNeice, Liz Penney, and Tina Stewart. You are amazing! Thank you for helping me get organized and for answering all my questions, big and small.
To the many dear friends I've found here, especially Rob Cockcroft and Pam Klaassen: I have been so blessed to know you, and I couldn't have asked for better people to journey with here in Hamilton. Thanks for sharing your in herent awesomeness with me.
To my amazing husband Zach Oaster, my mom and stepdad Marilynn and Gil Springman, my sister Niaomi and her husband Chad Curtis, and my husband's parents Nilene and John Oaster: You have been a source of balance when I needed it most. Thank you for your ever-present support.
IV &....--__TABLE OF CONTENTS!
Acknowledgments iv
List of Figures vii
List of Tables viii
Introduction 1
Chapter 1 A supernova review 2 1.1 Type Ia supernovae . . . . 2 1.2 Core-collapse supernovae . 9 1.3 Light echoes ...... 11 1.3.1 Nova Persei 1901 . 11 1.3.2 SN 1987 A . . . . . 12 1.3.3 Outside the galaxy 13 1.3.4 SuperMACHO and the LMC discoveries 14 1.3.5 The historical supernovae . . 15 1.3.6 Echo detectability and models 19 1.3.7 Surveying for light echoes 20
Chapter 2 Methods 21 2.1 The probability model 21 2.2 Absorption .. 24 2.3 Observations ...... 26
v Chapter 3 Results 28 3.1 Model results . 28 3.2 Observational results 29 3.3 Absorption ...... 29 3.4 Efficacy of the survey strategies 31
Chapter 4 Discussion & Conclusions 45 4.1 Extinction and the probability . 45 4.2 CO investigations 48 4.3 Conclusions ...... 49
Appendix A Code 55 A.1 Derror.m ...... 55 A.2 Cas_prob_commands.m 56
References 70
vi &...--____LIST OF FIGURE~
1.1 Supernova classification ...... 3 1.2 Combined supernova, CMB, galaxy cluster results 7 1.3 The layered structure of CC supernovae . 9 1.4 Kepler ellipsoid ...... 12 1.5 Couderc ellipsoid explanation . . 13 1.6 Apparent superluminal expansion 14
2.1 Kepler and Crab ellipsoids ... . 22 2.2 Extrapolated absorption .... . 25
3.1 Probability maps for the six remnants. 34 3.2 Tycho and Cas A echoes...... 37 3.3 Tycho and Cas A field probabilities 38 3.4 Tycho and Cas A maps, fields 39 3.5 Earth-dust absorption: Cas A . 40 3.6 Earth-dust absorption: Tycho 41 3.7 The effect of absorption: Cas A 42 3.7 The effect of absorption: Tycho 43 3.7 The effect of absorption: Cas A and Tycho . 44
4.1 Tycho and Cas A echo labels . . 46 4.2 CO line profile for echo #2729 . 52 4.3 Sample CO emission: Cas A 53 4.4 Sample CO emission: Tycho (1) 53 4.5 Sample CO emission: Tycho (2) 54 4.6 Sample CO emission: Tycho (3) 54
vii L...--____LIST OF TABLE~
1.1 Galactic supernovae and remnants . 16
3.1 The twelve scattered light echoes . 30 3.2 Strategy success rates ...... 31 3.3 Random-pointing success rates: Tycho 32 3.4 Random-pointing success rates: Cas A 32 3.5 Average probability near Tycho 33 3.6 Average probability near Cas A . . . . 33
viii L. Gaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
______INTRODUCTIOJ
Much of our current knowledge of the end stages of stars, the cosmological distance scale, and acceleration of the universe has been established through supernova observations. The outburst of SN 1987A in the Large Magellanic Cloud provided the opportunity to thoroughly observe a local supernova and trace the development of its remnant and light echoes from the earliest stages. Scattered-light echoes from supernovae have the potential to reveal informa tion about the explosion asymmetries and interstellar dust structure (Rest et al. 2005, 2008). Identifying echoes in significant numbers has been a chal lenge, however, and the optimum search strategy isn't obvious at present. In this work, I will introduce a new survey strategy that considers the dust distri bution, scattering function, flux-distance relation, and interstellar absorption and attempts to determine where light echoes are most likely to be found. The efficacy of this model will be evaluated based on observations performed in 2006 and 2007. Chapter 1 is a review of supernovae, the physics of their explosions, and their use in cosmology. I will also introduce the light echo phenomenon and its history along with the six historical, galactic supernova remnants that are considered in this thesis. Chapter 2 explains the development of the prediction model and the details of our observations, and in Chapter 3 the observational results are presented and analyzed. Chapter 4 contains the rates of success for each detection method and a discussion of future plans for similar work.
1 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
CHAPTERl ______~
IL..---__A SUPERNOVA REVIEW
Supernova observations are used to set the cosmological distance scale and determine the acceleration of the Universe (Wilson 1939; Perlmutter et al. 1998; Schmidt et al. 1998; Leibundgut 2008). The explosion mechanisms for the low- and high-mass regimes are summarized below, along with how each type has contributed to our knowledge of cosmology. Minkowski (1941) was the first to designate sub-groups for supernovae based on the absence ( "Type I") or presence ( "Type II") of hydrogen lines in the SN spectrum. These types were further subdivided based on the presence of helium and silicon lines (da Silva 1993); the current classification scheme is shown below in Figure 1.1 and will be referenced when discussing supernova types.
1.1 Type Ia supernovae
Explosion details When a white dwarf (WD) in a binary system accretes matter from its compan ion and exceeds 1.44 M 8 , electron degeneracy pressure is no longer sufficient to support the star against collapse (Nomoto & Sugimoto 1977; Hoeflich & Khokhlov 1996). A thermonuclear reaction begins at the center and spreads outward; if it propagates by a supersonic shock wave it is called a detona tion, whereas subsonic flame propagation is called deflagration (Hillebrandt & Niemeyer 2000). Detonation models are usually excluded for SNe Ia because they don't reproduce observed elements, but deflagration models are too slow to produce the supernova explosion (Ropke & Hillebrandt 2005). lntroduc
2 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Hydrogen? · Core-collapse · Thermonuclear yes no
~ ... Type II Type I
Silicon? .... ~ More H More He yes no
.... ~ I I Helium? IlL or liP lib Ia yes no
.... ~ Ib Ic
Fig ure 1.1 T he current supernova classification "tree," based on ob served li nes in the SN spectrum. Adapted from Fig. 1 of Leibundgut 2008.
ing turbulent combustion into the defl agration model is one solution to this problem: instabilities distort the fl ame surface and increase its surface area, using more fuel and accelerating the burning process (Ropke & Hillebrandt 2005). The entire star should be considered; many simulations only model one octant of the WD and assume spherical and mirror symmetries (Hillebrandt & Niemeyer 2000). The initial fl ame ignition has been the most difficult aspect to simulate and is still a very uncertain phenomenon. A common scenario (e.g. Ropke & Hillebrandt 2005; Khokhlov et al. 1993) is to start a flame near the center of the WD which produces hot ashes beneath layers of cooler, dense "fuel. " The buoyancy of the ashes below the fuel layer (Rayleigh-Taylor instability) , combined wit h the velocity shear present at the surface of ignited material (Kelvin-Helmholtz instability), produces turbulent motions that increase the fl ame surface area. Under these conditions, the deflagration process accelerates sufficiently to produce the supernova explosion (Ropke et al. 2007). There are many aspects of Ia explosions that are still actively studied. The large-scale features of the explosion are driven by buoyancy effects that cascade down to smaller scales (Ropke & Hillebrandt 2005). The turbulent fl ame velocity is then limited by velocity fluctuations over the smallest bubbles.
3 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Most models place these bubbles (the flame ignition) near the center of the star, but shifting this position may help in studies of the asymmetry of Ia explosions. Woosley et al. (2004) have proposed creating a one-sided ignition model where the flame begins near an outer edge and propagates across the white dwarf- to investigate the observed asymmetries. In addition, several SNe Ia spectra are noticeably polarized (e.g. Wang et al. 2003; Howell et al. 2001). There have been a few proposed explanations for this: the remnant could be atypically structured, the explosion may have been strongly anisotropic, or ejecta material might be interacting with a companion star or the surrounding ISM (Wang et al. 2003; Ropke & Hillebrandt 2005). In all of these cases, the polarization results from photons scattering off electrons generated either in the explosion or in the shock-heated circumstellar environment (Leonard et al. 2001). A true spherical explosion would produce no net polarization as the light would scatter equally in all directions. If the explosion or (heated) circumstellar environment were asymmetric, light would be scattered more in some directions than others - thus producing the net observed polarization (Leonard et al. 2001; Wang et al. 2003).
Supernovae and cosmology The standard cosmological model, which is the basic description of our Uni verse on the largest scales, makes two assumptions: that the Universe is homo geneous and isotropic, and that gravity is the dominant force on these scales (Narlikar & Padmanabhan 2001). This model is parametrized by:
1. The Hubble constant H0 , which describes the rate of expansion of the Universe,
2. The observed density of the Universe, n0 , in units of the critical density (p0 / Pc)· There are many sources that make up this observed density, but the major players are nM and nA - the portion contributed by all (baryonic +dark) matter sources and the cosmological constant, respec tively. The value of no determines the fate of the Universe; if no < 1 the Universe will forever expand, and no > 1 leads to eventual collapse (Narlikar & Padmanabhan 2001). Introducing a non-zero cosmological constant changes this condition slightly: if nA > 0 the Universe could still avoid collapse with no > 1 (see Figure 1.2). If we have a reliable method to measure distances to objects beyond our Galaxy, we can explore the expansion history of the Universe and set lim its on the cosmological parameters.
4 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Supernovae as distance indicators Any object used as a standard candle will have its magnitude drop predictably as a function of redshift (Leibundgut 2001):
m = 5log (~) + M + 25, z ~ 0.1 (1.1)
For a fixed absolute magnitude M, measuring the apparent magnitude m of an object at redshift z can be used to obtain a value for H0• However, knowing the Hubble constant isn't required to determine nA or nM because measuring
l:!:.m (= mz2 - mz1 ) of a standard candle at two sufficiently different redshifts z1 and z2 is enough to obtain the density values (Leibundgut 2001). Wilson (1939) first postulated that supernovae could be used as cosmic distance probes, and this was later expanded by Sandage's (1961) proposal that observing how standard candles dim with redshift could be useful in determin ing the distances to galaxies and other structures. Bright cluster galaxies had been previously used as distance indicators (Sandage 1961) until it was found that the effects of galaxy evolution were stronger than those from universal curvature or expansion (Schmidt et al. 1998). SNe Ia are desirable distance indicators because of their high intrinsic luminosities ((MB) = -19.5; Saha et al. 1997) and individual SN e are believed to be less prone to evolutionary effects on brightness because they are the thermonuclear explosions of white dwarfs.
Revealing the cosmic acceleration In the 1990s there were two research groups, the High-Z Supernova Search (Schmidt et al. 1998) and the Supernova Cosmology Project (Perlmutter et al. 1998), both attempting to find and analyze distant (z > 0.2) Ia supernovae. Their specific goals were to detect the supernovae, classify them spectroscop ically as Type Ia, and perform photometry in at least two different bands to measure their rate of decline (Schmidt et al. 1998). Their observations could reveal the cosmic acceleration if the SNe distances (obtained from their light curves) were considered as luminosity distances DL. Including time dilation and redshift, and expanding in z, we have
(1.2)
where q0 is the deceleration parameter and represents all sources that con tribute to the change of the Hubble constant; an empty universe has q0 = 0 while a flat universe has q0 = 0.5 (Schmidt et al. 1998). The Hubble Law is
5 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
the linear form of (1.2), and determining the Hubble constant H0 has been an ongoing problem in cosmology. While individual SNe Ia are not necessarily uniform - there appear to be intrinsic differences in luminosities, spectra, and light curve shapes - the rate of luminosity decline does correlate with their absolute magnitude at maximum light (Branch & Tammann 1992; Schmidt et al. 1998). This decline rate allows them to be normalized as a group by their light curve shape, and this corrects for much of the luminosity dispersion among the individual supernovae (Wilson 1939; Schmidt et al. 1998; Perlmut ter et al. 1998). Spectroscopy is required for classification, but SNe at z > 0.5 are difficult to classify because the distinctive Sill doublet (.A = 634.7 and 637.1 nm) is redshifted out of the optical range (Leibundgut 2001). The peak of the light curve is also required for normalization, and this can be hard to obtain even for nearby supernovae if there are gaps in the observations (e.g. poor weather conditions). It is therefore useful to compare the distant and local SNe Ia: those at z < 0.5 have light curves that decline 1.5 times faster than more distant ones, which is consistent with time dilation (Leibundgut 2001). The result from the High-Z and SOP groups was that the distant su pernovae are too faint to be in agreement with a "freely coasting, 'empty' universe" (Leibundgut 2001). The light curves and spectral evolution of SNe Ia are nearly uniform such that time dilation can be measured by compar ing distant and local supernovae; the results from this comparison agreed with an accelerating expansion with time. The uniformity of the cosmic mi crowave background and observations of dimming surface brightness of galaxies with increased redshift provide evidence supporting the accelerated-expanding universe model (Leibundgut 2001). The parameter q0 is degenerate for linear combinations of nA and nM, and Type Ia supernova data constrain the com bination nM- nA with nA > 0 (Knop et al. 2003, see Figure 1.2). The supernova results have been combined with measurements of the CMB (using WMAP data; Spergel et al. 2003) which indicate a fiat geometry, and galaxy clusters (using gas mass fractions from Chandra; Allen et al. 2002) which indicate a low matter density (Leibundgut 2008). The CMB data mostly depends on OM+ f2A while the galaxy cluster data constrains just OM (Knop et al. 2003). These three data sets, taken completely independently, constrain nM and nA to a very narrow probability region which agrees with an acceler ating, fiat universe (Leibundgut 2001, see Figure 1.2). Another major result was the constraint on the age of the Universe; the dynamical age t0 H0 ~ 1, so 1 1 t0 falls between 12.5 and 17 Gyr (for H0 = 75 to 55 km s- Mpc- ). This age is consistent with those of the oldest known stars, while the previous dynami cal age was not (Leibundgut 2001). One proposed explanation for the cosmic acceleration is a new form of dark energy with a negative pressure (Thrner &
6 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Supernova Cosmology ProJect 3
No Big Bang opergel et a '2 JO )
Allen 'al .o::: n&")
2
Supernovae
Clusters (~I -?. C(/ '// -1
0 2 3
Figure 1.2 The combined supernova, CMB, and galaxy cluster data tight ly constrain the probability region fo r nM and nA . Image from Figure 8 of Knop et a!. (2003).
Tyson 1999). Nearby SNe Ia can probe the linearity of the universal expan sion, while distant (z > 0.2) supernovae can give information on the evolution of peak luminosities. We need to know the absolute luminosity to obtain H 0 , but the expansion parameter q0 is not dependent on L and is assumed to be constant (Leibundgut 2001 ).
