The Spatial Distribution of Type Ia Supernovae within Host Galaxies

We study how type Ia supernovae (SNe Ia) are spatially distributed within their host galaxies, using data taken from the Sloan Digital Sky Survey (SDSS). This paper specifically tests the hypothesis that the SNe Ia rate traces the r-band light of the morphological component to which supernovae belong. A sample of supernovae is taken from the SDSS SN Survey, and host galaxies are identified. Each host galaxy is decomposed into a bulge and disk, and the distribution of supernovae is compared to the distribution of disk and bulge light. Our methodology is relatively unaffected by seeing. We find that, in galaxies dominated by disk light, SNe Ia trace light closely. The situation is less clear for bulges and ellipticals, because of resolution effects, but the available evidence is also consistent with the hypothesis that bulge/elliptical SNe Ia follow light.


INTRODUCTION
Type Ia supernovae (SNe Ia) are the thermonuclear detonations of carbon-oxygen white dwarfs (CO WDs) (Hoyle & Fowler 1960;Arnett 1969;Nomoto 1982), or possibly oxygen-neon white dwarfs (Marquardt et al. 2015;Augustine et al. 2019;Gal-Yam et al. 2022), which, by accretion of material from a binary companion, reach a critical mass and explode.Livio & Mazzali (2018) and others broadly classify the pathways to explosion into two channels: Single Degenerate (Whelan & Iben 1973), and Double Degenerate (Iben & Tutukov 1984).In the former case, a WD accretes material from a non-degenerate, close binary companion through Roche-Lobe transfer or through winds from a companion, until the WD reaches some critical mass and explodes.In the latter model, a binary WD system reaches a critical mass via disruption and accretion of one companion, or by coalescence.Double degenerate SN Ia ignition may even occur as a result of head-on collisions in dense stellar environments.The critical mass for explosion is usually taken to be the maximum mass of a WD, the Chandrasekhar (1931) mass, M Ch ≃ 1.4 M ⊙ ; however there exist mechanisms by which sub-Chandrasekhar mass explosions can occur (see Livio & Mazzali 2018;Ruiter 2020;Maoz et al. 2014 for these and other progenitor scenarios).
Each progenitor mechanism has strengths and weaknesses (Livio & Mazzali 2018), and at the present time it is unclear which of the many progenitor scenarios is dominant.(To quote Livio and Mazzali, "all the existing progenitor scenarios encounter difficulties".)Nor do there exist any observations of pre-explosion SNe Ia progenitors, or for that matter of post-explosion stellar remnants of SNe Ia (which might be expected for the single degenerate channel).This is the SN Ia "progenitor problem".
Given the observational challenges of identifying individual SNe Ia progenitor systems, one alternate approach to the progenitor problem is to study how SNe Ia are correlated with the environment in which they form or are found.For example, one can compare the radial distribution of supernovae with the radial distribution of stars.Most radial distribution studies have, however, been concerned with core collapse (CC) SNe, which are known to trace the spiral structure of their hosts (e.g.Maza & van den Bergh 1976).More recent studies have consistently shown that CC SN rates are closely related to near ultra-violet (nUV) and Hα luminosities of hosts, both of which are indicative of active formation of massive stars (e.g.Fruchter et al. 2006;James & Anderson 2006;Kelly et al. 2008;Raskin et al. 2008;Habergham et al. 2010;Perets 2014;Hakobyan et al. 2016;Aramyan et al. 2016;Hakobyan et al. 2017;Chakrabarti et al. 2018;Audcent-Ross et al. 2020;Schulze et al. 2021).
In contrast, little is known about the detailed spatial distribution of SNe Ia within their hosts.It is to be expected that SNe Ia will broadly track mass and light density; however, a complicating factor is that SNe Ia have a wide range of delay times (where the delay time is the time between the onset of the main sequence phase of a SN Ia progenitor, and its actual SN explosion).SNe Ia from the most massive progenitors may be found within a kpc of their formation region, but this is not true for most SNe Ia.
Compared to CC SNe, SNe Ia show a much broader spatial distribution relative to the spiral arms (Aramyan et al. 2016), and are also further from the plane of the disk (Hakobyan et al. 2009(Hakobyan et al. , 2012(Hakobyan et al. , 2014)).Taking the spiral arms and the disk to be the locations of active star formation, the broad conclusion from this series of papers is that SNe Ia occur at significant distances from the locations of star formation, as expected for long delay times.
Similarly, Anderson et al. (2015) have applied a method of pixel statistics developed in Anderson & James (2008) and Fruchter et al. (2006) to show that SN Ia distributions do not correlate with the nUV and Hα light associated with active star formation, thus arguing against a dominant young or 'prompt' progenitor pathway.They found the best correlation with the underlying B-band light, indicative of an intermediate age progenitor population.On the other hand, in early type host galaxies, studies have shown that SNe Ia track the light distribution of their old, evolved stellar populations, indicating long delay times, of order several gigayears (Förster & Schawinski 2008;Barkhudaryan et al. 2019;Hakobyan et al. 2020;Audcent-Ross et al. 2020).
How would the SNe Ia rate be expected to depend on host galaxy surface brightness?To answer this question, consider the delay-time distribution, DTD(t), which is defined as the number of supernovae from a burst of star formation as a function of the age of the burst (normalized per unit mass of the burst).There exist arguments that DT D ∼ t −1 , where t is the age of a stellar popu-lation (Maoz et al. 2014;Heringer et al. 2019).In this case, and somewhat remarkably, the SNe Ia rate per unit r-band luminosity, SN R/L r , can be shown to be nearly constant over a wide range of stellar population color, metallicity, or star formation history (Heringer et al. 2017(Heringer et al. , 2019)).The reason for this is the (somewhat fortuitous) cancellation of the effects of M/L (which increases with increasing color) and SNR (which decreases with increasing age and hence color).However, for a steeper DTD∼ t −1.5 there is some weak g − r host color dependence of SN R/L r .Given that the DTD power-law index probably lies between −1 and −1.5 (Heringer et al. 2019), and that the color gradients in bulges and disks are small, the intrinsic SN Ia rate and light intensity should track each other quite closely, at least within a single morphological component.
The goal of our paper is to test the hypothesis that SN Ia track light, using a subsample of the ∼2000 SNe Ia from the SDSS-II Stripe 82 Supernova Survey (Sako et al. 2018).This well-characterized supernova survey includes host galaxies with a wide variety of morphological types and star formation histories, making them ideal for testing whether SNe trace light.Typical SN Ia host galaxies in this survey have redshifts z ≃ 0.2 − 0.3, apparent magnitudes r ≃ 18 − 20, and sizes of a few arcsec.Seeing and optics blur the distribution of light in such hosts, so it is necessary to fit the underlying distribution of light with one or two component models, prior to comparing with the distribution of supernovae (whose positions are less affected by image quality considerations -e.g.Appendix C.3).
However, it is necessary to differentiate between the true, intrinsic SN Ia rate, and what is observed.The observed rate in a particular galaxy depends on a variety of astrophysical effects (stellar age being crucial; metallicity and binary frequency also play a role), galaxy orientation (which modulates the effects of dust obscuration), and observational incompleteness.While it is difficult to estimate the astrophysical and geometrical corrections to derive an intrinsic SN Ia rate, it is important to note that our analysis is sensitive only to radial gradients in the SN R/L ratio.In addition, SN R/L is analyzed in different components (bulge and disk) separately.Both of these analysis techniques will minimize (though not eliminate) astrophysical and orientation effects.Observational incompleteness is amenable to numerical experiments, and these are discussed both in the analysis sections ( § §5-7) and also Appendix C.1.
We have organized the paper as follows: In Section 2, we describe the SDSS-II stripe 82 Supernova Survey (Sako et al. 2018) used for this study, and the quantitative metric used to identify SN Ia host galaxies.§3.1 describes two independent galaxy morphology fitting methods, GIM2D (Simard et al. 2002) and imfit (Erwin 2015) that we used to map the underlying host galaxy light distributions.We test for a correlation between the SN Ia distribution and the light distribution of the host galaxies using the L inc /L acc metric, which is defined and described in §3.2.Our large sample size permits us to study the L inc /L acc distribution in statistically significant subsets of our primary sample, as described in §3.4.An initial analysis is presented in §4.We provide results for disk dominated hosts in §5, for bulge dominated hosts in §6, and for the mixed bulge+disk sample in §7.§8 is devoted to a discussion of our main findings, and our conclusions are summarized in §9.The Appendices provide more details on the fitting of galaxy structural parameters (and their associated errors), on the calculation of L inc /L acc , and on biases in the light profile fitting (especially incompleteness effects).