Other explanations for SN dimming Evolution of peak luminosity has been considered as a non-cosmological ex planation for the observed dimming of distant supernovae (Leibundgut 2008). The standard model is the t hermonuclear explosion of a 1.44 Jv!0 carbon oxygen white dwarf, and no change is expected in the nuclear physics over time. According to Leibundgut (2001 ) shell burning could vary with age: the younger (more massive) WDs are expected to be brighter than older WDs, yet the more distant (younger) SN e I a are fainter. He mentions (citing Falco et al. 1999) that t he distant observed SNe appear bluer than local SNe; if
7 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
they are intrinsically bluer, then the extinction derived (and consequently the distances) would be too small. This color difference would present a coun terexample to the established correlation between light curve shape and color, where both the distant and local SNe agree well in light curve parameters. Drell et al. (2000) claimed that differences in composition leads to lower lu minosities in distant supernovae compared to the local SNe, but this is not supported by the data (Leibundgut 2001): while the range of SN luminosities appears narrower at large distances, the scatter isn't necessarily smaller. Dust effects have also been postulated to account for the dimming. "Gray" dust that scatters all visible wavelengths equally could produce the observed absorption if grain sizes are kept above 1.0 J-Lm, and this could bring the supernova results into agreement with an open universe (Leibundgut 2001). Introducing the necessary quantity of gray dust to explain the SN dimming would also produce a strong cosmic far-IR background since the radiation would be thermalized (Leibundgut 2001). A faint IR background has been discovered (Hauser & Dwek 2001); however, there is a limit on the amount of dust that can produce this background and continued studies are pushing this limit lower (Leibundgut 2001). The last issue with this proposal is the presence of bluer SN e Ia at higher redshifts, since by definition gray dust should not affect the color of the supernovae. Finally, the effects of gravitational lensing have been explored in light of the supernova results. Most distant sources are dimmed somewhat as light is scattered by massive objects along the way, but sources with a line of sight passing sufficiently near a very deep potential well will undergo substantial lensing with apparent amplification (Leibundgut 2001). Unfortunately, we can't measure this effect directly from the supernova observations- but the effect of lensing can be estimated for different matter distributions. Taking the extreme case and dividing all matter in the Universe into compact ("point mass") objects, distant sources are dimmed by a maximum of 0.15 mag for z = 0.5 and nM = 1 (Leibundgut 2001). When matter is clumped instead into galaxy-sized objects, the dimming effect is nearly eliminated (Leibundgut 2001). To summarize, non-cosmological hypotheses for the supernova dimming aren't well supported by the data. When the CMB and galaxy cluster data are added to the SN results, it appears much more likely that the observed dimming is due to cosmic acceleration than to the scenarios mentioned above.
8 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
1.2 C ore-collapse supernovae
When a massive (M > 8M8 ) star depletes the hydrogen and helium fuel in its core, heavier elements are progressively fused until the core contains iron (Burrows 1996; Leibundgut 2008). At this stage the star contains layers of newly synthesized material interlaced with layers of burning helium, carbon, oxygen, and silicon (Figure 1.3); since iron fusion is not an exothermic process the star will no longer be stable against gravitational collapse.
To surface
H, He envelope
Hydrogen burning
Figure 1.3 The layered structure of core-collapse supernovae. Image from Figure 15.10 of Carroll & Ostlie (2007).
As burning continues in the star, material from the outer layers falls down onto
the iron core. Eventually the core mass exceeds the 1.44 M8 Chandrasekhar limit, collapses until the iron photodisintegrates into neutrons and a - particles, and forms neutrinos when protons and electrons merge under the dense, hot conditions (Woosley & Janka 2005). Most of the neutrinos escape the star and carry away much of the binding energy, ending core collapse. A shock front propagates through the (still infalling) outer layers which, at much later times, is capable of setting off or halting external star formation (Burrows 1996; Leibundgut 2008) and affecting the shape of the host galaxy (Leibundgut 2008). Core-collapse SN e are classified spectroscopically based on the presence of hydrogen/helium lines (Type II) or absence of silicon lines (Type Ib and Ic) in their spectrum at maximum light, depending on which elements are still present in the outer layers of the star at the time of explosion (see Figure 1.1) . The peak of a Ib/Ic light curve is determined by the amount of nickel synthesized in the explosion, its width is from the (in)ability of photons to escape the interior of the supernova, and the overall shape is dominated by the 56 Ni --+ 56 Co --+ 56 Fe reaction (Hamuy 2003). Type II SN e are divided into two classes based on their light curves (Lei bundgut & Suntzeff 2003):
9 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
1. Type liP supernovae have a plateau shortly after maximum light where their temperature and luminosity remain nearly constant. This plateau is caused by the ionization of hydrogen in the envelope from the shock wave which increases the opacity until the gas cools enough to recombine. 2. Type IlL SNe light curves decline linearly after maximum light, which implies a lower degree of ionization in the envelope. Heavy elements created during the explosion are propelled into the ISM (at velocities between 4000-7000 km/s) and can enrich other stars during forma tion (Hamuy 2003). Hamuy investigated the properties of 24 Type liP and seven Type Ib/Ic supernovae to compare their expected physical and observed properties and degree of nucleosynthesis. He used an approximation of Weaver & Woosley's (1980) model: small progenitors with high energies were expected to synthesize the greatest amount of metals by the relation E )1/4 Toe ( - Ro3 (1.3)
where Ro is the progenitor radius and E is the explosion energy. Typical SN energies are on the order of 1051 ergs, or one foe (Nomoto et al. 2007). The Type liP supernovae in Hamuy's sample released energies between 0.6 and 5.5 foe, with progenitor radii between 40-600 R8 and envelope masses of 14-56 M8 . The supernovae produced .0016-0.26 M8 of nickel; the result from his data was that SN e with high ejecta velocities and bright plateaus produced the most nickel mass (Hamuy 2003). Additionally, the more massive progenitors produced higher-energy explosions; the explosion energy E depends on the mass contained in the envelope. Similar relations were found for the Type Ib/Ic SNe in his sample (Hamuy 2003). The expanding photosphere method (EPM), first proposed by Kirshner & Kwan (1974) to obtain distances to Cepheid variables, has been successfully applied to Type liP SNe. It is related to the Baade-Wesselink method for variable stars: by finding the ratio of the star's radius at two different points in its pulsation and using the radial velocity of the star's photosphere, we can solve for the two radii and obtain the star's distance (Wesselink 1946). Following Kirshner & Kwan (1974), in the EPM the angular size () of the star from Earth is first determined by its temperature and flux density. If we know when the expansion started (t 0 ), and the velocity v of material in the photosphere is found at time t, then the radius of the star is given by
R = v(t - to) + Ro (1.4)
where R0 is the star's initial radius. The distance D to the Cepheid is then
10 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
found by D = v2(t2- t1) + Ro(l- v2jv1) (1.5) 02- (01v2/v1) In the case of SN 1999em, its EPM distance of 11.5 ± 1.0 Mpc (Dessart & Hillier 2006) is consistent with the previously-obtained Cepheid distance to its host galaxy of 11.7 ± 1.0 Mpc (Leonard et al. 2003). There also exists a correlation between the luminosity of a liP SN 0 33 0 04 and its ejecta velocity, Vp rv Lp · ± · , measured spectroscopically from the Fell 516.9 nm line during the plateau phase (Hamuy & Pinto 2002). This correlation allows their luminosities to be normalized and the supernovae can be used as standard candles for distance calculations, often with precisions better than 20% (Leibundgut 2008). (For comparison, Schmidt et al. (1998) claimed distance uncertainties of 5% or better when comparing the distant and local SN samples.) The v - L correlation is based directly on the physics of the plateau phase of a Type liP and is therefore independent of the distance ladder (Leibundgut 2008).
1.3 Light echoes
A supernova explosion sends a "light front" outward in all directions, and we observe the portion along our line of sight when we see the outburst directly. Interstellar dust can scatter other portions of this light front back toward Earth (Zwicky 1940) which would arrive (as an echo) later than the outburst light due to the extra travel time involved. The set of all possible echo locations at a time t after the SN explosion forms an ellipsoid with the Earth and supernova at the focii (see Figure 1.4). Echoes from novae (non-supernova explosions) have been well studied, and Nova Persei is discussed below as an example. The echoes from SN 1987A were used to deduce its circumstellar dust structure (Sugerman et al. 2005), while those from extragalactic supernovae can place their own signature on the supernova's light curve and spectrum (Welch et al. 2007; Sugerman 2005). This thesis is concerned with Galactic supernovae observed in the pre-telescopic era, supported by historical astronomy records (with the exception of Cas A) and later identified with a remnant at the same position.
1.3.1 Nova Persei 1901 GK Persei (or Nova Persei) was observed in outburst in 1901, and six months afterward a "light halo" was observed in its vicinity (Flammarion & Antoniadi 1901). It appeared as a kind of nebulosity around the nova, rapidly expanding
11 L . Oaster • M. Sc. Thesis Dept. of Physics & Astronomy, McMaster University
1000 ~ 0 N -1000
5000
x(ly) y(ly)
Fig ure 1.4 Kepler ellipsoid: D = 16300 ly, t = 403 years, b = +6.84° T he size and "roundness" of the ellipsoid is determined by the elapsed time since exploding and the distance from Earth (blue diamond). T he galactic latitude of the SN (red star) will affect where light echoes appear; those exploding near the Galactic plane (shown in tan) may produce more echoes near the rem nant due to the increased amount of dust.
outward. Kapteyn (1901 ) was the first to propose that this halo was actually light from the nova illuminating the surrounding dust , and Couderc (1939) later refined this theory. He presented the echo geometry and proposed that layers of dust along our line of sight were responsible for t he light halo around t he nova; see Figure 1. 5. Couderc also explained the apparent superluminal motion of the echoes. The expansion appeared to be faster t han t he speed of light, and t his was due to the amount of forward motion in t he echo with respect to the observer (Figure 1.6). After t his initial phase, Nova Persei continued to be a target of study and still undergoes smaller periodic outbursts (e.g. Bianchini et al. 1982; Brat et al. 2006).
1.3.2 SN 1987A SN 1987A in the Large Magellanic Cloud was the brightest supernova observed in the sky since Kepler's SN in 1604, and its neutrino burst provided the first detections of neut rinos from supernovae (McCray 1993). The explosion energy E was estimated to be 1.5 foe (McCray 1993) and it reached a peak magnitude V ~ 3. 0 (Catchpole et al. 1987). 1987 A is the only supernova whose evolution has been fully traced at all wavelengths (Arnett et al. 1989) . Within a few years of exploding a triple-ring nebula appeared around t he SNR, and light echoes were used to map out the surrounding dust structures (Sugerman et al. 2005) . The echoes' surface brightnesses put constraints on the composition
12 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
...... To Earth To Earth
Figure 1.5 The paraboloid model proposed by Couderc. The orientation of an intersecting dust layer with respect to our line of sight can affect the apparent location of echoes from the nova or supernova.
and density of the dust: based on its grain size and composition (astronomical silicate, PAHs, etc.) , different frequencies of light will scatter with different efficiencies which produces echoes of varying colors and surface brightnesses (Sugerman et al. 2005) . The light echoes from 1987A are still being studied as they continue to illuminate the surrounding dust (Sugerman 2003; Rest et al. 2005).
1.3.3 Outside the galaxy Scattered light echoes can be detected even in extragalactic supernovae. SN 2002hh was observed in NGC 6946 by several groups from October 2002 to July 2006 (Li 2002; Filippenko et al. 2002; Pozzo et al. 2006; Welch et al. 2007). It was of type liP and stayed near its peak for three months. Later work showed that the Her emission line had retained its P Cygni profile (Welch et al. 2007). The presence of a bright optical echo was contributing to the spectrum in such a way that the H-alpha line profile was a combination of the original supernova and the late-time remnant spectra. The echo was only 4 mag fainter than the SN at maximum (Welch et al. 2007; Pozzo et al. 2006) , so the strong spectrum "contamination" is not surprising. SN 2003gd was a supernova discovered in M74/ NGC 628 (Evans & McNaught 2003; Sugerman 2005). It exploded near a spiral arm, and small echo "arclets" became visible 0.23"- 0.38" away from the SN during HST ob servations in 2004 (Sugerman 2005). Their V-band magnitude was 24.2, which
13 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Figure 1.6 The apparent superluminal expansion of light echoes, adapted from Panagia and Bond (website accessed 2006). An observer first notes
the location of an echo D 1 away from the SNR. The echo expands by a distance d during each time interval while the observer sees an expansion of b.D1...... 2 and b.D2...... 3 , both larger than d.
is 11 magnitudes below the supernova's peak (Sugerman 2005 ; Hendry et al. 2005) . Using both the echoes and echo candidates (those needing further ob servation), their positions were consistent with being scattered by a strongly inclined dust sheet 60° from our line of sight. However, a perpendicular-dust sheet model could fit the observations when only the known echoes were used; more observations were needed to determine the actual dust geometry (Sug erman 2005). A later study (van Dyk et al. 2006) concluded the scattering dust was 113 pc in front of the remnant - in the ISM , not the circumstell ar environment - but did not mention the inclination of the dust sheet. Light echoes, then, can reveal the structure of the scattering dust (when found in sufficient numbers) for both local and extragalactic supernovae.
1.3.4 SuperMACHO and the LMC discoveries The SuperMACHO project was a microlensing survey using the Large Magel lanic Cloud as a set of background sources to estimate the amount of mass in compact halo objects (Rest et al. 2005) . A difference-imaging technique was used in order to reveal any stars varying in brightness over several epochs. Images of the same sky area were taken over five years and then subtracted in order to detect the flux increase that would signal a microlensing event.
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In addition to their intended targets, the subtraction routine uncovered faint, extended objects that appeared to move over time. The proper motions and arclet structure were inconsistent with a MACHO detection; these were, in fact, optical light echoes from previously unseen supernovae in the LMC. Af ter observing enough epochs, it was possible to trace the origin of each moving arclet to a small number of supernova remnants. A single-template differ ence image method was used to recover the well-defined echoes surrounding SN1987 A (Rest et al. 2005). For the purposes of this work (described later in Chapter 2), the same differencing technique was sufficient in our search for Galactic light echoes. The images are first aligned by comparing the "bright" areas (e.g. stars) and examining for any possible shift or rotation in position, and a reference frame with the best seeing is chosen for the initial subtraction (e.g. Alard & Lupton 1998; Alard 2000). The subtraction routine divides the image into (for example) four smaller regions, solves for a convolution kernel and uses it to difference each region from the corresponding region in the reference frame. The resulting image has all regions with constant flux subtracted away while those with varying flux will be brighter or darker, and in the SuperMACHO case the echo flux appeared as a series of moving filaments in each subtracted image.
1.3.5 The historical supernovae Many light echoes found in the LMC were from supernovae at least as old as the ancient supernovae in the Galaxy which were noted for their appearances as "guest stars" in historical astronomy records and later matched with radio sources as radio astronomy developed in the 1940's-'50s. We concentrated on six historical supernovae in this work: two of them (Tycho and Cas A) were observed in 2006 and 2007 and provided tests for the echo probability model described in Chapter 2, while the other four (SN 1006, SN 1181, Crab, and Kepler) were studied but not observed for the purpose of testing this model.
SN 1006 The earliest of the sample occurred in A.D. 1006; it was mentioned in as tronomy records worldwide as a remarkably bright new star appearing in the constellation Lupus (Clark & Stephenson 1977, hereafter CS77). Japanese and Arab records described the star as bright enough to illuminate the sky around it and cast shadows on the ground, much like the full moon (Stephenson & Green 2002, hereafter SG02). By most accounts it was visible for nearly two years, setting and rising twice with the seasons. It also had an unpredictable
15 L. Gaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Table 1.1 The Galactic supernovae and their remnants. Distance esti mates are from Green (2006).