SN Ia Sample
Our SN Ia sample is taken from the Sloan Digital Sky Survey (SDSS) Stripe 82 Supernova Survey (Sako et al. 2014(Sako et al. , 2018)), henceforth referred to as the S82-SNe survey.The survey strategy was to use SDSS to repeatedly observe the same region of sky every alternate night during a three month observing window (Sep-Nov) over three years (2005 to 2007).The imaging was conducted within Stripe 82, covering an area of ∼275 deg 2 in a 2.5 • wide belt centered on the celestial equator and spanning −50 • ≤ RA ≤ 59 • .The survey efficiency and the selection biases have been investigated in detail by Dilday et al. (2008Dilday et al. ( , 2010a)).
A total of 10258 transients were discovered during the full survey.The light curves of the transients were classified with the Photometric SN IDentification (PSNID) (Sako et al. 2011).For our analysis, we use only the three broad groups of Type Ia supernovae: SNIa (499 spectroscopically confirmed Type Ia SNe); zSNIa (824 photometrically classified SNe Ia with spectroscopic redshifts from their host galaxies); and pSNIa (624 photometrically classified SNe Ia with photometric redshifts only).The total number of SNe Ia is 1947.We omit a fourth class of uncertain identifications (SNIa?, 41 objects) from our study.The completeness of the SN Ia sample is discussed in §3.4 and Appendix C.1.

Supernova -Host Galaxy Matching
In the crowded galaxy field typical of a deep imaging survey, the galaxy with the closest angular distance (CAD) to a supernova is not always its host.Gupta et al. (2016) and Gagliano et al. (2021) discuss the chal-lenges of identifying the host galaxies of transients in large imaging surveys such as the S82-SNe.
We have adopted an algorithm similar to those of Sullivan et al. (2006) and Sako et al. (2018) for SN-host galaxy matching.Photometric and geometric properties of galaxies are taken from the SDSS-DR7 photometric catalogs (Abazajian et al. 2009) for the Stripe 82 footprint.We define a dimensionless quantity ρ 25 , which is the projected separation of a supernova from a galaxy, measured in units of the r-band 25 mag arcsec −2 isophotal radius at the position angle of the supernova (d/d DLR in the terminology of Sako et al. (2018)).The host galaxy of a particular supernova is taken to be that galaxy with a minimum value of ρ 25 .
We select only hosts whose ρ 25 separations lie below ρ crit 25 = 1.959, for which 8% of randomly positioned (unhosted) SNe would be assigned to a host.A simulation shows that ∼ 0.2% of objects would be assigned to the wrong host.Most importantly, 97% of our host identifications agree with those of Sako et al. (2018) for r host < 20 mag.Also of interest is the fact that 86.5% of our host identifications for the full host catalog agree with the galaxy with the closest angular distance.

ANALYSIS
Our technique for comparing SNe Ia and host galaxy light (the L inc /L acc method described in §3.2) relies on obtaining an estimate of the intrinsic light distribution of host galaxies, affected as little as possible by seeing.To do this, we fit a seeing-convolved model to each host galaxy, as described below.

Host Fitting
SDSS SNe Ia hosts at z ≃ 0.2 are fairly similar (in both median luminosity, and also, as we shall see, median bulge to total light ratio) to the nearby spiral galaxy M31.Scaling from the properties of M31 (Courteau et al. 2011), we find that M31-like galaxies at z = 0.2 have a bulge half-light radius (HLR) of 0.3 arcsec, and a disk HLR of 3 arcsec.This illustrates the importance of seeing effects on host galaxies (and especially their bulge components) at the typical redshifts of SDSS SNe.To test whether SNe trace galaxy light, it is necessary to compare the spatial distribution of supernovae with the true, intrinsic light distribution of galaxies, unaffected by seeing1 .
Each host galaxy was fitted using two different galaxy light modelling programs, to provide added confidence in the fits.The programs used were GIM2D (Simard et al. 2002), and imfit (Erwin 2015).These programs fit multiple component galaxy models to seeing-degraded image data, convolving each model with a pre-determined seeing profile to match the observations.GIM2D uses the Metropolis algorithm (Metropolis et al. 1953) to search for the best fit in parameter space; imfit allows the use of a variety of best-fit search algorithms, including a genetic algorithm.
The fitting functions used with each program were one or two elliptical Sérsic (Sérsic 1963) profiles of the form (1) with profile parameters r e (half-light radius), I e (surface brightness at r = r e ), Sérsic index n, and with constant (i.e.spatially invariant) shape and orientation parameters (ellipticity b/a and position angle P A).The value of b n is calculated as in Graham & Driver (2005).
Most of the fits were made with a two component (disk plus bulge) model, where the disk is exponential (n = 1), the bulge has a de Vaucouleurs profile (n = 4), and r e , I e , P A, and b/a are fitted separately for each component 2 .At the typical redshift of SDSS SNe Ia (z ≃ 0.2 − 0.3), a typical bulge is only a few pixels in effective radius, and hence it is not feasible to fit for the bulge Sérsic index n bulge for the two component models (see Bottrell et al. 2019, §4.5).The assumption n bulge = 4 is discussed further in Appendix C.4.In addition to the two-component fitting, we made an additional fit to each host using a single Sérsic component, allowing its index n to vary.
We used r-band Stripe 82 stacked images (Annis et al. 2014) for both GIM2D and imfit fits.The PSFs and other properties of these stacked images are discussed in Appendix A.1.More information on the use of GIM2D and imfit is provided in Appendices A.2 and A.3, where the determination of errors in the fitted model parameters is also discussed.The most important error parameter is σ B/T , which is used in the definition of the primary sample ( §3.4).The reliability of the models is tested by comparing the L inc /L acc results of the two programs in Appendix B.2.
Biases in the profile fitting are discussed in Appendix C. Most of the results that we discuss in § §5-7 refer to GIM2D; the results with imfit are similar, except where noted.
2 For an exponential disk (n = 1), half light radius re and exponential disk scale-length β are related through re = 1.678β.shows a SN (in red) superimposed on its host galaxy.The red shaded area is the area used to calculate Linc; its boundary is defined by an isophote passing through the SN; this isophote is not necessarily elliptical.Lacc is computed within an elliptical boundary with semi-major axis a = 1.959a25 and semi-minor axis b = 1.959b25.This is the area within which a SN can be matched to its host; see §2.2.Note that Linc and Lacc are computed using the fitted model light distribution.
The middle panel shows one simulation of 100 SNe in which SNe trace light (bin size 0.01).The lower panel shows 30 simulations of 100 SNe, plotted as cumulative distributions.

Do Supernovae Trace Light? The L inc /L acc Metric
It is a remarkable fact that, for a stellar population of a given color, the SN Ia rate per unit r-band luminosity, SN R/L r , is almost independent of age and star formation history (Heringer et al. 2017(Heringer et al. , 2019; see §1).One would therefore expect the distribution of supernovae to follow light quite closely.
To test this hypothesis, we use the L inc /L acc technique (e.g.Fruchter et al. 2006;Raskin et al. 2008;Audcent-Ross et al. 2020, and references therein).We take the host galaxy isophote passing through the coordinates of a SN, and sum up the fitted light, L inc , included (enclosed) within that isophote (see Fig. 1, upper panel).We also sum the "accessible" fitted light, L acc , included within the elliptical aperture that was used for SN-host matching (with axes 1.959a 25 and 1.959b 25see §2.2). (L acc is typically 80-90% of the total light of the galaxy.)Since L inc /L acc is a ratio of enclosed light, objects that are stochastically sampled from the light distribution are uniformly distributed in L inc /L acc , if supernovae follow light.It follows that multiple supernovae in different host galaxies will also be uniformly distributed as in Fig. 1, and the cumulative distribution of supernovae with L inc /L acc is linear (Fig. 1, bottom panel).Simple statistical tests ( §3.3) can be applied to Fig. 1 to test the null hypothesis that supernovae trace light.
L inc and L acc are calculated from the fitted GIM2D and imfit models.The calculations are complex because the L acc aperture does not necessarily have the same shape and orientation as the host galaxy components, and because the L inc isophote is not elliptical for a two component model (unless both components have exactly the same b/a and P A).For this reason we used two independent programs (written by CP and KT) to calculate and check the results; these programs agree to better than ±0.001 in L inc /L acc .
Although L inc /L acc was described above in terms of total light, one can use it to test whether supernovae follow the light of only one component, by computing L inc /L acc using bulge light or disk light alone.For the total light calculation one can also allow for a different SN R/L r in the bulge and disk.
Appendix B.1 discusses two shortcuts that can be used in the calculation of L inc /L acc ; Appendix B.2 compares the L inc /L acc values computed using GIM2D and imfit.