SN SNR Remnant discovered Distance SN 1006 G327.6+14.6 1965 2.2 kpc SN 1054 Crab Nebula 1844, 1949 2.0 kpc SN 1181 3C58 1971 3.2 kpc Tycho 3C10 1952 3.1 kpc Kepler 3C358 1948 6.0 kpc ? Cas A 1948 3.4 kpc
visual appearance; a European account was quoted as saying the star was "sometimes contracted, sometimes diffused, and moreover sometimes extin guished" (CS77). A supernova remnant was discovered at the same location asSN 1006 by Gardner & Milne (1965). Their paper described the SNR as a radio source with a shell-type structure, and van den Bergh later detected the remnant at optical wavelengths. He compared the filament morphology to that of SN 1572, a known Type I supernova (under the old Minkowski classification), and said the 1006 event was likely a Type I as well (van den Bergh 1975). Due to its brightness, position above the plane of the galaxy, and lack of neutron star in the remnant, SN 1006 is currently thought to be of Type Ia (Lozinskaia & Chugai 1987).
SN 1054: "The Crab supernova" The guest star of A.D. 1054 in the constellation Taurus wasn't as bright as SN 1006; all known records of its discovery are from China and Japan (CS77). It was also visible for approximately two years, of which 23 days were visible in daylight. It had dimmed to -4 mag when it ceased to be visible in daylight hours (SG02). Mayall and Oort (1942) established that the Crab Nebula is the rem nant of SN 1054. Line emission photographs show that the nebula is expand ing, and its expansion timescale is in agreement with A.D. 1054 as a starting epoch. Additionally there is a pulsar located within 10" of the center of the nebula; Bolton & Stanley (1949) were the first to show that the cosmic radio source Taurus A was associated with the optical Crab Nebula. The pulsar has since been confirmed in optical and X-ray wavelengths (SG02). The existence of the pulsar implies that SN 1054 was a core-collapse supernova, since SNe
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Ia do not produce neutron stars.
SN 1181 Another occurred in the year 1181 A.D. in the asterism Chuanshe, now in Cassiopeia (SG02). Its brightness faded over 180 days from 0.0 to +5.5 mag, which is in good agreement with the visual limit of the asterism (CS77). Chi nese records stated the guest star's first appearance and duration in the sky, while Japanese records confirmed its date of appearance and location. The most probable remnant of this supernova is the radio source 3C58. It resides less than 1o away from the fifth star in Chuanshe, later identified as SAO 12076, and its central pulsar has a characteristic age consistent with being "born" in 1181 (SG02). SN 1181 is also understood to be a core-collapse supernova due to the presence of the central pulsar (Lozinskaia & Chugai 1987).
SN 1572: "Tycho's supernova" Tycho Brahe took note of a new bright star in Cassiopeia in 1572. He carefully tracked its position over several months, and in his report De nova stella he writes, "... after the lapse of several months it has not advanced by its own motion a single minute from that place in which I first saw it" (SG02). It was just bright enough to be seen during daylight hours; its peak magnitude was estimated at -4.0. Its color changed from bright white to faint red before fading from view (SG02). Brown and Hazard identified the central radio source in 1952; others confirmed the location of the source at radio wavelengths and also observed its optical and X-ray emission (SG02). The SNR is approximately 8 arcmin in diameter. It has a limb-brightened shell in both radio and X-ray emission and faint optical filaments near the sharp radio edge (SG02). X-ray spectroscopy of the remnant indicates that Tycho's supernova was a Type Ia event (Badenes et al. 2007).
SN 1604: "Kepler's supernova" The SN of 1604 was widely observed and because it appeared low in the sky, multiple records mention its noticeable scintillation. European astronomy was advancing, and the position of the guest star was recorded with much higher precision (1') than the 1o precision of the East Asian records (SG02). Kepler carefully observed the new star to determine if it was fixed or moving relative to the background stars. After establishing that it was indeed stationary, he continued to observe the star until it faded from view one year later (SG02).
17 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Kepler's measurement of the star's position was only 4' away from the optical SNR discovered by Baade (1943). Baldwin & Edge (1957) first detected radio emission from the remnant and revealed a limb-brightened shell much like Tycho's SNR. There is some clumpy optical emission near the edge and center, and Tuohy et al. (19(9) detected its X-ray emission and reported a thermal spectrum across the remnant. Studies of the SNR have given conflicting results for its expansion: op tical and radio observations indicate significant deceleration, but X-ray mea surements show little, if any, deceleration (SG02). This causes an appreciable uncertainty in the age and distance of the SNR. Radio studies of its proper motion were combined with optical studies of its shock velocities to obtain a distance of 2.9 kpc to the SNR. .But the inconsistency of expansion rates across different wavelengths may introduce a systematic error to this distance calculation, now approximately 3.4-6.4 kpc (SG02). As for its type, based on the presence of Fe emission and lack of 0 emission in Chandra observations, it appears to be of Type Ia (Badenes et al. 2007).
Cassiopeia A Besides the Sun, Cas A is the strongest source in the sky at most radio fre quencies (Penzias & Wilson 1965). It was detected by multiple researchers in the 1940s as radio astronomy progressed, and when Minkowski (1959) iden tified filamentary optical emission it was recognized as a supernova remnant. Its expansion velocity is consistent with a SN explosion between 1662-1672 (Fesen et al. 2006), making it the youngest known supernova. (That is, until the discovery by Reynolds et al. (2008) that the Galactic SNR Gl.9+0.3 is approximately 200 years younger than Cas A.) No mention of a supernova or guest star appears anywhere in European or East Asian records in this time period. The question has been raised with some controversy that Flamsteed may have inadvertently observed Cas A in 1680 when surveying that region of the sky. The uncertainty on the supernova's age is such that it may have still been visible after peak luminosity during that time, but the lack of other sightings indicate that the outburst light was probably absorbed by interstel lar dust (SG02). Either way, its strong radio emission makes it a worthwhile target to study among the other Galactic supernovae. Recently Krause et al. (2005, 2008) have discovered moving infrared "echoes," the signature of dust absorbing and re-radiating echo flux, using 24-J-Lm data from Spitzer. In 2003 and 2004 they located several !R-emitting filaments moving away from the Cas A remnant (Krause et al. 2005). After selecting a recently brightened dust cloud in a later Spitzer image, they ob tained optical spectra of it with the Subaru telescope. From the spectra and
18 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
through comparison with SN 1993J (Matheson et al. 2000) they found that Cas A had a Type Ilb profile (Krause et al. 2008). This is the first definitive spectroscopic classification of a historical Galactic supernova.
1.3.6 Echo detectability and models Zwicky first wrote in 1940 about the possibility of scattered-light echoes af fecting supernova observations: "... if the image of a supernova a long time after maximum should sud denly and irregularly brighten up, a spectroscopic investigation might reveal that this increase in brightness is due to scattered light of the same spectral distribution as that which was previously observed in the light of the supernova near maximum." (emphasis added)
van den Bergh (1965) later investigated the echo detectability by compar ing the absolute magnitude of Nova Persei (peak M = -8.7) and a typical (hydrogen-absent) Type I SN (M = -19). We note that the supernova will 10 3 be 10.3 magnitudes, or 2.512 · ~ 12,000 times brighter than Nova Persei. If we consider the r-2 flux dependence and take
fN = 2.512(MN-MsN) = D~N (1.6) fsN DN
then DsN ~ 100DN and we see that the SN can illuminate interstellar dust clouds 100 times farther than Nova Persei (van den Bergh 1965). Additionally, while Nova Persei's light halo was visible for 20 months after the explosion, the supernova's light halo could persist for 2000 months (or 167 years) after its explosion. The result is a "thicker" light front which would produce higher surface-brightness echoes than Nova Persei. Based on the photographic plate limits, van den Bergh concluded that the light front of a supernova would still be observable 1000 years after the initial outburst (1965). He then explored the possibility of observing light echoes from nebulae as well as around the other Galactic supernovae. For example, echoes from the Crab SNR would be faint and spread over a large sky area while those from Cas A would be hard to detect with then-current technology, with a ''visual surface brightness close to the plate limit of the 48-inch Schmidt" due to the high absorption levels (1965). Patat (2005) compared the detectability of echoes from Type Ia and Type II SNe; Type Ia SNe are intrinsically brighter, but their progenitors are older stars and are not likely to be surrounded in dusty circumstellar environments ( CSEs). Core-collapse SNe would have plenty of dust in their
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CSEs, so the likelihood of observing an echo from these supernovae should be greater (Patat 2005). Chevalier (1986) constructed models of optical and infrared echo light curves using two Type II supernovae, SN 1979c and 1980k, to test their success. For the optical scattering case the light curves were calculated for moderate optical depths (7 = 0.3) when considering single scattering, and then modified to include multiple scattering for 7 = 1.0 and 3.0 (1986). The infrared light curves included emission from the dust as it was heated by the expanding light shell, and assumed an optically thin scenario. Chevalier used the single scat tering, or "instantaneous injection," approximation (1986) on the assumption that the supernova was observed at a time much larger than the timescale of the initial explosion. The main feature of his model was a rapidly decaying op tical light curve and a noticeable late-time infrared excess from the added dust grain emission. When applied to SN 1979c the model did not accurately fit its known light curves; Chevalier concluded that the continuum observations either a) were of scattered light echoes and at least one of the model assump tions had broken down, or b) another source was responsible for the observed continuum and very little was due to scattered light (1986). SN 1980k was fit well with the model light curves; the late-time infrared excess was present and a similar exponential decay was observed at the late timescale, though it was possible to explain the decay using energy input from radioactivity (1986) rather than with the light echo assumption. The author noted that additional UV spectroscopy could help distinguish between these two scenarios.
1.3.7 Surveying for light echoes It has long been postulated that echoes from even very old (1000-year) su pernovae could still be visible (van den Bergh 1965), and the SuperMACHO group's success with echo detection in the LMC indicates that we have the means to detect and analyze faint echoes. The current problem is in locating them; their appearance hinges on having the "right" amount of interstellar dust located at some ideal location relative to Earth and the SNR. By consid ering the echo geometry and accounting for the interstellar extinction, I have attempted to predict the best survey locations for the six Galactic supernovae mentioned above. Our methods and observations are provided in the next chapter.
20 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
CHAPTER2 ______~ ~~------METHODS
We were granted a total of 13.5 nights on the Mayall 4m telescope at Kitt Peak National Observatory (KPNO) in October and December in both 2006 & 2007 to search for echoes from Cas A, Tycho, and SN 1181. An algorithm to focus our observations became necessary as the mosaic imager at the telescope covers just 36' x 36' (Jacoby et al. 2004); in contrast, a sample calculation for Tycho predicts echoes residing 6.5° away from the SNR (Rest et al. 2005). This algorithm would have to include the effects of absorption, distance, and echo scattering direction on the strength of the echo flux at a particular location on the sky. I have compiled a script using Matlab that combines these effects to form a relative probability of detecting an echo for a given right ascension and declination.
2.1 The probability model
The script1 first converts the input (celestial) coordinates into galactic longi tude l and latitude b, then into 3D Cartesian coordinates centered on the Sun with the positive x-direction pointing toward the SNR and z = 0 defining the Galactic plane. Light echoes can potentially be found anywhere on the (imag inary) ellipsoid defined by the time elapsed since the supernova's outburst and the location of both the sun and supernova remnant. There must be dust present on the ellipsoid to reflect the outburst light back to Earth. The disk of the Galaxy is approximately 50 pc thick and contains most of the dust, so the intersection of the Galactic plane with this
1Cas_prob_commands.m; see Appendix
21 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
1000 ~ 0 N -1000
5000
x(ly) y(ly)
1000 ~ 0 N -1 000
5000
-5000 -1 .{).8 X 10 y(ly) X (ly)
Figure 2.1 Kepler (top) and Crab (bottom) elli psoids. T he Galactic plane is shown in tan .
ellipsoid is calculated. The four major effects on t he echo flux that this model considers are as follows:
1. T he flux density follows an inverse square law with increasing distance. T he line-of- sight distance from the SNR to any point on t he ellipsoid is calculated from standard ellipsoid geometry, and the dust- Earth distance is simply t he radius of the SN light front ( ct) minus the SN- dust distance:
2 a(1 - e ) TSNR-dust = , Tdust-Earth = ct - TSNR-;:iust (2. 1) 1 + e cos
22 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
2. The dust density in the Galactic plane decreases as e-z/H, where H is the scale height of the thin disk (50 pc).
3. The distribution of interstellar dust grain sizes (0.001 to 10 microns) leads to scattering in the forward direction (Draine & Lee 1984). Refer encing (Sugerman et al. 2005), a scattering functionS(>-., B) is calculated:
S(>-., B) = JQsc(>-., a)o-(g, B)f(a) da (2.3)
This scattering function incorporates the Henyey-Greenstein phase func tion (g B) - 1 - g2 (2.4) ' - (1 + g2 - 2g cos B)3/2 where g represents the amount of forward scattering (taken to be 0.6 in Rest et al. (2005) and this work) and Bis the scattering angle. Addition ally S(>-., B) contains the dust grain cross-section o-, scattering efficiency
Q8 c, and size distribution function f(a), which were obtained from Draine & Lee (1984) using the "astronomical silicate" composition assumption. The scattering angle is calculated for a given point on the ellipsoid, and S(>-., B) "weights" the probability of echo detection at each point.
4. The echo flux is absorbed by dust in the Galactic plane. We estimate the absorption Av in magnitudes along both segments of the light echo's path: from the supernova to the intersecting dust, and the dust to Earth. This absorption is converted into a fractional drop in flux by taking !::,.F "' 10-0.4A.
Each of these functions will have various constant prefactors. This model estimates the relative strength of the echo flux over the entire ellipsoid by multiplying all of these functions together:
1 0 4 · P(RA, Dec) "' _2 (e-zfH)(1o- A)S(>-., B) (2.5) (rsNR-dust + rdust-Earth) Any overall factor appearing in a given function will affect each location equally, so it can be disregarded. In this way, the relative flux strength can be viewed as a relative probability of detecting the flux at a particular location in the sky.