Statistical tests
The null hypothesis that SNe follow light can be tested with the well-known Kolmogorov-Smirnov test applied to the cumulative L inc /L acc diagram (Fig. 1).While this is a simple calculation, it is mainly sensitive to deviations in the middle of the range 0 < L inc /L acc < 1.We therefore also use the Anderson-Darling test, which has greater sensitivity overall, but especially at the ends of the distribution.(The reader is referred to Feigelson & Babu 2020 for a discussion and comparison of these two tests.)We have tested our probability calculations with Monte Carlo simulations, and find them to be in excellent agreement with statistical tables for these 2 tests.We have also experimented with boot-Figure 2. Galaxy isophotal size in arcsec vs. host r band magnitude.The size is the circularized radius of the µr = 25 mag/arcsec −2 isophote as given in SDSS DR7.The small grey points are a sample of 93000 galaxies brighter than r = 21 mag in Stripe 82 with 0 o < α < 10 o .Squares are host galaxies, colour coded according to type: black: SNIa; blue: zSNIa; red: pSNIa.The shaded area represents our host cut at r < 20 mag.
strapping the distributions of L inc /L acc values, and find that the bootstrapped KS test results are in good agreement with the original KS test provided that the data supports SN R ∝ L.

Primary Sample
We now consider the properties of the host galaxies and supernovae, in order to determine which objects to use in our analysis.
There exists a surprisingly tight correlation between host galaxy isophotal angular size and apparent magnitude (Fig. 2).At r host = 20 mag, the isophotal radius (at µ r = 25 mag arcsec −2 ) of hosts is typically 2 arcsec.(For reference, the surface brightness corresponding to S/N=1 in one arcsec 2 of the sky-noise-limited Annis et al. stacks is µ r ≃ 27.3 mag arcsec −2 ).Two component fits at r > 20 mag become noisy, due to the smaller number of resolution elements in the image; this is especially true for bulges.The noisiness of the fits at r > 20 mag is also due to the lower signal-to-noise ratio of the data: for GIM2D, the typical error in fitted bulge to total light ratio B/T at r host > 20 mag is σ B/T > 0.07.Furthermore, below r host = 20 mag we observe poorer consistency between the GIM2D and imfit results.
For the above reasons we use only those SN hosts with r ≤ 20 mag.(In § §5-7 we also consider the effects of limiting the host sample to r ≤ 19 mag.)Of the 657 host galaxies with r ≤ 20 mag and ρ 25 < 1.959 that were fitted by GIM2D, 28 could not be fitted with imfit.A further 29 objects were flagged by visual inspection because their fits showed strong residuals, but most of these were eliminated by other cuts on the data (all but 5 for GIM2D, 9 for imfit).
Another cut that can be made is on SN maximum brightness, which is related to completeness.The primary sample uses r max SN < 22 mag, which corresponds to S/N≃ 7 for a point source detection in the Stripe 82 individual scans (Dilday et al. 2008); fainter that this magnitude it is clear that the SN Ia data are incomplete.(The effects of completeness are addressed in more detail in Appendix C.1.)The number of objects left after this cut is 614 (576) for GIM2D (imfit).We have experimented with brighter values of limiting r max SN , and discuss these results in § §5-7 below.
A magnitude cut in r max SN corresponds to a redshift cut, because SNe Ia are approximately standard candles.At a given host apparent brightness, higher redshift corresponds to higher intrinsic luminosity and larger physical size, so that the angular diameter, and hence the quality of the fit, is not too strongly redshift dependent.
Our primary sample contains all SN Ia types (SNIa, zSNIa, pSNIa).Eliminating the pSNIa class (photometrically classified SNe, photometric redshifts) removes 30 objects, and makes no difference to any of the results.We eliminate the noisiest fits using the cuts χ 2 r < 2 and σ B/T < 0.07 (where χ 2 r is the reduced χ 2 of the fit, and σ B/T is the 1σ error in B/T ), leaving a total of 412 (433) objects for the GIM2D (imfit) fits.
Table 1 summarizes the constraints that define the primary sample of objects.This is for the primary sample of objects, and the GIM2D analysis.The dot-dash lines are Sérsic models: blue: n = 1 (disk); green: n = 2.5; red: n = 4 (bulge).The Sérsic profiles have been corrected for missing light at ρ25 > 1.959, and hence do not pass through (R/HLR = 1, f = 0.5).
Finally we make a small adjustment to the fits to eliminate poorly constrained components: strongly diskdominated systems (B/T < 0.05) are transformed into pure disks (B/T = 0), and strongly bulge-dominated systems (B/T > 0.95) are transformed into pure bulges (B/T = 1).

A FIRST LOOK AT SUPERNOVAE AND HOST
GALAXY LIGHT Before we turn to the L inc /L acc analysis, we first examine what can be said about supernovae and host galaxy light distributions as a function of angular radius.
To remove the effects of physical size and distance, we normalize each SN radial offset by the half-light angular radius, HLR, of its host.(This HLR is derived from the fits, and is therefore not strongly affected by seeing.)The results are shown in Fig. 3 for both bulge-and diskdominated galaxies, and are compared with 3 pure Sérsic profiles.The main conclusion to be drawn from Fig. 3 is that SNe Ia in bulge-and disk-dominated galaxies have very different spatial distributions.SNe Ia in disks roughly follow the n = 1 Sérsic profile for the light; SNe Ia in bulges follow a more centrally concentrated Sérsic profile, with n ≃ 2.5 − 3 (though n = 4 is not as good a match to the data).
Nevertheless the normalization by HLR (Fig. 3) has problems: the analysis does not include the effects of galaxy ellipticity, or of multiple components.We therefore turn to the L inc /L acc analysis, first looking at diskdominated galaxies.

DISK GALAXIES
The L inc /L acc metric provides a convenient and rigorous way of comparing the distribution of supernovae with the underlying (i.e.unaffected by seeing) distribution of light in galaxies.L inc /L acc takes into account galaxy size, ellipticity, and orientation, and furthermore uses only supernovae and light with ρ 25 ≤ 1.959 (i.e.within the isophote at which supernovae and hosts are considered to be associated).
Fig. 3 shows that a single-component Sérsic model with n = 1 is a reasonable description of disk-dominated galaxies.We use a cutoff B/T < 0.2 on the primary sample, and examine the distribution of L inc /L acc for these disks in Fig. 4a.The SN radial distribution matches the light distribution in the disk.Under the null hypothesis that the SN radial distribution and disk light distribution are drawn from the same parent distribution, the p-values for the Kolmogorov-Smirnov and Anderson-Darling tests are p KS = 0.482, p AD = 0.415meaning that the null hypothesis is acceptable.Not surprisingly, SNe Ia in disk galaxies do not follow the bulge light distribution.The light distributions for this figure are from GIM2D fits; the results using imfit fits are very similar.The same result is obtained using a single component model fit restricting the Sérsic index to n < 1.5; SNe follow light for such a selection of objects.
Could the use of an upper limit B/T < 0.2, with its small but not non-negligible bulge contribution, have affected these results?To check this, we tried two other limits on B/T (< 0.1, < 0.05) to define the diskdominated sample.The results are very similar to those for B/T < 0.2.
We also try 2 techniques for removing the effects of residual bulge or nuclear light, while reducing the problem of incompleteness near the cores of galaxies (if it exists -e.g.Appendix C.1).We analyze supernovae and host light only in the range 0.1 < L inc /L acc < 0.9 (where the upper limit also excludes a few SNe at very large radii); and we consider only SNe with R > 0.75 arcsec (i.e.outside the seeing disk).The result of applying both of these constraints is shown in Fig. 4b.Again, the match to the "SNe follow light" hypothesis is excellent (p KS = 0.642, p AD = 0.643).Using either of the above 2 constraints alone likewise produces excellent agreement, for both GIM2D and imfit fits.
In Fig. 4c we compare the radial distributions (in disk dominated host galaxies, B/T < 0.2) of Sako class SNIa objects (spectroscopically confirmed, N = 60) with Sako's zSNIa and pSNIa classes (photometric supernovae, N = 76, of which 67 are zSNIa).There is a difference between these two samples, in the sense that the spectroscopically-confirmed sample is underrepresented at small separations.This is a clear and expected (see Appendix C.1) signature of incompleteness at small separations in the Sako spectroscopicallyconfirmed (SNIa) sample.Now consider the full SNIa+zSNIa+pSNIa sample.If there were a significant radial variation in SN completeness, then noticeable differences would be expected in the L inc /L acc plots for different values of the limiting SN magnitude at maximum light, r max SN .In Fig. 4d we vary the limiting r max SN for our primary sample of disk galaxies (B/T < 0.2).The results for r max SN < 22, < 21.5 and < 21 mag are, either from KS and AD tests relative to the expected 1:1 relation, or from KS 2 sample tests, statistically indistinguishable from one another or from the null hypothesis (probabilities 40-90%, using either GIM2D or imfit data).These tests therefore show no evidence for any radial variation in SN Ia completeness, and furthermore all support the null hypothesis that "SNe trace light" in disk galaxies.
Radial gradients are known to exist in the mean ages (Muñoz-Mateos et al. 2007;Casasola et al. 2017) and metallicities (Kruit & Freeman 2011) of stellar populations in galaxy disks.Age is known to affect the rate of SNe Ia through the delay-time distribution (DTD -e.g.Heringer et al. 2019), and metallicity may also affect the SN Ia rate (e.g.Kistler et al. 2013).One might therefore expect a small additional gradient in the SN radial distribution relative to the distribution of galaxy light.This could result in a scale length (or effective radius) for the supernova radial distribution different from that of the light.
To test this hypothesis, we have tried recalculating L inc /L acc using several different disk scale lengths.Fig. 4e shows the results of scaling all GIM2D disk scale lengths by a factor of 1.1× and 0.83× (using the approximation in Appendix B.1).These models differ from the canonical (unscaled) model in Fig. 4a at the 95% confidence level: there is only a ∼5% KS or AD probability that these models agree with the "SNe follow light" hypothesis.We therefore conclude that, at the 95% confidence level, the scale length of the SN distribution is the same as that of disk light to within about 15%.The same conclusion is drawn using imfit data.
An alternate approach is to recalculate L inc /L acc with different values of Sérsic n disk (again using the approximation in Appendix B.1).It can be seen from Fig. 4f that n = 0.5 and n = 1 (p KS , p AD > 0.40) match the data best, in agreement with the n = 1 exponential disk  that was used in the fitting process for the light.Increasing n to 1.5 results in p KS = 0.122, p AD = 0.058; n = 2 can be rejected at the 99% confidence level if SNe trace light.In other words, there is no evidence supporting the hypothesis that the radial distribution of SNe Ia follow a different Sérsic profile from that of the disk, as might have been expected had gradients in metallicity and stellar age affected SN rates.
Finally, we consider the effects of using a sample of disk-dominated host galaxies with r < 19 mag.The results are almost indistinguishable from those for the Primary Sample with r < 20.