23 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
2.2 Absorption
There have been measurements of the line-of-sight absorption in the solar neighborhood, and we used the results from Arenou et al. (1992) to estimate the absorption along the path from us to a point on the ellipsoid. Their work had produced a three-dimensional model of the interstellar extinction within 1 kpc of the Sun. The entire sky was divided into nearly 200 regions: each was 10-30° in Galactic longitude land latitude b, with smaller box sizes used near the Galactic plane (b = 0°). Each region had its own quadratic absorption function Av(r) with a parameter Ro (not the distance to the Galactic center) that gave the limiting distance over which each function was valid, typically between 0.5-1.0 kpc (Arenou et al. 1992). I wanted to extrapolate these absorption models out to R ~ 4 kpc; none of the supernovae of interest are farther away than 4 kpc, so it seemed to be a reasonable upper limit in this context. Because these Av(r) functions were proportional to -r2 they turned over beyond some peak R; this is not a physical effect as the total absorption will never decrease with increasing distance. Exponential functions were fit to each region using the EzyFit toolbox for Matlab (Moisy 2006). After plotting each quadratic function the start and end points r0 and rmax were chosen; the toolbox then generated the exponential fit and plotted it over the existing curve. The quality of the fit was visibly different if too much of this turnover was included, so each quadratic function had three exponential fits generated:
1. rmax = peak R,
2. rmax = R' = 80% of the peak R,
3. Tmax = Ro (with Ro < R).
A small script2 was written to evaluate the percent difference of the first derivative between the original function and each of the three fits. I decided the 5% difference level would be sufficient for this probability model, and in nearly all cases R' gave the best fit - both visually and within the 5% level in slope. Integrating each fit along the line of sight gave the total absorption between Earth and a given point on the ellipsoid. An example is given in Figure 2.2 for the original and extrapolated absorption function toward SN 1181. Determining the absorption behavior through the ISM is still an uncer tain process; if we instead assume linear absorption and fit the Arenou function toward SN 1181 from R = 0 to 1 kpc, the total (foreground) extinction at 4
2Derror.m; see Appendix
24 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
SN 1181 absorption functions 3.5 r---.----,----,----,----,----,----,------,
_ Quadratic absorption function (Arenou et al. 1992) ___ ~~(r)~enti~IJ ~ bs6*e-e · 91872 " r ) + 2.0636 ------using 1 kpc cutoff ------
F igure 2.2 T he original Arenou eta!. (1992) absorption function Av(r) in the direction of SN 1181 and the exponential fi t generated within Matlab. The exponential form was integrated along the line of sight from Earth to the intersecting dust to give the total absorption in magnitudes for that location.
kpc is now approximat ely seven magnitudes instead of the 3 mag estimate from t he exponent ial fit in Figure 2.2. This may or may not reflect t he actual absorbing behavior through the dust layers, but we do not have line-of-sight "d umpiness" estimates at t hese extended distances. In almost all cases a lin ear fi t to the A v (r ) functions would provide a higher absorption estimate than an exponent ial fit t hat fl attens at large distances, so I have adopted the latt er in t his work. 2 While most of the 198 functions were of the form - ar , there were 26 functions that did increase with increasing r; the minimum Av was located very near r = 0 kpc in these cases. Applying an exponential fi t to these functions between r = 0 and R 0 or R' produced unrealistic absorption values (upwards of 150 magnitudes). Instead the function was fit from r = 0 to 4 kpc and capped at 5-15 magnitudes, using neighboring regions to determine a reasonable maximum value. The absorption along the line of sight from the supernova to t he dust , where no local constraints are available, also needed to be included in this
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model. I assumed a basic form for the absorption:
Av(r) rv a:rlzl <5o pc + .Brizl >5o pc (2.6)
I selected a: = 1.2 and .B = 0.4 mag/kpc; assuming that conditions near the Sun are similar to those near the SNR, these values produced extinction levels that were consistent with the Arenou functions for similar lines of sight. Us ing the Tycho SNR as an example, the maximum absorption value using the extrapolated Arenou fits was 7.2 magnitudes, compared with 4.5 magnitudes using (2.6). Converting the absorption into a fractional drop in flux and com bining with the previous calculations, this forms a relative probability of the detectability of the echo at a specific location. This model can be used used in two different ways. First, it can loop over the grid defined by the user's input range of right ascension and decli nation, providing a visual map of the relative probability for the entire area. The resolution of this map can be set by the user; a typical computation time for a 360 x 360-element grid was between four and five hours on my 700MHz laptop. The map was most useful for gauging where to set the outermost RA and declination boundaries for the coordinate list given the time of year and the observatory's location. Because the field of view at the telescope is only 36' x 36' using the mosaic CCD imager, it is important to identify which areas of the sky are, a priori, most likely to contain light echoes. Secondly, it can also perform these same calculations over a user-specified list of coordinates and generate the resulting probability for each point. The computational time was much shorter in this case (approximately twenty minutes for a given list). When finished, the coordinate list was sorted in order of decreasing probability.
2.3 Observations
To generate a coordinate list for the KPNO observations, we first obtained a list of previously targeted fields for each SNR from Armin Rest of the Super MACHO group (private communication, 10/2007). The list contained 4290 individual paintings, 0.3-0.5 degrees on a side and tiled continuously over the sky, and was used to choose fields based on IRIS (re-processed IRAS; Miville Deschenes & Lagache 2005) maps in the previous search strategy. In order to narrow this list down to a reasonable size to observe over three or four nights (approximately 200 entries), our probability script was run over the fields to find the relative predicted success associated with each pointing. Initially, the 200 highest-probability points were selected from each original list for observation. As a sanity check, the visual map was referenced to see where these points were in relation to the SNR and the Galactic plane.
26 L. Oaster Ill M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Because each of the Arenou absorption fits covered a range of 10 - 15° in galactic latitude and longitude, they produced a visible boundary in the prob ability map between adjacent cells. This boundary was most pronounced near the plane where the absorption tends to increase quickly, which meant groups of apparently high-probability locations were adjacent to low points. Assum ing a smoother boundary is actually present, we could potentially disregard useful fields if we simply take the highest relative-probability points from the top of the coordinate list. To compensate for the discontinuity, the closest low-probability points to each boundary were also included in the final list. (In future work, a linear smoothing process could be applied across adjacent fields.) Finally, this list was divided into smaller sections to be observed each night, and re-sorted to minimize the slewing time at the telescope. The results from our observations are given in Chapter 3.
27 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
CHAPTER3 ______~
L..-I______RESULTS
3.1 Model results
I generated a whole-sky probability map for each of the six historical super nova remnants, which is shown in Figure 3.1. These maps were then used to determine the field observation list for Cas A, Tycho, and SN 1181. We ob served in this region because the three SNR.s are projected on the sky so near each other. However, SN 1181 is so much older than the other two remnants that we expect its echoes to be fainter; no SN 1181 echoes have been detected to date. We will concentrate on Tycho and Cas A for the remainder of this work. The fields observed in October and December 2006 were chosen based on "bright" (higher-intensity) areas in 100-J.Lm IRIS (re-processed IRAS) maps of the Galactic plane (Miville-Deschenes & Lagache 2005). Beginning in September 2007 I added a second target list of coordinates based on the prob ability maps. Since Cas A and Tycho are in the same sky region, the two maps were overlaid and 46 points were chosen from the highest-probability areas. I then obtained from Armin Rest (CTIO, CfA) the list of 4290 pre-generated fields mentioned in Chapter 2, appropriately sized for the KPN0 4m Mosaic Imager and tiled over the Tycho/Cas A region. A probability was generated for each field, and from these I selected 178 to add to the list. A total of (46 + 178 =) 224 points were submitted to the observing queue for Oct./Dec. 2007 based on this model. In Figure 3.1 we can see two general trends among all the maps:
1. The Galactic plane is more highly favored relative to the rest of the sky.
28 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
2. The probability drops in the direction of the supernova, where the line of sight between us and the intersecting dust is the longest (i.e. behind the SNR).
Both of these trends arose after applying absorption to the model; this is discussed further in §3.3 regarding the effect of the interstellar extinction on the model's predictions.
3.2 Observational results
In all we observed 255 IRIS-based and 31 model-based fields at least twice, which is the minimum number of times required to difference the images and detect any echoes. Echoes were located from Cas A in December 2006, and twelve fields (six from each remnant) were confirmed with visible echo arclets after completing all the observing runs. These were reported by Rest et al. (2008), and an image from that article is shown in Figure 3.2. All points observed at least twice are plotted in Figures 3.4 and 3.3. It is clear that echoes were found in low-probability areas of the model reported here; none were located in the paintings submitted from this model. Table 3.1 contains all twelve echoes, their coordinates (by field ID), and their relative probabilities. The efficacy of this model is discussed in §3.4.
3.3 Absorption
The line-of-sight absorption illustrated in Figure 3.7 significantly lowered the relative probability for most of the paintings near the SNRs (see Table 3.1). Each absorption function was defined by Arenou et al. (1992) over an area 10°X10° in l and b (for b ~ 0) and 30°X30° for high-latitude (I bI > 60°) areas (Arenou et al. 1992). The consequence of defining the absorption over such coarse regions was that a single function could be applied to both the Galactic plane and the supernova remnant, when in reality it is likely that the absorption varies significantly within that 10° region (Figures 3.5 and 3.6). Most of the echoes were located in seemingly low-probability areas. We would expect the maximum level of absorption in the plane due to the exponentially decaying dust distribution with increasing Ib I, but three of the twelve echoes (#3024, #3325, and #3826) were found less than 0.5° off the plane. A second effect of applying this extinction method was mentioned ear lier: the entire Galactic plane is favored even in areas where the echoes would be back-scattered. Due to the long path length and inefficient scattering we don't expect to find light echoes so far away from the supernovae, and this
29 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Table 3.1 The twelve fields containing scattered-light echoes reported by \ ~est et al. (2008), with coordinates given in degrees. Probabilities P are "--assigned based on the associated supernova (a) including and (b) excluding the extinction along both paths. For example, in field #2116 its P of 0.11 was 0.4% of the maximum with absorption included, and its P of 405 was 2.6% of the maximum with absorption excluded- increasing 2.2% relative to the maximum.
Field ID RA Dec SNR Pa % ofmaxa pb % ofmaxb b.%b--a #2116 345.54 56.71 Cas A 0.11 0.4 405 2.6 2.2 #2729 348.23 64.77 Cas A 0.03 0.1 16 0.1 0.0 #3024 354.03 61.67 Cas A 5.4 18.0 9600 61.4 43.4 #3117 356.69 57.33 Cas A 0.05 0.2 83 0.5 0.3 '-)#3824 364.29 61.67 Cas A 2.3 7.8 1500 9.5 1.7 ~3826 364.21 62.91 Cas A 5.3 17.5 2200 13.9 -3.6 #3325 357.75 62.29 Tycho 9.6 42.3 5900 49.3 7.0 --""/#4022 366.88 60.43 Tycho 0.43 1.9 250 2.0 0.1 .c--')#:4430 372.95 65.39 Tycho 1.4 6.2 710 5.9 0.3 #4523 373.30 60.80 Tycho 1.8 8.0 630 5.2 -2.8 #4821 376.73 59.81 Tycho 0.86 3.8 200 1.7 -2.1 #5717 386.35 57.33 Tycho 0.91 4.0 52 0.4 -3.6
behavior is more accurately shown in the model with absorption excluded. If we consider Figures 3.7e and 3.7f (page 44), more echoes are found in "bright" areas in the non-absorption model. Our results are consistent with the suggestion that the presence of dust may be a stronger indicator of light echo location than its absorbing properties, such that the foreground extinction could be excluded (Armin Rest, 2008, priv. comm.). Relative to the maximum, seven of the twelve echo fields increased in probability when the foreground extinction was ignored- in the absence of a higher-resolution absorption model than Arenou et al. (1992), this version of the probability model may give better predictions for echo searches. Regarding the background (SN-dust) extinction, my estimate was 1.2 and 0.4 mag kpc-1 within/outside 50 pc of the plane, respectively. By definition this varies continuously over the sky and does not reproduce the expected "dumpiness" of the dust, and the maximum absorption from this method was checked for consistency against that from the foreground extinction. Because the foreground extinction would' not comprise much of the total extinction for paintings near the SNRs (providing 3% and 8% of the total Av toward Cas A and echo field #2116, respectively), and in light of the issues encountered with the Arenou-based absorption model, I felt it more instructive to compare the geometric effects against the total absorption than to view each absorption
·30 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
effect individually.
3.4 Efficacy of the survey strategies
From the total number of fields observed at least twice from each survey strat egy (IRIS and this model) and the number of echoes found, I have summarized the rates of success in comparison with those expected from using randomly placed fields near the Cas A and Tycho supernova remnants. The results are shown in Tables 3.2-3.6. Using the IRIS strategy, 5% of all multiply-observed fields contained echoes. In comparison, the best return for a random-pointing strategy was 0.6% for Cas A when searching within the area bounded by dust at Dnear = DsNR - 2000 ly (Green 2006). It appears that the IRIS strategy does reveal more echoes than that expected by chance. No echoes were found at the coordinates suggested from the model. It is possible that the low number of (multiply-observed) model-based fields is a factor in the lack of echo detection: with a total of (255 + 31 =) 286 fields observed and 12 of them containing echoes, this gives a total success rate of ex 4.2%. If this rate of success is applied to just the 31 model-based fields, we expect to find echoes in only one of them. However, given the placement of all the model-based fields with respect to the echo-containing IRIS fields (see Figure 3.4) it seems equally likely that the model suppresses areas that are actually good places to search for echoes. We cannot fully address my model's efficacy without a better estimation for the absorption filling factors within a grid square.
IRIS fields: 255 Model fields: 31 Echoes: 12 Success rate (IRIS): 4.7% Success rate (model): 0.0%
Table 3.2 The success rates of each survey strategy.
31 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Tycho Dust at SNRa Nearer dustb Radius on sky 3.2° 10.1° Area (sq. deg.) 32.2 321 FOV (sq. deg.) 0.36 0.36 No. of fields 89 890 No. of echoes 0 4 Success rate 0.00% 0.45%
Table 3.3 Estimated success rates if pointing randomly on the sky within the area bounded by intersecting dust at a) the same distance and b) 2000 ly closer than Tycho. Dust radii are from Green 2006.
Cas A Dust at SNRa Nearer dustb Radius on sky 1.80 6.3° Area (sq. deg.) 10 120 FOV (sq. deg.) 0.36 0.36 No. of fields 28 350 No. of echoes 0 2 Success rate 0.00% 0.58%
Table 3.4 Estimated success rates if pointing randomly on the sky within the area bounded by intersecting dust at a) the same distance and b) 2000 ly closer than Cas A. Dust radii are from Green 2006.
32 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Tycho Absorption No absorption < Pa > 3.4 3000 %of Pmax 15% 25% < pb > 1.8 1100 %of Pmax 8.0% 8.9%
Table 3.5 The average probability contained in the circular regions (a) and (b) as defined in Table 3.3. Comparison with the maximum P when including/ excluding the absorption is also shown.
Cas A Absorption No absorption < Pa > 0.41 6600 %of Pmax 1.4% 42% < pb > 0.99 1900 %of Pmax 3.3% 12%
Table 3.6 The average probability contained in the circular regions (a) and (b) as defined in Table 3.4. Comparison with the maximum P when includingjexcluding the absorption is also shown.
33 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
(a) SN 1006.
350 300 200 100 .. '" RA (deg) "'
(b) SN 1054 (Crab).
Figure 3.1 Probability maps for the six remnants, with the SNR shown in yellow. T he RA has been extended beyond 360° in some cases for visual clari ty.
34 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
(c) SN 1181.
.,, 400 ,.. 300 ,.. 100 .. ... RA (deg) ""
(d) Tycho's SN.
Figure 3.1 Probability maps for the six remnants, with the SNR shown in yellow. TheRA has been extended beyond 360° in some cases for visual clarity.
35 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
80 20
18 " ,.
12 ~ ~ 0 10 ~
3SO 300 200 1~0 100 '" RA(deg) "
(e) Kepler's SN.
eo ... 350 300 250 200 150 100 RA(deg)
(f) Cas A.
Figure 3.1 Probability maps for the six remnants, wi t h t he SNR shown in yell ow. The RA has been extended beyond 360° in some cases for visual clarity.
36 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
(a) Vector directions for all echoes. Each pointing contained several echo arclets, and t he inclination of the scattering dust causes some variation in the observed proper motion.
(b) The proper motion of each echo, averaged to one per fie ld.
Figure 3 .2 Tycho and Cas A echoes, from Fig. 2 of Rest et al. 2008.
37 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University
Tycho - Field Probabilities (relative) 80 31 .110
75 27.853
24.197 70 .. .. 20.740 t. 17.284 ~ 13.828 55 10.371
50 Ul15
40 0.002 ... 420 380 320 300 RA (deg) Cas A- Field Probabilities (relative) 80 75 70 20.740 65 ~oo 17.284 ~ ....:. 13.828 55 10.371 50 8.815 •• 3.459 40 0.002 440 420 340 300 RA(deg) Figure 3.3 All fi elds observed at least twice. IRAS-based fie lds are marked with crosses, model-based fi elds are circles, and those containing echoes are squares. The inner/outer circles represent intersecting dust at the same distance, and 2000 ly closer than, each supernova (Green 2006). All points are color-coded by their relative probability (scale at right), and the echoes have been highlighted according to their respective supernovae. 38 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University RA (deg) Figure 3 .4 Tycho and Cas A probability maps (both overlaid ) with all fields observed at least twice. T he IRIS-based fields are shown with white squares, while red circles are the fields from this model. 39 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University 80 60 " ... ...... ... 300 l50 200 100 .. ''" RA (deg) '" RA(...) Figure 3.5 The extrapolated absorption (in magnitudes) from Arenou et a!. (1992) applied to Cas A. For clarity, the RA has been extended beyond 360° in the top panel; echoes from Cas A are shown in the lower panel. Boundaries between adj acent absorption fun ctions are clearly visible, and this may not be an accurate representation of the true absorption effects near the supernova. 40 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University RA (cleg) Figure 3.6 T he extrapolated absorption (in magnitudes) from Arenou et a!. (1992) applied to Tycho. For clarity, the RA has been extended beyond 360° in the top panel; echoes from Tycho are shown in the lower panel. T he same artificial behavior from Figure 3.5 is seen here. 41 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University (a) Cas A: No absorption (b) Cas A: Absorption included Figure 3.7 The effect of omitting (top) and including (bottom) absorp tion on the relative probabilities. Echoes from Cas A are also plotted. 42 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University (c) Tycho: No absorption (d) Tycho: Absorption included Figure 3.7 The effect of omitting (top) and including (bottom) absorp tion on the relative probabilities. Echoes from Tycho are also plotted . 43 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University (e) Combined: No absorption (f) Combined: Absorption included Figure 3.7 The effect of omitting (top) and including (bottom) absorp tion on the relative probabilities. Echoes from Cas A and Tycho are also plotted. 44 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University CHAPTER 4------. lo1scuss10N & coNCLUSIONS The suggestion that light echoes from 1000-year-old supernovae might still be detectable was first made almost 70 years ago (Zwicky 1940), and current wide-field CCDs and difference imaging capabilities make the detection of such echoes a practical reality. The current problem is developing a reasonably efficient method of finding them. In the course of this work I generated six probability maps to predict where echoes from the Galactic SNRs would be located, and we were able to test two of them (Tycho and Cas A) against observations and compare the models' success rates with an IRIS-based survey strategy. In this chapter, I will discuss the results from Chapter 3 regarding the model's predictive capabilities and absorption effects. I will also describe some preliminary work on CO emission and light echo location that may prove useful for future surveys. 4.1 Extinction and the probability To determine the strength of the absorption effects on my model, I removed the extinction from the Cas A and Tycho models (§3.3) and generated new relative probabilities for all multiply-observed paintings. Seven of the twelve echoes increased in probability relative to the maximum, and the average prob ability in all four circular regions defined in §3.4 also increased. This section outlines how each echo field probability was altered when the absorption was added/excluded. For reference, Figure 4.1 (page 46) is Figure 3.3 with the echo IDs added. 45 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University 31 .110 75 27.853 24.1517 70 ··. 20.740 . . . 17.284 ~ - • • eoo ,. • • ~~: " ~ ( "1>6'0 ·•• • • ••• . . ~~ ~ .:· ~ 13.828 55 / • .. 10.371 50 5717 4821 8.815 45 d'q,po 3.459 00 ~~------~.20~------~------~-~------~-~------~------~------~ 0.002 320 300 RA (deg) Cas A- Field Probabilities (relative) 80 31.110 75 . ·3826 27.853 24.187 70 ··. 20.740 65 t. 17.284 .! 13.828 55 10.371 50 6.915 d"yo 45 3.459 co 40 0.002 440 420 400 300 300 RA (dog) Figure 4.1 Tycho and Cas A echo labels. Highlighted in blue are fields that in creased in P , while yellow-highlighted fields decreased in P when absorption was removed; see Figure 3.7(f) for the position of the echoes wi th respect to the Galactic plane. 46 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University Results for individual fields Removing the absorption caused echo #3024 to increase 43% in probability P relative to the maximum (see Table 3.1); this is one of the three echoes mentioned in §3.3 that was very near the Galactic plane. As for the other two, #3325 increased by 7% but #3826 decreased by 3.3% when the absorption was omitted. From the dust density at b = 0° we might have expected these three echoes to increase the most in P by removing the extinction; instead one of them decreased and the other increased only modestly. The largest gains in P among all the echoes after removing the extinction were seen in two of these fields, #3024 and #3325. In contrast, four of the echo fields were located in relatively low-probability areas regardless of the presence of extinction in the model (see Fig. 3.7). After removing the extinction and comparing with the maximum P: • #2729 did not change. • #3117 increased by 0.3% . • #4821 and #5717 decreased by 2% and 4%, respectively. Based on the model these fields have a low probability of containing echoes, and yet echoes were present. Finally, of the remaining five echoes, the probability for #4523 de creased by 2.8% and the rest increased by less than 2.5% in P after removing the extinction. Most of these were neither in the plane nor very far away, and on average they were the least affected by changing the absorption. Because both versions of the model produce low relative probabilities for these fields, there is no indication that the model would favor these locations for echoes. It would appear that the geometric effects alone would not have pre dicted the presence of all the echoes found during this search; however, the benefit of including the absorption in this model remains an open question. The average P over the two circular regions defined around each SN did in crease for both supernovae when removing the extinction (page 33), which might be an argument in favor of omitting the absorption at this time. We know from IRIS data (Miville-Deschenes & Lagache 2005) that the dust den sities can vary rapidly over scales of arcminutes on the sky, and the Arenou et al. (1992) absorption functions have averaged this "clumpy" behavior over each 10 x 10-degree element in their (l,b) grid. It could be that the present (non-clumpy) absorption model is sufficiently inaccurate that my model has better predictive capabilities without it. There have been earlier absorption models with similarly-sized regions (Neckel et al. 1980) and smaller-scale stud ies in specific regions of the sky (e.g. de Geus et al. 1989), but the study by Arenou et al. is the best available all-sky absorption model we could use in this work. Still we are lacking an adequate explanation for the persistently low 47 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University probabilities for the four echo fields mentioned above, which further illustrates the difficulty in predicting where light echoes will appear. 4.2 CO investigations After the echoes were located, Christine Wilson (McMaster University, priv. comm.) suggested that I check for any possible correlation between the lo cations of the twelve echoes and CO emission (v = 115.27 GHz) within the Galaxy. Radio observations of this molecule can be used to determine both the temperature and the velocity of the gas, and CO happens to be a good ''tracer" of interstellar dust (e.g. Lada et al. 1gg4; Alves et al. 1ggg). If significant CO emission was found in the same direction as an echo, the peak velocity in the line profile could be taken as the radial velocity v;. of the gas at that position. From Fich et al. 1g8g (based on Oort 1g27): SRo ) Vr = ( R- So sinlcosb (4.1) 2 2 2 1 2 R = (d cos b + R0 - 2R0 dcos bcos l) 1 (4.2) where S is the circular velocity of an object around the Galactic center and R and d are the object's distances from the Galactic center and the Sun. In this case, b is sufficiently close to zero such that cos b ~ 1 which is a valid assumption for the six historical supernovae. Using the Galactic rotation curve S(R) (Clemens 1g85) and adopting the IAU standard values of S 0 = 220 km/s and R0 = 8.5 kpc at the Sun's location (Kerr & Lynden-Bell 1g86), the distance d to the gas - and consequently the scattering dust, if associated with the gas - can be obtained. If we are observing within the solar circle (0° < l 48 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University Heyer et al. 1998) that contained the Galactic coordinates of ten of the twelve echoes. (Echoes #3117 and #5717 fell outside the range of b covered by the survey.) I then generated velocity line profiles for the ten echoes using the kvis routine within the program Karma. Figure 4.2 is the profile for echo #2729, and no emission is detected above the noise level. The other echoes' profiles were equally "empty," indicating no detectable excess CO at those locations. From this preliminary work, a distinct Vr cannot be obtained from the CO at any of the echo positions. After finding this result, I randomly selected ten non-echo, IRIS-based fields near Tycho and Cas A (within the Dnear annulus from §3.4) and examined them for CO emission. If there were molecular clouds along the line of sight to the intersecting dust, we might expect this to correlate with the foreground dust (absorption) and thus anti-correlate with light echo position. Four of the ten paintings showed clear CO emission, which could be statistically significant given that none of the twelve echo fields had any CO emission. The four line profiles, peak Yr, and approximate distances are shown in Figures 4.3-4.6. To estimate the distance to each CO-emitting cloud, I used (4.1) and (4.2) and assumed a flat rotation curve. These distances ranged from 3.8-7 kpc from the Sun; for comparison, Cas A and Tycho are estimated to be at distances of 3.5 and 3.1 kpc, respectively (Green 2006). Each of these clouds lie along the line of sight but behind the supernovae with the possible exception of the 3.8 kpc cloud near Cas A; if its circular velocity isn't well matched with the flat rotation curve approximation, its distance could be as low as 3.0 kpc for 8 = 210 km/s (for example). This could place the cloud in the foreground with respect to the supernova - at only 1° away from Cas A this would agree with the finding that the outburst was heavily obscured. There appears to be some potential here for finding clouds along the path between us and the intersecting dust, and the distance estimates could be used to eliminate (current) IRIS paintings if the gas/dust cloud is positioned well behind the SNR. If the CO emission does relate to light echo position for intervening clouds, then we could use this information when choosing new paintings in future surveys as well. 4.3 Conclusions The major results of this work can be summarized as follows: 1. All twelve echoes located from observations were in low-probability fields based on the model developed for this thesis. 