BULGES AND ELLIPTICALS
It is more difficult to perform an L inc /L acc analysis for galaxies with prominent bulge components (and ellipticals).The main problem is resolution: at the typical redshifts of the SDSS SN Survey, bulges of hosts are only marginally resolved (e.g.§3.1).SN incompleteness may also be a problem in the inner regions of bulges because of their higher surface brightness: they are both noisier due to photon noise, and also are more likely to have artifacts and substructure in the subtracted images which are used for SN detection.(Evidence of incompleteness for inner bulge SNe may be visible in Fig. 3, though the overall completeness of the survey is high (Dilday et al. 2010b, see also Appendix C.1).) With these caveats in mind, we now look at L inc /L acc for bulge-dominated galaxies with B/T > 0.7 (Fig. 5a).p KS and p AD are lower for these galaxies (≲ 0.1).However, if the bulge analysis is compromised by resolution effects, then it makes sense to restrict the analysis to the largest objects, which are better resolved.The constraint HLR ≡ r ef f > 1 arcsec (i.e.including only the largest 50% of bulges, corresponding to err(r ef f )/r ef f ≲ 0.1) improves the probabilities (GIM2D: p KS , p AD = 0.348, 0.356; imfit: p KS , p AD = 0.550, 0.516).This is shown in Fig. 5b; results are similar for HLR > 1.2 arcsec.What about the single component models?Looking only at the objects with Sérsic n > 3, the KS and AD probabilities are ≲ 0.10.However, including a constraint HLR > 1 arcsec improves the agreement with the SNe traces light hypothesis (GIM2D: p KS , p AD = 0.662, 0.428; imfit: 0.719, 0.463).This is shown in Fig. 5c.
The KS and AD probabilities can also be raised to > 0.7 by restricting the analysis to 0.1 < L inc /L acc < 0.9(0.7)for GIM2D (imfit).This is probably a signature of misfitting of the r ef f value, presumably due to resolution effects, and possibly also incompleteness in the inner regions.It may also be due to SNe being scattered out of the inner regions due to errors in the coordinates (Appendix C.3). Changing the value of n bulge does not improve the agreement between GIM2D and imfit; nor does adding in the effects of the (relatively small amount of) disk light.For single component models with Sérsic index n > 3, there is no statistically significant difference between the results for r < 19 and r < 20 (2 sample KS test p-value 0.66, or 0.79 if the sample is restricted to 0.1 ≤ L inc /L acc ≤ 0.9).For two component models with B/T > 0.7, a similar result is obtained if the samples are restricted to HLR > 1 arcsec or SNe with ∆θ > 0.75 arcsec (2 sample KS p ≥ 0.80).
We conclude that the evidence from the largest 50% (HLR > 1 arcsec) of bulge-dominated hosts is consistent with the hypothesis that Type Ia supernovae and galaxy light have the same spatial distribution.

GALAXIES WITH PROMINENT BULGES AND DISKS
In this section we discuss objects intermediate between disk-dominated ( §5) and bulge-dominated ( §6) galaxies.These objects have 0.2 < B/T < 0.7.
The GIM2D analysis of these objects is shown in Fig. 6a.This figure shows that L inc /L acc calculated from bulge or disk light alone does not match the 1:1 relation; in other words, supernovae must occur in both components.On the other hand, L inc /L acc measured for total light provides a somewhat better match to the SN distribution (p KS , p AD = 0.399, 0.228).The agreement is substantially worse for the imfit analysis (p KS , p AD = 0.069, 0.059), as was also the case for objects dominated by bulge light ( §6).
We have tested whether the host magnitude limit affects these results for hosts with intermediate B/T.A KS two sample test shows that results for hosts with r < 19 are indistinguishable from the primary sample (r < 20).
The agreement between L inc /L acc and cumulative SN numbers for 0.2 < B/T < 0.7 can be improved in at least three ways, two of which we saw in bulge-dominated galaxies ( §6).First, the analysis can be restricted to 0.1 <L inc /L acc < 0.9, excluding SNe in the inner and outer regions.The result of this is shown in Fig. 6b; the total light fit is much better (GIM2D: p KS , p AD = 0.930, 0.677, imfit: p KS , p AD = 0.420, 0.389).Second, the analysis can be restricted only to the largest objects (Fig. 6a).For objects with HLR >1 (>1.5) arcsec, the GIM2D probabilities are p KS , p AD = 0.684, 0.419 (0.966, 0.934).Third, the bulge SN rate per unit light can be decreased relative to the disk rate.We defer a discussion of this to §8.As discussed in §1, for plausible delay-time distributions, the SN Ia rate per unit r-band luminosity, SN R/L r , is expected to be more or less constant over a wide range of stellar population color, metallicity, and star formation history (Heringer et al. 2017(Heringer et al. , 2019)).It is therefore not surprising that the rate of SNe Ia and light track each other closely in disks ( §5).
The situation with bulges is more complex.As already noted, at the typical redshifts of the SDSS SN Survey, bulges of hosts are only marginally resolved, so the bulge parameters r ef f , µ ef f , and b/a are poorly constrained, especially with an admixture of disk light.Furthermore, nuclear light may contaminate the bulge fitting, resulting in an underestimate of r ef f .Our use of GIM2D and imfit requires that bulges be fitted with a fixed Sérsic index n = 4, because of the small number of pixels in the bulge; but real bulges and ellipticals actually possess a wide range of n values (Appendix C.4).Finally, astrometric errors may scatter SNe out of the region L inc /L acc ≲ 0.1 in bulges (Appendix C.3).
In §6 we noted that, for bulge-dominated host galaxies, bulge light and SNe Ia followed the same profile if the sample of host galaxies is restricted to the largest, best-resolved ∼ 50% of objects (HLR ≳ 1 arcsec).This is true for both the 2-component and 1-component models.
What about the objects which are intermediate between bulge-dominated and disk dominated (0.2 < B/T < 0.7)?Based on the above discussion, one might have expected SNe to follow total light.However, in §7 we noted a lack of SNe in the inner regions relative to the total light L inc /L acc profile.This discrepancy is ameliorated if the sample of hosts is restricted to HLR ≳ 1 arcsec, or if the sample of SNe is restricted to 0.1 < L inc /L acc < 0.9.
Another explanation is possible: the rate of SNe Ia per unit r-band luminosity, SN R/L r , in bulges may be different from that in disks, and would hence result in a lower than expected number of supernovae in the in-ner regions of galaxies relative to the disk rate.We have already discussed the fact that this is unlikely for a delay-time distribution DTD ∼ t −1 .However given the relatively large color difference between bulges and disks (typically ∆(g − r) ≃ 0.2), a significant difference between bulge and disk SN R/L r might be expected for DTD≃ t −1.5 or steeper, or for a DTD with a cutoff (cf.Heringer et al. (2019), Fig. 7).
Fig. 6c shows the result of scaling the bulge SN rate relative to the disk rate, for L inc /L acc calculated using total light.(This plot is computed using the approximation in Eq.B1.)As expected, for small bulge rate scaling factor f , L inc /L acc (tot) approaches that for disk light; for larger f , L inc /L acc (tot) approaches the bulge results.The best scaling factor is f ≃ 0.5 ± 0.2, where p KS , p AD = 0.858, 0.802 at f = 0.5, and the quoted error range corresponds to p ≃ 0.5.This is for the GIM2D analysis; the results for imfit are similar.
Referring to Fig. 7 of Heringer et al. (2019), if bulges are old, then this result is consistent with a DTD ∼ t −1.5 , or a power-law DTD with a cutoff (a steeper power-law slope) at t > 1 − 2 Gyr.However, as already mentioned, this result can also be explained as due to resolution effects; the correct explanation must await host (and supernova) observations with better resolution.

Astrophysical Effects
Here we consider how the properties of supernovae, or of their host galaxies, affect the conclusions regarding the spatial distribution of supernovae.First we examine the effects of SN light curve width or stretch (s or x1) and color at maximum light (c).These photometric parameters are used by the SALT2 model to empirically determine the absolute magnitude of a SN Ia (Guy et al. 2007;Sako et al. 2018); although both parameters are required by SALT2, c is significantly more sensitive in determining absolute magnitude.
There exists a great deal of work on the properties of SN Ia host galaxies, and especially on how these properties relate to SN Ia light curves (e.g.Murakami et al. (2023) and references therein).Earlier work by, for example, Sullivan et al. (2006) (see also Maoz et al. 2014 and references therein) showed that the stretch parameter s correlates with host galaxy sSFR: SNe in galaxies with high sSFR tend to have broad light curves with stretch s > 1, whereas elliptical and bulge-dominated spiral hosts with low or zero sSFR have an s distribution dominated by SNe with s < 1.Furthermore, SNe Ia with s < 1 appear to be associated with galaxy mass, whereas s > 1 SNe Ia appear to be prompt objects associated with star formation.These conclusions apply to the integrated properties of galaxies; the question arises whether they may also apply within galaxies.In other words, are the spatial distributions of s < 1 and s > 1 SNe (or c < 0 and c > 0 SNe) different?Fig. 7 compares the GIM2D L inc /L acc distributions for disk galaxies (B/T < 0.2) hosting red (c > 0) and blue (c < 0) SNe Ia.Formally these two distributions are in acceptable agreement (p = 0.36 from a 2-sample KS test).Of interest, however, is the lack of blue SNe Ia in the inner regions of disk-dominated galaxies: of the 14 SNe Ia at L inc /L acc < 0.1, only 2 have c < 0. Under the null hypothesis that red and blue SNe possess equal likelihood (which is suggested by the overall numbers of supernovae in each category), the probability of finding ≤ 2 red or blue objects is only p = 0.013.Similar results are found with imfit.A much weaker effect exists for the blue SNe Ia in hosts with mixed morphology (0.2 < B/T < 0.7): the null hypothesis that SNe trace total light in these hosts has p KS , p AD = 0.225, 0.119), with no statistically significant difference between the red and blue subsamples.No color separation is seen for the bulge-dominated hosts (B/T > 0.7).
Could the SN color effect in disks be due to incompleteness in the cores of galaxies?Typical blue SNe Ia are 0.5-1 mag brighter than typical red SNe Ia, in the opposite direction of what would be expected from incompleteness.
This anomaly could be due to a gradient in some property in these galaxies, coupled with a dependence of c on this property.For example, if SN c depends on mean stellar age ⟨τ ⟩, and ⟨τ ⟩ is different in the cores of disk galaxies compared to their outer regions, then the observed effect (missing blue SNe in the core) could be a consequence.We have examined the L inc /L acc distributions for c < 0 and c > 0 SNe Ia for hosts with even smaller amounts of bulge light (B/T < 0.1 and B/T < 0.05).The results are almost identical, although with lower significance because of the smaller numbers of objects involved.We conclude that, if c depends on SN age, it is not the age difference between bulges and disks that is causing this effect.
Finally, the presence of significant amounts of dust in the central regions of disk galaxies could obviously redden core SNe Ia.
No significant differences are seen in the radial distribution of SNe Ia that depend on the stretch parameter s.However, as noted by Sullivan et al. (2006), there are very significant differences in the numbers of low stretch (s < 1 : n = 96), and high stretch (s > 1 : n = 29) SNe Ia in bulge dominated (B/T > 0.7) galaxies.This is a ∼ 4σ difference compared to the equal numbers of s < 1 and s > 1 objects observed in all SDSS host galaxies.
Do our conclusions on the radial distribution of SNe Ia depend on the properties of the host galaxies?We have examined the following host galaxy properties in the Sako et al. (2018) database: absolute magnitude, mass, g − r color, mean age, specific star formation rate, and number of companions.(Many of these properties are correlated.)For each property, we have divided the SNe into two roughly equal-sized samples, and have compared the L inc /L acc distributions for bulge-dominated, disk-dominated, and intermediate hosts, using a 2 sample KS or AD test.No significant differences were seen, except in one case: intermediate B/T (0.2 < B/T < 0.7) hosts divided according to whether the mean stellar pop- ulation age is < 5 Gyr or > 5 Gyr (Fig. 8).Here we see a strong difference between the two samples, in the sense that older intermediate B/T galaxies have disproportionately more SNe Ia in their outer regions: from a KS 2 sample test the probability that these 2 samples are drawn from the same distribution is only 6%.
However, this result should be viewed with some caution.Determining star formation history and mean age is difficult from broadband photometry alone (Conroy 2013), even with the addition of UV and IR fluxes to SDSS ugriz (Gupta et al. 2011).In addition, galaxy age is observed to correlate with B/T in the SDSS host sample; this is expected because bulges are older than disks.Dividing the data by B/T rather than by age in fact produces a plot similar to Fig. 8. Thus it is likely that B/T is the driver of the age effect.