2. When the extinction was removed, the majority of echo fields increased 49 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University in P along with the average relative probability near each supernova. This may indicate that our absorption model is not sufficiently accurate for this application. 3. Other than the lack of CO emission, there does not appear to be a singular trend connecting all twelve echoes. They were found both in the Galactic plane and several degrees away from it, but the echo fields that increased the most in P when removing the extinction were less than 0.5° off the plane. From observations and the non-absorption version of my model, it appears that the best places to search are within b ~ 2.5° of the Galactic plane and/or within the Dnear annulus around each supernova. Examining Figure 4.1 nearly all of the echoes fit this pattern for both Cas A and Tycho; this leaves #5717 and #4821 unaccounted for by either search pattern. This could give us some new options for our surveys: we could a) search a strip within 2.5° of the Galactic plane and add other paintings within the larger boundary around each supernova, or b) combine these search regions with the IRIS data, effectively weighting the fields by how much dust is present at each pointing. Either option would be useful for finding new echoes from the other four historical supernovae or for re-surveying the Tycho/Cas A region to further investigate the dust structure (e.g. Sugerman et al. 2005; Rest et al. 2008). Future work Based on the early CO results, it may be worthwhile to investigate the CO emission and cloud distances in the 274 fields observed that didn't contain echoes. Some of the paintings fall outside the range covered by the Heyer survey (1998), but there may still be enough fields to establish a CO-light echo correlation. The only other historical supernova covered in the Heyer survey is SN 1181, while the all-sky CO survey by Dame et al. (2001) captures all of the supernova remnants except for SN 1006 at l = 325.6°, b = 14.5°. At this time there does not appear to be a CO survey that would cover SN 1006. Regardless of which data are used, the choice of rotation curve will affect the cloud distance estimates: in the Cas A pointing mentioned earlier, a 5% drop in 8 caused a 24% decrease in distance. An error of that size could place a foreground cloud in the background (or vice versa), so the rotation curve should be carefully applied when obtaining distance estimates. Additionally, following Krause et al. (2008) a similar method of detect ing optical counterparts to previously-located IR echoes could be used here as a survey strategy. There appear to be Spitzer maps in the data archives that 50 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University cover all six Galactic SNR positions, and this could complement the IRIS based method for finding echoes. At present, the IRIS strategy has been the most successful with a 5% echo detection rate and is more efficient than searching randomly-placed fields around each supernova. Ifmolecular gas emission is sufficiently anti-correlated with echo position, it may be more effective for future surveys to combine the CO data with the IRIS maps (as opposed to this model) and search in the two zones mentioned earlier. It would be useful to re-examine my model's predic tive capabilities if an improved all-sky absorption scheme becomes available. In the meantime, now that we have definite echo positions we can go back to obtain spectra- and classify a supernova using light observed by Tycho Brahe in 1572 and our research group in 2006. 51 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University Glon: 112.675° Glat: +3.8999998° g E"' .,... s"'p., ~ Ill .,Ill Q ~ .!:P... -0.5 o:l 0 -50 -100 Velocity (km/s) Figure 4.2 The CO line profile for echo #2729. The rms noise level of the brightness temperature (i.e. intensity) was 0.16 K; the intensity would have to exceed 0.5 K (3a) for a given velocity to be considered a detection. For comparison, the strongest signals in the data peaked around 20 K. This line profile is representative of those generated at the other nine (available) echo positions. 52 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University Glon: 111.385" Glat: -2.9199996" 0 -50 -100 Velocity (km/s) Figure 4.3 CO emission in a non-echo field near Cas A. The rms noise is 0.16 K; the peak Vr is -39.4 km/s (a 31a detection) and the distance to the CO-emitting cloud is 3.8 kpc. Glon: 121.27° Glat: :t.8·064999783" g 1.5 " 1.0 ~... "p. 13 E-<" " ~bO ·t:: -0.5 Ill -l.oL--_.__...... _...__....-..-!o~--'---'--...... _-'---_-*5""o_.____...._.....__.....___....,1:1::-oo,..-...... _~__.-_,___~150 Velocity (km/s) Figure 4.4 CO emission in a non-echo field near Tycho. The rms noise is 0.16 K; the peak Vr is -51.8 km/s (a 5a detection) and the distance to the CO-emitting cloud is 4.8 kpc. 53 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University Gion: 120.11' Glat: +3.2899999" 4 g 3 - 1 ~~~--~--~~0~_.--~--~~--~-5~0--._~--~--~--~1~00~~--~~--~---1~50 Velocity (km/s) Figure 4.5 CO emission in a non-echo field near Tycho. The rms noise is 0.16 K; the peak Vr is -69.9 km/s (an lla detection) and the distance to the CO-emitting cloud is 7.0 kpc. Gion: 120.48" Giat: -1.105' 4 g 3 0 -50 -100 Velocity (km/s) Figure 4.6 CO emission in a non-echo field near Tycho. The rms noise is 0.16 K; the peak Vr is -56.7 km/s (a 14a detection) and the distance to the CO-emitting cloud is 5.4 kpc. 54 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University APPENDIX A______-.. I.______CODE A. I Derror.m %Takes in two functions funct1 (assumed to be quadratic) % and funct2 (the exponential fit), and the peak of the %quadratic function. Calculates the first derivative (slope) %of each, then finds the percent difference of the two slopes. function percent = Derror(funct1, funct2, Peak) Dfunct1 = diff(funct1); Dfunct2 = diff(funct2); percent= 100 * abs(Dfunct1-Dfunct2)/(0.6*(Dfunct1+Dfunct2)); figure; ezplot(percent, [0 Peak]); 55 L. Gaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University A.2 Cas_prob_commands.m % Cas A parameters % % This script finds symbolic expressions for x,y,z in terms of 1 and b, and then % calculates the total effect on flux based on each point's location in space. %A plot is generated at the end, using RA and dec instead of 1 and b. % Define symbolic and numeric variables. syms 1 b x y z D = 11410; A = 5868.5; B = 1375.59705219225; % in ly long= 111.735*(pi/180); lat = -2.13*(pi/180); t c 327; %Set up the three equations; reduce it down to an equation with x,l,b only y = X*tan(l); z = sqrt(xA2 + yA2)*tan(b); eq = ( ((x*cos(lat)-z*sin(lat}-D/2}/A)A2 + (y/B)A2 + ((x*sin(lat)+z*cos(lat))/B)A2 ) - 1; x = solve(eq,x); root = 3; [x,hov] = simple(simplify(x(root))); %== If using GRID: %==a====== [RA,Dec] = meshgrid(0:360, -90:.5:90); [lL,bB] = sky2gal(RA,Dec); el = lL; bee = bB; lL = (lL ·* (pi/180)) - long; bB = bB ·* (pi/180); % Initialize the arrays so they don't grow while in the loop. Prob = zeros(size(lL)); R = zeros(size(lL)); %Calculate the probability over the GRID (minus absorption): % substitute the numeric lL, bB values in for the (symbolic) 1 and b. tic for i = 1:size(lL,1) for k =1:size(lL,2) i,k 1 = lL(i,k); b = bB(i,k); X= subs(x); Y = X*tan(l); Z = (sqrt(XA2 + YA2))*tan(b); R(i,k) = (XA2 + YA2 + ZA2)A(1/2); % Nov find original x,y,z (origin at center of ellipsoid) xi = X*cos(lat) - Z*sin(lat) - D/2; y1 = Y; z1 = X*sin(lat) + Z*cos(lat); dotcos = ((0.5*D)-x1)/(sqrt((x1-(0.5*D))A2 + y1A2 + z1A2)); dist = A*(1-(D/(2*A))A2) I (1+(D/(2*A))*(-dotcos)); scatterfn =(0.64/(1.36-1.2*dotcos)A(3/2))*pi*0.00016967; DistSq = distA(-2); 56 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University ScHeight = exp(-abs(Z/(50*3.26))); Prob(i,k) = scatterfn*DistSq*ScHeight*1E13; % scale up to have "readable" values end end R = (R./3.26) ./ 1000; toe %======%== If using POINTS: %======[pointlL,pointbB] = sky2gal(pointRA,pointDec); el = pointlL; bee = pointbB; pointlL = (pointlL ·* (pi/180)) - long; pointbB = pointbB ·* (pi/180); % Initialize the arrays so they don't grow while in the loop. Prob = zeros(size(pointlL)); R = zeros(size(pointlL)); % Calculate probability for individual POINTINGS: for i = 1:length(pointlL) i 1 = pointlL(i); b = pointbB(i); X(i) = subs(x); end X= X'; Y = X.*tan(pointlL); Z = (sqrt(X.A2 + Y.A2)).*tan(pointbB); R = (X.A2 + v.A2 + z.A2).A(1/2); % Now find original x,y,z (origin at center of ellipsoid) x1 = X.*cos(lat) - Z.*sin(lat) - D/2; y1 = Y; z1 = X.*sin(lat) + Z.*cos(lat); dotcos = ((0.5*D)-x1)./(sqrt((x1-(0.5*D)).A2 + y1.A2 + z1.A2)); dist = A*(1-(D./(2*A))A2) ./ (1+(D/(2*A)).*(-dotcos)); scatterfn =(0.64./(1.36-1.2.*dotcos).A(3/2))*pi*0.00016967; DistSq = dist.A(-2); ScHeight = exp(-abs(Z./(50*3.26))); Prob = scatterfn.*DistSq.*ScHeight; R = (R./3.26) ./ 1000; %/ll/lll//ll/lll/lll//ll/ll/////l///l//l/1//l///l////l///l///l///l//l///l///l///ll/l//// %/ll/lll/lll/1/l//l/1///l//ll//ll//ll//ll/l/l/lll/ll/l/lll/lll/l////////ll/l//////////l/ absBank_el 57 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University syms r; AA = (1 - 0.27815*exp(-8.8765*r)) - 0.72434; AB = (1- 0.25724*exp(-14.466*r)) - 0.74606; AC = (1- 0.26989*exp(-11.413*r)) - 0.73263; AD = (1 - 0.3332*exp(-14.858*r)) - 0.67016; AE = (1- 0.268*exp(-19.206*r)) - 0.74431; AF = (1 - 0.42643*exp(-19.867*r)) - 0.57738; AG = (1 - 0.22961*exp(-11.76*r)) - 0.77244; AH = (1 - 0.27566*exp(-13.329*r)) - 0.72669; AI= (1- 0.19389*exp(-8.1016*r)) - 0.80813; AJ = (1- 0.23815*exp(-7.5045*r)) - 0.76352; AK = (1 - 0.26106*exp(-8.8608*r)) - 0.74081; AL = (1 - 0.23284*exp(-11.051*r)) - 0.76914; BA = (1 - 0.28467*exp(-10.621*r)) - 0.71717; BB = (1- 0.24689*exp(-8.673*r)) - 0.7656; BC = (1- 0.102*exp(-20.227*r)) - 0.89848; BD = (1- 0.16441*exp(-6.9797*r)) - 0.83683; BE = (1 - 0.60483*exp(-3.016*r)) - 0.49866; BF = (1- 0.22094*exp(-13.564*r)) - 0.78087; BG = (1 - 0.2669*exp(-12.306*r)) - 0.73628; BH = (1 - 0.31311*exp(-11.29*r)) - 0.6893; BI = (1 - 0.16921*exp(-11.232*r)) - 0.83227; BJ = (1- 0.22862*exp(-8.923*r)) - 0.77282; CA = (1- 0.27967*exp(-7.7966*r)) - 0.72243; CB = (1 - 0.20477*exp(-8.068*r)) - 0.79704; CC = (1- 0.21666*exp(-4.5305*r)) - 0.78493; CD= (1- 0.20709*exp(-10.316*r))- 0.79475; CE = 29.396*exp(0.49417*r) - 35.762; Y. limited to 10 CF = 13.771*exp(0.49371*r) - 16.736; Y. limited to 10 CG = (1 - 0.33363*exp(-10.631*r)) - 0.66915; CH = (1- 0.28087*exp(-7.5632*r)) - 0.72122; CI = (1- 0.24896*exp(-8.542*r)) - 0.76361; CJ = (1- 0.26382*exp(-10.698*r)) - 0.74794; CK = (1 - 0.2277*exp(-11.686*r)) - 0.77401; DA = (1 - 0.68524*exp(-6.2342*r)) - 0.41871; DB = (1 - 0.63612*exp(-6.6133*r)) - 0.36903; DC= 3.3329*exp(0.18861*r) - 3.3396; DD = (1 - 0.54737*exp(-1.1865*r)) - 0.46864; DE = (1 - 0.32274*exp(-6.9628*r)) - 0.68012; DF = (1- 0.33277*exp(-3.7626*r)) - 0.66996; DG = 6.0846*exp(0.43297*r) - 6.8437; Y. limited to 10 DH = 24.956*exp(0.47703*r) - 29.635; Y. limited to 10 DI = 8.1611*exp(0.2164*r) - 8.1908; DJ = (1 - 0.39629*exp(-3.0124*r)) - 0.60684; DK = (1- 0.19881*exp(-6.7156*r)) - 0.80318; DL = (1 - 0.23685*exp(-6.2526*r)) - 0.76606; DM = (1 - 0.45637*exp(-4.8004*r)) - 0.6482; DN = (1 - 0.69076*exp(-2.6375*r)) - 0.41525; DO = (1 - 0.39436*exp(-5.7965*r)) - 0.60954; DP = (1 - 0.39445*exp(-4.4744*r)) - 0.60939; DQ = (1 - 0.37912*exp(-4.249*r)) - 0.62448; EA = (1 - 0.96233*exp(-2.9672*r)) - 0.047951; EB = (1 - 1.3176*exp(-2.6688*r)) + 0.30925; EC = (1 - 1.2305*exp(-2.7213*r)) + 0.17178; ED= (1- 1.7612*exp(-1.1675*r)) + 0.73723; EE = 6.6694*exp(0.26507*r) - 6.7035; 58 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University EF = (1 - 2.5534*exp(-0.3583*r)) + 1.4075; EG = (1- 0.77565*exp(-2.1288*r)) - 0.33461; EH = 1.9159*exp(0.40291*r) - 2.0978; EI = (1 - 1.3258*exp(-1.4129*r)) + 0.31417; EJ = (1 - 0.95392*exp(-2.4081*r)) - 0.1372; %Cas A EK = (1 - 1.4771*exp(-0.88205*r)) + 0.2703; % Cas B EL = (1- 1.0867*exp(-1.7961*r)) + 0.0061; % Cas C EM = (1 - 11.111*exp(-0.13921*r)) + 10.1; EN = (1 - 1.6328*exp(-1.2777*r)) + 0.61664; EO = 8.3506*exp(0.29422*r) - 8.9182; EP = (1- 3.1178*exp(-2.4373*r)) + 2.0898; % Crab A EQ = (1 - 1.3209*exp(-4.0034*r)) + 0.092327; % Crab SNR ER = (1- 0.77073*exp(-1.099*r)) - 0.23446; ES = 2.3044*exp(0.38007*r) - 2.3065; % Crab B ET = 0.6874*exp(1.0104*r) - 0.70777; %limited to 15 EU = (1 - 0.66806*exp(-1.5862*r)) - 0.39916; EV = (1 - 0.23103*exp(-2.022*r)) - 0.77114; EW = (1 - 0.43474*exp(-2.3151*r)) - 0.67963; EX= (1 - 0.50101*exp(-2.1938*r)) - 0.51193; EY = (1- 0.74757*exp(-1.3869*r)) - 0.28229; EZ = (1 - 0.85898*exp(-1.1071*r)) - 0.1493; EAA = (1 - 0.64413*exp(-2.4792*r)) - 0.36191; EAB = (1 - 1.3216*exp(-2.1367*r)) + 0.30952; EAC = (1 - 1.2812*exp(-2.7382*r)) + 0.26989; EAD = (1 - 0.8656*exp(-2.6803*r)) - 0.14171; EAE = (1 - 0.75604*exp(-2.5858*r)) - 0.25157; EAF = (1 - 0.84599*exp(-2.5931*r)) - 0.16269; EAG = (1 - 0.67312*exp(-2.388*r)) - 0.35551; EAH = (1 - 0.51911*exp(-2.6089*r)) - 0.50725; FA = (1 - 2.0933*exp(-1.4142*r)) + 1.0662; FB = (1 - 3.2771*exp(-1.0737*r)) + 2.2417; FC = (1- 3.8368*exp(-1.2334*r)) + 2.7988; FD = (1- 20.547*exp(-0.12671*r)) + 19.526; FE = (1 - 2.6603*exp(-0.96557*r)) + 1.554; FF = (1 - 14.166*exp(-0.09608*r)) + 13.16; FG = (1- 4.1224•exp(-0.46423*r)) + 3.0946; FH = (1 - 4.0328*exp(-0.60055*r)) + 2.9988; FI = (1- 8.8145*exp(-0.17676*r)) + 7.7966; FJ = (1 - 2.648*exp(-0.9908*r)) + 1.6073; %Cas D FK = (1 - 3.6242*exp(-0.97264*r)) + 2.6115; %CasE FL = (1 - 3.7953*exp(-0.71307*r)) + 2.7879; % Cas SNR FM = (1 - 3.0776*exp(-1.0075*r)) + 2.0635; % Cas F FN = (1 - 3.096*exp(-0.91872*r)) + 2.0636; FO = (1 - 3.0452*exp(-1.1822*r)) + 1.8233; FP = (1- 2.4258*exp(-1.7942*r)) + 1.1427; % Crab C FQ = (1 - 2.5683*exp(-0.97622*r)) + 1.3366; % Crab D FR = (1 - 2.39978*exp(-0.91661*r)) + 1.0713; % Crab E FS = (1 - 3.4601*exp(-0.56353*r)) + 2.2866; % Crab F FT = (1 - 3.0264*exp(-0.36454*r)) + 2.0201; % Crab G FU = (1 - 2.3333*exp(-0.54874*r)) + 1.3255; %Crab H FV = (1- 6.5464*exp(-0.14666*r)) + 4.806; % Crab I FW = (1- 1.169*exp(-0.77386*r)) + 0.015129; FX = (1- 1.1634*exp(-0.7915*r)) - 0.11218; FY = (1- 0.79264*exp(-1.0817*r)) - 0.39402; FZ = (1 - 1.9299*exp(-0.5051*r)) + 0.37517; FAA = 18.694*exp(0.065152*r) - 20.134; FAB = (1 - 1.5142*exp(-0.50711*r)) + 0.4822; FAC = -42.434*exp(-0.017656*r) + 42.344; FAD = (1 - 1.4683*exp(-0.80912*r)) + 0.45351; FAE = (1- 2.2968*exp(-0.74761*r)) + 1.2188; FAF = (1- 2.1277*exp(-1.0804*r)) + 0.89849; 59 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University FAFF = (1 - 1.4104*exp(-1.0237*r)) + 0.28372; FAH = (1 - 2.2306*exp(-0.89604*r)) + 1.2079; FA! = (1 - 2.1495*exp(-1.1767*r)) + 1.1302; FAJ = (1 - 2.5927*exp(-0.9295*r)) + 1.3675; GA = (1 - 1.4718*exp(-2.245*r)) + 0.41986; GB = 29.064*exp(0.44457*r) - 33.192; %limited to 10 GC = (1 - 1.2369*exp(-3.3972*r)) + 0.22418; GD = (1 - 0.72243*exp(-3.2061*r)) - 0.28074; GE = (1 - 1.4808*exp(-1.0734*r)) + 0.469; GF = (1 - 0.39435*exp(-2.8354*r)) - 0.60748; GG = (1 - 0.41084*exp(-1.5289*r)) - 0.66972; GH = 37.597*exp(0.037196*r) - 38.067; GI = (1 - 2.242*exp(-2.1661*r)) + 1.2222; GJ = (1- 3.857*exp(-0.71273*r)) + 2.825; GK = 13.552*exp(0.030687*r) - 11.946; GL = (1 - 1.8749*exp(-2.5255*r)) + 0.65924; GM = (1 - 6.5747*exp(-0.39182*r)) + 5.5709; :,( Crab J GN = (1 - 2.4817*exp(-0.82867*r)) + 1.4814; %Crab K GO= (1 - 0.7749*exp(-1.8609*r)) - 0.40681; %Crab L GP = (1 - 0.52117*exp(-1.8627*r)) - 0.53706; %Crab M GQ = (1- 0.31773*exp(-1.2598*r)) - 0.75359; %Crab N GR = (1 - 0.3551*exp(-2.3009*r)) - 0.65063; GS = (1 - 0.68228*exp(-3.1128*r)) - 0.32147; GT = (1 - 0.57981*exp(-2.9475*r)) - 0.55849; GU = (1 - 0.63436*exp(-1.5369*r)) - 0.37027; GV = (1 - 0.4843*exp(-1.9179*r)) - 0.54759; GW = (1- 0.50638*exp(-1.7661*r)) - 0.47143; GX = (1 - 0.53098*exp(-2.3216*r)) - 0.49419; GY = (1 - 0.68808*exp(-2.2609*r)) - 0.3544; GZ = (1- 0.93749*exp(-2.1485*r)) - 0.11886; GAA = (1 - 0.73726*exp(-1.8596*r)) - 0.29461; GAB= (1 - 1.2671*exp(-2.1761*r)) + 0.10521; GAC = (1 - 3.9904*exp(-0.92541*r)) + 2.9505; GAD = (1 - 1.3519*exp(-2.0984*r)) + 0.34058; HA = (1 - 1.3158*exp(-5.1593*r)) + 0.306; HB = (1 - 0.97356*exp(-4.9669*r)) - 0.032653; HC = (1 - 0.64949*exp(-1.7113*r)) - 0.3437; HD = (1- 0.15847*exp(-5.7262*r)) - 0.84246; HE = 1.221*exp(0.24338*r) - 1.2518; HF = (1 - 0.033698*exp(-12.551*r)) - 0.96642; HG = (1 - 0.21716*exp(-2.8097*r)) - 0.78401; HH = 33.654*exp(0.48322*r) - 42.744; % limited to 10 HI= 3.7866*exp(0.49922*r) - 4.689; % limited to 10 HJ = (1- 0.25408*exp(-9.6037*r)) - 0.74691; HK = (1 - 0.38767*exp(-3.933*r)) - 0.61555; HL = (1 - 0.69026*exp(-2.5189*r)) - 0.31503; HM = (1 - 0.43231*exp(-3.9775*r)) - 0.5711; HN = (1 - 0.40397*exp(-5.3462*r)) - 0.59965; HO = (1 - 0.47908*exp(-5.9554*r)) - 0.52506; HP = (1 - 1.3094*exp(-4.4307*r)) + 0.29911; IA = (1 - 0.46317*exp(-6.9056*r)) - 0.53992; IB = (1 - .10096*exp(-17.42*r)) - 0.89947; IC = (1 - 0.14194*exp(-13.193*r)) - 0.85912; ID = (1 - .061094*exp(-23.557*r)) - 0.93928; IE= 3.163*exp(0.5182*r) - 3.9904; %limited to 10 IF = (1 - .096077*exp(-16.872*r)) - 0.90438; IG = 12.203*exp(.50273*r) - 15.023; % limited to 10 IH = (1 - 0.045509*exp(-29.498*r)) - 0.95471; II= (1- 0.22433*exp(-7.4193*r)) - 0.77708; 60 L. Oa.ster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University IJ = (1 - 0.35303*exp(-8.2944•r)) - 0.64901; IK = (1 - 0.49764*exp(-6.1966•r)) - 0.50651; IL = 10.049*exp(0.29433*r) - 10.331; %limited to 10 JA = (1 - 0.078326*exp(-18.681*r)) - 0.92197; JB = (1- 0.19227•exp(-12.66*r)) - 0.80867; JC = (1 - 0.085971*exp(-16.658*r)) - 0.91431; JD = (1 - .012274*exp(-3.795*r)) - 0.9877; JE = (1 - 0.21957*exp(-6.832*r)) - 0.78158; JF = 11.806*exp(0.47696*r) - 14.312; % limited to 5 JG = (1- 0.028755*exp(-20.766*r)) - 0.97132; JH = (1- 0.25816*exp(-15.116•r)) - 0.74279; JI = (1- 1.1416•exp(-1.1197*r)) + 0.13591; KA = (1 - 0.61577*exp(-1.2192•r)) - 0.38809; KB = 5.1994•exp(0.35641•r) - 5.492; KC = (1- 0.18352•exp(-6.4534•r)) - 0.81747; KD = 13.87*exp(0.49818•r) - 16.986; %limited to 5 KE = (1 - 0.082878*exp(-9.6529•r)) - 0.91626; KF = (1- 0.12424*exp(-10.705•r)) - 0.87609; KG = (1 - 0.081064*exp(-15.492*r)) - 0.9192; KH = (1 - 0.068801*exp(-15.307•r)) - 0.93162; KI = (1- 0.13075*exp(-4.3712•r)) - 0.86968; KJ = (34.521*exp(0.49936*r)) - 42.35; X limited to 5 KK = (1 - 0.14756•exp(-5.2949•r)) - 0.85291; KL = (1 - 0.050452*exp(-16.069•r)) - 0.9497; absrp = zeros(size(el)); for j = 1:size(el,1) for k = 1:size(el,2) j,k if bee(j,k) >= -90 &t bee(j,k) < -60 if el(j,k) >= 0 &t el(j,k) < 29 absrp(j,k) = double(int(AA,O,R(j,k))); elseif el(j,k) >= 29 && el(j,k) <57 absrp(j,k) = double(int(AB,O,R(j,k))); elseif el(j,k) >= 57 && el(j,k) < 85 absrp(j,k) = double(int(AC,O,R(j,k))); elseif el(j,k) >= 85 &t el(j,k) < 110 absrp(j,k) = double(int(AD,O,R(j,k))); elseif el(j,k) >= 110 && el(j,k) < 150 absrp(j,k) = double(int(AE,O,R(j,k))); elseif el(j,k) >= 150 &t el(j,k) < 180 absrp(j,k) = double(int(AF,O,R(j,k))); elseif el(j,k) >= 180 && el(j,k) < 210 absrp(j,k) = double(int(AG,O,R(j,k))); elseif el(j,k) >= 210 &t el(j,k) < 240 absrp(j,k) = double(int(AH,O,R(j,k))); elseif el(j,k) >= 240 &t el(j,k) < 270 absrp(j,k) = double(int(AI,O,R(j,k))); elseif el(j,k) >= 270 &t el(j,k) < 300 absrp(j,k) = double(int(AJ,O,R(j,k))); elseif el(j,k) >= 300 && el(j,k) < 330 absrp(j,k) = double(int(AK,O,R(j,k))); else absrp(j,k) = double(int(AL,O,R(j,k))); end 61 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University elseif bee(j,k) >= -60 && bee(j,k) < -45 if el(j,k) >= 0 && el(j,k) < 30 absrp(j,k) = double(int(BA,O,R(j,k))); elseif el(j,k) >= 30 && el(j,k) < 60 absrp(j,k) = double(int(BB,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 110 absrp(j,k) = double(int(BC,O,R(j,k))); elseif el(j,k) >= 110 && el(j,k) < 180 absrp(j,k) = double(int(BD,O,R(j,k))); elseif el(j,k) >= 180 && el(j,k) < 210 absrp(j,k) = double(int(BE,O,R(j,k))); elseif el(j,k) >= 210 && el(j,k) < 240 absrp(j,k) = double(int(BF,O,R(j,k))); elseif el(j,k) >= 240 && el(j,k) < 270 absrp(j,k) = double(int(BG,O,R(j,k))); elseif el(j,k) >= 270 && el(j,k) < 300 absrp(j,k) = double(int(BH,O,R(j,k))); elseif el(j,k) >= 300 && el(j,k) < 330 absrp(j,k) = double(int(BI,O,R(j,k))); else absrp(j,k) = double(int(BJ,O,R(j,k))); end elseif bee(j,k) >= -45 && bee(j,k) < -30 if el(j,k) >= 0 && el(j,k) < 30 absrp(j,k) = double(int(CA,O,R(j,k))); elseif el(j,k) >= 30 && el(j,k) < 60 absrp(j,k) = double(int(CB,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 90 absrp(j,k) = double(int(CC,O,R(j,k))); elseif el(j,k) >= 90 && el(j,k) < 120 absrp(j,k) = double(int(CD,O,R(j,k))); elseif el(j,k) >= 120 && el(j,k) < 160 absrp(j,k) = double(int(CE,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 160 && el(j,k) < 200 absrp(j,k) = double(int(CF,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 200 && el(j,k) < 235 absrp(j,k) = double(int(CG,O,R(j,k))); elseif el(j,k) >= 235 && el(j,k) < 265 absrp(j,k) = double(int(CH,O,R(j,k))); elseif el(j,k) >= 265 && el(j,k) < 300 absrp(j,k) = double(int(CI,O,R(j,k))); elseif el(j,k) >= 300 && el(j,k) < 330 absrp(j,k) = double(int(CJ,O,R(j,k))); else absrp(j,k) = double(int(CK,O,R(j,k))); end elseif bee(j,k) >= -30 && bee(j,k) < -15 if el(j,k) >= 0 && el(j,k) < 20 62 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University absrp(j,k) = double(int(DA,O,R(j,k))); elseif el(j,k) >= 20 && el(j,k) < 40 absrp(j,k) = double(int(DB,O,R(j,k))); elseif el(j,k) >= 40 && el(j,k) < 80 absrp(j,k) = double(int(DC,O,R(j,k))); elseif el(j,k) >= 80 && el(j,k) < 100 absrp(j,k) = double(int(DD,O,R(j,k))); elseif el(j,k) >= 100 && el(j,k) < 120 absrp(j,k) = double(int(DE,O,R(j,k))); elseif el(j,k) >= 120 && el(j,k) < 140 absrp(j,k) = double(int(DF,O,R(j,k))); elseif el(j,k) >= 140 && el(j,k) < 160 absrp(j,k) = double(int(DG,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 160 && el(j,k) < 180 absrp(j,k) = double(int(DH,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 180 && el(j,k) < 200 absrp(j,k) = double(int(DI,O,R(j,k))); elseif el(j,k) >= 200 && el(j,k) < 220 absrp(j,k) = double(int(DJ,O,R(j,k))); elseif el(j,k) >= 220 && el(j,k) < 240 absrp(j,k) = double(int(DK,O,R(j,k))); elseif el(j,k) >= 240 && el(j,k) < 260 absrp(j,k) = double(int(DL,O,R(j,k))); elseif el(j,k) >= 260 && el(j,k) < 280 absrp(j,k) = double(int(DM,O,R(j,k))); elseif el(j,k) >= 280 && el(j,k) < 300 absrp(j,k) = double(int(DN,O,R(j,k))); elseif el(j,k) >= 300 && el(j,k) < 320 absrp(j,k) = double(int(DO,O,R(j,k))); elseif el(j,k) >= 320 && el(j,k) < 340 absrp(j,k) = double(int(DP,O,R(j,k))); else absrp(j,k) = double(int(DQ,O,R(j,k))); end elseif bee(j,k) >= -15 && bee(j,k) < -5 if el(j,k) >= 0 && el(j,k) < 10 absrp(j,k) = double(int(EA,O,R(j,k))); elseif el(j,k) >= 10 && el(j,k) < 20 absrp(j,k) = double(int(EB,O,R(j,k))); elseif el(j,k) >= 20 && el(j,k) < 30 absrp{j,k) = double(int(EC,O,R(j,k))); elseif el(j,k) >= 30 && el(j,k) < 40 absrp(j,k) = double(int(ED,O,R(j,k))); elseif el(j,k) >= 40 && el(j,k) <50 absrp(j,k) = double(int(EE,O,R(j,k))); elseif el(j,k) >=50 && el(j,k) < 60 absrp(j,k) = double(int(EF,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 80 absrp(j,k) = double(int(EG,O,R(j,k))); elseif el(j,k) >= 80 && el(j,k) < 90 absrp(j,k) = double(int(EH,O,R(j,k))); elseif el(j,k) >= 90 && el(j,k) < 100 absrp(j,k) = double(int(EI,O,R(j,k))); 63 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University elseif el(j,k) >= 100 tt el(j,k) < 110 absrp(j,k) = double(int(EJ,O,R(j,k))); elseif el(j,k) >= 110 tt el(j,k) < 120 absrp(j,k) = double(int(EK,O,R(j,k))); elseif el(j,k) >= 120 tt el(j,k) < 130 absrp(j,k) = double(int(EL,O,R(j,k))); elseif el(j,k) >= 130 tt el(j,k) < 140 absrp(j,k) = double(int(EM,O,R(j,k))); elseif el(j,k) >= 140 tt el(j,k) < 150 absrp(j,k) = double(int(EN,O,R(j,k))); elseif el(j,k) >= 150 tt el(j,k) < 160 absrp(j,k) = double(int(EO,O,R(j,k))); elseif el(j,k) >= 160 tt el(j,k) < 180 absrp(j,k) = double(int(EP,O,R(j,k))); elseif el(j,k) >= 180 tt el(j,k) < 190 absrp(j,k) = double(int(EQ,O,R(j,k))); elseif el(j,k) >= 190 tt el(j,k) < 200 absrp(j,k) = double(int(ER,O,R(j,k))); elseif el(j,k) >= 200 tt el(j,k) < 210 absrp(j,k) = double(int(ES,O,R(j,k))); elseif el(j,k) >= 210 tt el(j,k) < 220 absrp(j,k) = double(int(ET,O,R(j,k))); if absrp(j,k) > 15 absrp(j,k) = 15; end elseif el(j,k) >= 220 tt el(j,k) < 230 absrp(j,k) = double(int(EU,O,R(j,k))); elseif el(j,k) >= 230 tt el(j,k) < 240 absrp(j,k) = double(int(EV,O,R(j,k))); elseif el(j,k) >= 240 tt el(j,k) < 250 absrp(j,k) = double(int(EW,O,R(j,k))); elseif el(j,k) >= 250 tt el(j,k) < 260 absrp(j,k) = double(int(EX,O,R(j,k))); elseif el(j,k) >= 260 tt el(j,k) < 270 absrp(j,k) = double(int(EY,O,R(j,k))); elseif el(j,k) >= 270 tt el(j,k) < 280 absrp(j,k) = double(int(EZ,O,R(j,k))); elseif el(j,k) >= 280 tt el(j,k) < 290 absrp(j,k) = double(int(EAA,O,R(j,k))); elseif el(j,k) >= 290 tt el(j,k) < 300 absrp(j,k) = double(int(EAB,O,R(j,k))); elseif el(j,k) >= 300 tt el(j,k) < 310 absrp(j,k) = double(int(EAC,O,R(j,k))); elseif el(j,k) >= 310 tt el(j,k) < 320 absrp(j,k) = double(int(EAD,O,R(j,k))); elseif el(j,k) >= 320 tt el(j,k) < 330 absrp(j,k) = double(int(EAE,O,R(j,k))); elseif el(j,k) >= 330 tt el(j,k) < 340 absrp(j,k) = double(int(EAF,O,R(j,k))); elseif el(j,k) >= 340 tt el(j,k) < 350 absrp(j,k) = double(int(EAG,O,R(j,k))); else absrp(j,k) = double(int(EAH,O,R(j,k))); end elseif bee(j,k) >= -5 tt bee(j,k) < 5 if el(j,k) >= 0 tt el(j,k) < 10 absrp(j,k) = double(int(FA,O,R(j,k))); elseif el(j,k) >= 10 tt el(j,k) < 20 absrp(j,k) = double(int(FB,O,R(j,k))); 64 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University elseif el(j,k) >= 20 && el(j,k) < 30 absrp(j,k) = double(int(FC,O,R(j,k))); elseif el(j,k) >= 30 && el(j,k) < 40 absrp(j,k) = double(int(FD,O,R(j,k))); elseif el(j,k) >= 40 && el(j,k) <50 absrp(j,k) = double(int(FE,O,R(j,k))); elseif el(j,k) >= 50 && el(j,k) < 60 absrp(j,k) = double(int(FF,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 70 absrp(j,k) = double(int(FG,O,R(j,k))); elseif el(j,k) >= 70 && el(j,k) < 80 absrp(j,k) = double(int(FH,O,R(j,k))); elseif el(j,k) >= 80 && el(j,k) < 90 absrp(j,k) = double(int(FI,O,R(j,k))); elseif el(j,k) >= 90 && el(j,k) < 100 absrp(j,k) = double(int(FJ,O,R(j,k))); elseif el(j,k) >= 100 && el(j,k) < 110 absrp(j,k) = double(int(FK,O,R(j,k))); elseif el(j,k) >= 110 && el(j,k) < 120 absrp(j,k) = double(int(FL,O,R(j,k))); elseif el(j,k) >= 120 && el(j,k) < 130 absrp(j,k) = double(int(FM,O,R(j,k))); elseif el(j,k) >= 130 && el(j,k) < 140 absrp(j,k) = double(int(FN,O,R(j,k))); elseif el(j,k) >= 140 && el(j,k) < 150 absrp(j,k) = double(int(FO,O,R(j,k))); elseif el(j,k) >= 150 && el(j,k) < 160 absrp(j,k) = double(int(FP,O,R(j,k))); elseif el(j,k) >= 160 && el(j,k) < 170 absrp(j,k) = double(int(FQ,O,R(j,k))); elseif el(j,k) >= 170 && el(j,k) < 180 absrp(j,k) = double(int(FR,O,R(j,k))); elseif el(j,k) >= 180 && el(j,k) < 190 absrp(j,k) = double(int(FS,O,R(j,k))); elseif el(j,k) >= 190 && el(j,k) < 200 absrp(j,k) = double(int(FT,O,R(j,k))); elseif el(j,k) >= 200 && el(j,k) < 210 absrp(j,k) = double(int(FU,O,R(j,k))); elseif el(j,k) >= 210 && el(j,k) < 220 absrp(j,k) = double(int(FV,O,R(j,k))); elseif el(j,k) >= 220 && el(j,k) < 230 absrp(j,k) = double(int(FW,O,R(j,k))); elseif el(j,k) >= 230 && el(j,k) < 240 absrp(j,k) = double(int(FX,O,R(j,k))); elseif el(j,k) >= 240 && el(j,k) < 250 absrp(j,k) = double(int(FY,O,R(j,k))); elseif el(j,k) >= 250 && el(j,k) < 260 absrp(j,k) = double(int(FZ,O,R(j,k))); elseif el(j,k) >= 260 && el(j,k) < 270 absrp(j,k) = double(int(FAA,O,R(j,k))); elseif el(j,k) >= 270 && el(j,k) < 280 absrp(j,k) = double(int(FAB,O,R(j,k))); elseif el(j,k) >= 280 && el(j,k) < 290 absrp(j,k) = double(int(FAC,O,R(j,k))); elseif el(j,k) >= 290 && el(j,k) < 300 absrp(j,k) = double(int(FAD,O,R(j,k))); elseif el(j,k) >= 300 && el(j,k) < 310 absrp(j,k) = double(int(FAE,O,R(j,k))); elseif el(j,k) >= 310 && el(j,k) < 320 absrp(j,k) = double(int(FAF,O,R(j,k))); elseif el(j,k) >= 320 && el(j,k) < 330 absrp(j,k) = double(int(FAFF,O,R(j,k))); 65 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University elseif el(j,k) >= 330 && el(j,k) < 340 absrp(j,k) = double(int(FAH,O,R(j,k))); elseif el(j,k) >= 340 && el(j,k) < 350 absrp(j,k) = double(int(FAI,O,R(j,k))); else absrp(j,k) = double(int(FAJ,O,R(j,k))); end elseif bee(j,k) >= 5 && bee(j,k) < 15 if el(j,k) >= 0 && el(j,k) < 10 absrp(j,k) = double(int(GA,O,R(j,k))); elseif el(j,k) >= 10 && el(j,k) < 30 absrp(j,k) = double(int(GB,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 30 && el(j,k) < 40 absrp(j,k) = double(int(GC,O,R(j,k))); elseif el(j,k) >= 40 && el(j,k) <50 absrp(j,k) = double(int(GD,O,R(j,k))); elseif el(j,k) >= 50 && el(j,k) < 60 absrp(j,k) = double(int(GE,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 70 absrp(j,k) = double(int(GF,O,R(j,k))); elseif el(j,k) >= 70 && el(j,k) < 80 absrp(j,k) = double(int(GG,O,R(j,k))); elseif el(j,k) >= 80 && el(j,k) < 90 absrp(j,k) = double(int(GH,O,R(j,k))); elseif el(j,k) >= 90 && el(j,k) < 100 absrp(j,k) = double(int(GI,O,R(j,k))); elseif el(j,k) >= 100 && el(j,k) < 120 absrp(j,k) = double(int(GJ,O,R(j,k))); elseif el(j,k) >= 120 && el(j,k) < 130 absrp(j,k) = double(int(GK,O,R(j,k))); elseif el(j,k) >= 130 && el(j,k) < 140 absrp(j,k) = double(int(GL,O,R(j,k))); elseif el(j,k) >= 140 && el(j,k) < 160 absrp(j,k) = double(int(GM,O,R(j,k))); elseif el(j,k) >= 160 && el(j,k) < 170 absrp(j,k) = double(int(GN,O,R(j,k))); elseif el(j,k) >= 170 && el(j,k) < 200 absrp(j,k) = double(int(GO,O,R(j,k))); elseif el(j,k) >= 200 && el(j,k) < 210 absrp(j,k) = double(int(GP,O,R(j,k))); elseif el(j,k) >= 210 && el(j,k) < 230 absrp(j,k) = double(int(GQ,O,R(j,k))); elseif el(j,k) >= 230 && el(j,k) < 240 absrp(j,k) = double(int(GR,O,R(j,k))); elseif el(j,k) >= 240 && el(j,k) < 250 absrp(j,k) = double(int(GS,O,R(j,k))); elseif el(j,k) >= 250 && el(j,k) < 260 absrp(j,k) = double(int(GT,O,R(j,k))); elseif el(j,k) >= 260 && el(j,k) < 270 absrp(j,k) = double(int(GU,O,R(j,k))); elseif el(j,k) >= 270 && el(j,k) < 280 absrp(j,k) = double(int(GV,O,R(j,k))); elseif el(j,k) >= 280 && el(j,k) < 290 absrp(j,k) = double(int(GW,O,R(j,k))); elseif el(j,k) >= 290 && el(j,k) < 300 absrp(j,k) = double(int(GX,O,R(j,k))); 66 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University elseif el(j,k) >= 300 && el(j,k) < 310 absrp(j,k) = double(int(GY,O,R(j,k))); elseif el(j,k) >= 310 && el(j,k) < 320 absrp(j,k) = double(int(GZ,O,R(j,k))); elseif el(j,k) >= 320 && el(j,k) < 330 absrp(j,k) = double(int(GAA,O,R(j,k))); elseif el(j,k) >= 330 && el(j,k) < 340 absrp(j,k) = double(int(GAB,O,R(j,k))); elseif el(j,k) >= 340 && el(j,k) < 350 absrp(j,k) = double(int(GAC,O,R(j,k))); else absrp(j,k) = double(int(GAD,O,R(j,k))); end elseif bee(j,k) >= 15 && bee(j,k) < 30 if el(j,k) >= 0 && el(j,k) < 20 absrp(j,k) = double(int(HA,O,R(j,k))); elseif el(j,k) >= 20 && el(j,k) < 40 absrp(j,k) = double(int(HB,O,R(j,k))); elseif el(j,k) >= 40 && el(j,k) < 60 absrp(j,k) = double(int(HC,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 80 absrp(j,k) = double(int(HD,O,R(j,k))); alseif el(j,k) >= 80 && el(j,k) < 100 absrp(j,k) = double(int(HE,O,R(j,k))); elseif el(j,k) >= 100 && el(j,k) < 140 absrp(j,k) = double(int(HF,O,R(j,k))); elseif el(j,k) >= 140 && el(j,k) < 180 absrp(j,k) = double(int(HG,O,R(j,k))); elseif el(j,k) >= 180 && el(j,k) < 200 absrp(j,k) = double(int(HH,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 200 && el(j,k) < 220 absrp(j,k) = double(int(HI,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 220 && el(j,k) < 240 absrp(j,k) = double(int(HJ,O,R(j,k))); elseif el(j,k) >= 240 && el(j,k) < 260 absrp(j,k) = double(int(HK,O,R(j,k))); elseif el(j,k) >= 260 && el(j,k) < 280 absrp(j,k) = double(int(HL,O,R(j,k))); elseif el(j,k) >= 280 && el(j,k) < 300 absrp(j,k) = double(int(HM,O,R(j,k))); elseif el(j,k) >= 300 && el(j,k) < 320 absrp(j,k) = double(int(HN,O,R(j,k))); elseif el(j,k) >= 320 && el(j,k) < 340 absrp(j,k) = double(int(HO,O,R(j,k))); else absrp(j,k) = double(int(HP,O,R(j,k))); end elseif bee(j,k) >= 30 && bee(j,k) < 45 if el(j,k) >= 0 && el(j,k) < 20 absrp(j,k) = double(int(IA,O,R(j,k))); 67 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University elseif el(j,k) >= 20 && el(j,k) <50 absrp(j,k) = double(int(IB,O,R(j,k))); elseif el(j,k) >=50 && el(j,k) < 80 absrp(j,k) = double(int(IC,O,R(j,k))); elseif el(j,k) >= 80 tt el(j,k) < 110 absrp(j,k) = double(int(ID,O,R(j,k))); elseif el(j,k) >= 110 tt el(j,k) < 160 absrp(j,k) = double(int(IE,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 160 && el(j,k) < 190 absrp(j,k) = double(int(IF,O,R(j,k))); elseif el(j,k) >= 190 tt el(j,k) < 220 absrp(j,k) = double(int(IG,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end elseif el(j,k) >= 220 && el(j,k) < 250 absrp(j,k) = double(int(IH,O,R(j,k))); elseif el(j,k) >= 250 tt el(j,k) < 280 absrp(j,k) = double(int(II,O,R(j,k))); elseif el(j,k) >= 280 && el(j,k) < 320 absrp(j,k) = double(int(IJ,O,R(j,k))); elseif el(j,k) >= 320 && el(j,k) < 340 absrp(j,k) = double(int(IK,O,R(j,k))); elseif el(j,k) >= 340 tt el(j,k) < 360 absrp(j,k) = double(int(IL,O,R(j,k))); if absrp(j,k) > 10 absrp(j,k) = 10; end end elseif bee(j,k) >= 45 && bee(j,k) < 60 if el(j,k) >= 0 && el(j,k) < 60 absrp(j,k) = double(int(JA,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 90 absrp(j,k) = double(int(JB,O,R(j,k))); elseif el(j,k) >= 90 && el(j,k) < 110 absrp(j,k) = double(int(JC,O,R(j,k))); elseif el(j,k) >= 110 && el(j,k) < 170 absrp(j,k) = double(int(JD,O,R(j,k))); elseif el(j,k) >= 170 tt el(j,k) < 200 absrp(j,k) = double(int(JE,O,R(j,k))); elseif el(j,k) >= 200 && el(j,k) < 230 absrp(j,k) = double(int(JF,O,R(j,k))); if absrp(j,k) > 5 absrp(j ,k) = 5; end elseif el(j,k) >= 230 tt el(j,k) < 290 absrp(j,k) = double(int(JG,O,R(j,k))); elseif el(j,k) >= 290 && el(j,k) < 330 absrp(j,k) = double(int(JH,O,R(j,k))); else absrp(j,k) = double(int(JI,O,R(j,k))); end elseif bee(j,k) >= 60 && bee(j,k) < 90 68 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University if el(j,k) >= 0 && el(j,k) < 30 absrp(j,k) = double(int(KA,O,R(j,k))); elseif el(j,k) >= 30 && el(j,k) < 60 absrp(j,k) = double(int(KB,O,R(j,k))); elseif el(j,k) >= 60 && el(j,k) < 90 absrp(j,k) = double(int(KC,O,R(j,k))); elseif el(j,k) >= 90 && el(j,k) < 120 absrp(j,k) = double(int(KD,O,R(j,k))); if absrp(j,k) > 5 absrp(j,k) = 5; end elseif el(j,k) >= 120 && el(j,k) < 150 absrp(j,k) = double(int(KE,O,R(j,k))); elseif el(j,k) >= 150 && el(j,k) < 180 absrp(j,k) = double(int(KF,O,R(j,k))); elseif el(j,k) >= 180 && el(j,k) < 210 absrp(j,k) = double(int(KG,O,R(j,k))); elseif el(j,k) >= 210 && el(j,k) < 240 absrp(j,k) = double(int(KH,O,R(j,k))); elseif el(j,k) >= 240 && el(j,k) < 270 absrp(j,k) = double(int(KI,O,R(j,k))); elseif el(j,k) >= 270 && el(j,k) < 300 absrp(j,k) = double(int(KJ,O,R(j,k))); if absrp(j,k) > 5 absrp(j,k) = 5; end elseif el(j,k) >= 300 && el(j,k) < 330 absrp(j,k) = double(int(KK,O,R(j,k))); else absrp(j,k) = double(int(KL,O,R(j,k))); end end end end for i = 1:size(absrp,1) for k = 1:size(absrp,2) i,k if absrp(i,k) < 0 absrp(i,k) = 0; end end end Y. end absBank_el Y. /l//l/l/1/llll/l/111/l//l//l//l//l//llllll/ll/ll/ll/ll/ll//l//l/l/1/lll/ll/ll/ll/lllll/ Y. l/1/l/ll/ll/ll////l//l//lll/1/lllllllll/ll/lllll//l//l//ll/l//l//l//l//lll//llll/ll/11/ Y. Find absorption from SNR to ellipsoid point. SN_R = ((D+t)/3260) - R; theta= asin( R.*sin(abs(bB-lat))./SN_R ); alpha= abs(theta) + abs(lat); SNPlane_R = 0.1 ./ sin(alpha); SNOuter_R = SN_R - SNPlane_R; SN_Absrp = 1.2 * SNPlane_R + .4*SNOuter_R; 69 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University for i = 1:size(bee,1) for j = 1:size(bee,2) if SN_Absrp(i,j) > 15 SN_Absrp(i,j) = 15; end end end % Combine the two absorption effects, calculate flux drop &probabilities. Cas_Complete_Absrp = absrp + SN_Absrp; Cas_Complete_Flux = 10.·(-0.4.*Cas_Complete_Absrp); Cas_Complete_Prob = Cas_Complete_Flux ·* Prob; % *all* effects worked in Cas_TotalProb = (1o.·(-0.4.*absrp)).*Prob; %ignoring SN->dust extinction (for testing) %Plot the resulting GRID probabilities (in RA and Dec) surf(RA,Dec,Cas_Complete_Prob,'EdgeColor' ,'none'), colormap pink view ( [90 0] ) set(gca,'Color',[O 0 0]) % Plot resulting POINT probabilities plot3(pointRA,pointDec,Cas_Complete_Prob, 'o','LineStyle','none') view ( [90 0] ) 70 L. Oaster • M.Sc. Thesis Dept. of Physics & Astronomy, McMaster University L...--_____ BIBLIOGRAPH~ Alard, C. 2000, A&AS, 144, 363 Alard, C., & Lupton, R. H. 1998, ApJ, 503, 325 Allen, S. W., Schmidt, R. W., & Fabian, A. C. 2002, MNRAS, 334, Lll Alves, J., Lada, C. J., & Lada, E. A. 1999, AJ, 515, 265 Arenou, F., Grenon, M., & Gomez, A. 1992, AA, 258, 104 Arnett, W. D., Bahcall, J. N., Kirshner, R. P., & Woosley, S. 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