CONCLUSIONS
We have used the SDSS SN Survey to study the radial distribution of SNe Ia in their host galaxies.Each host galaxy has been decomposed into a seeing-convolved bulge plus disk model, and the distribution of SNe has been compared with the distribution of model light enclosed by an isophote passing through each SN ( §3.2).
We find that the rate of SNe Ia in disk-dominated galaxies is proportional to the r-band surface brightness of the disk.Not surprisingly, a Sérsic index n = 1 (i.e. an exponential disk) provides the best fit for the distribution of supernovae.There is no evidence that the disk scalelength (or equivalently r ef f ) is different for the radial distributions of SN Ia and light.
The situation with bulge-dominated and elliptical galaxies is less clear because of resolution effects.Nevertheless, the evidence from the larger B/T > 0.7 galaxies also shows that the SN Ia rate is proportional to light.
For host galaxies with intermediate B/T (0.2 ≤ B/T ≤ 0.7), it is clear that SNe Ia occur in both the disk and bulge components.There is a lack of SNe Ia in the inner regions of these galaxies.This could be due to incompleteness or resolution effects, as was the case for bulge-dominated galaxies.It could also be due to a smaller SN rate per unit light in bulges (as compared with disks).
There is evidence for missing blue (SALT2 c < 0) Type Ia supernovae in the cores of disk-dominated galaxies.This is possibly due to the effects of central dust.
One major limitation in this work is the lack of resolution of bulges and ellipticals at redshifts z ≃ 0.2 − 0.4 in SDSS data.In the future we plan to avoid this limitation by using HST imaging for the host light fitting, and by redoing the analysis on a lower redshift transient survey such as the Zwicky Transient Facility (ZTF)3 .errors were computed from the distributions of bootstrap fitted values.Unlike GIM2D, imfit does not fit B/T ; rather, it is calculated from 6 other fitted parameters.σ B/T is simply calculated by estimating B/T separately for each bootstrap iteration, and using the marginal distribution of these values.
For the total light calculation one can also allow for a different SN R/L r in the bulge and disk by simple scaling: where tot refers to total light, b is bulge, d is disk, and f is the factor by which SN R/L r in the bulge is scaled relative to that in the disk.Strictly speaking a full (time-consuming) integration is required to do this correctly, but numerical experiments show that eqn.B1 is a good approximation for most galaxies.
In the analysis that starts in §5, we look at the effects of varying the model scale lengths and Sérsic indices on L inc /L acc .To do this exactly would be a great deal of work in recomputing the time-consuming L inc /L acc integrations.Again there is a simple intuitive approximation, as follows.1. Calculate α inc = R inc /R ef f that corresponds to L inc for the component of interest.2. In the same way find α acc for L acc .3. Calculate the new α ′ inc = α inc /β, and the same for α ′ acc .Here β is the amount by which the original R ef f is scaled.4. Recompute L inc and L acc using the new α ′ inc and α ′ acc , and the new Sersic index n ′ . 5. Calculate the new value of L inc /L acc .Numerical experiments show that this approximation for perturbing models is adequate for our purposes.

B.2. Comparison of GIM2D and imfit Fitting
How well do GIM2D and imfit models agree?To answer this question, we compare computed L inc /L acc values (Fig. 9), because these are the principal diagnostics used in this paper.The sample of objects is the "Primary Sample" discussed in §3.4.L inc /L acc for the disk component, and also total light (the sum of the bulge and disk components) agree very well between the two programs.Not surprisingly, however, the bulge light component shows considerable scatter, because bulges are small and poorly resolved in these data ( §3.1).The scatter in the fitted value of B/T is σ B/T = 0.08, assuming each program contributes equally to the disagreement.We use this as a rough estimate of the systematic error in the fitted value of B/T .(For B/T = 0.5 the statistical 1σ error reported by the fitting programs is typically about ±0.07 for r = 20, and ±0.033 for r = 19.)Also shown is a comparison of L inc /L acc for the single component (pure Sérsic) models: GIM2D and imfit are in excellent agreement for these fits.In principle the overall completeness of our SN Ia sample, or of our host galaxies, should have no effect on our methodology, provided there is no radial gradient of SN completeness in our host galaxies.But radial gradients in completeness must exist, at least in the sub-sample of spectroscopically confirmed SDSS supernovae (Sako class SNIa): as discussed in Sako et al (2008), fainter SNe Ia (r ≳ 20.5 mag) are selected for follow-up according to a weight that depends on the amount of host galaxy light contamination.(The inclusion of photometric SNe Ia (classes zSNIa and pSNIa) compensates for this radial effect.)Dilday et al. (2008) discuss SDSS SN survey detection efficiency using artificial SNe injected into the survey data.According to Dilday et al. (2010b ApJ 715), "Analysis of these artificial SN Ia from the three observing seasons of the SDSS-II Supernova Survey does not show evidence for significant loss of efficiency near the cores of galaxies." To test this, we have compared the redshift distributions and SN maximum light distributions for SNe inside and outside the seeing disk.We find that the redshift distribution of inner (R < 0.75 arcsec) SNe is shifted ∼ 0.02 to larger z, and the apparent magnitude distribution is shifted by ∼ 0.1 mag to fainter magnitudes.These two effects are small, and in the opposite direction of what would be expected if the detection efficiency were lower in the central regions of host galaxies.
In the analysis ( § §5-7) we try three additional tests for incompleteness in the cores of galaxies: (1) we restrict the analysis to galaxies hosting brighter supernovae; (2) we compare L inc /L acc distributions excluding objects at small L inc /L acc ; and (3) we consider only supernovae outside the cores of galaxies, using a host-SN angular separation restriction (∆θ > 0.75 arcsec), and calculating L inc /L acc starting at this inner boundary.
Incompleteness may be more of a problem in the cores of bulge components.Because bulge cores have higher surface brightness, photon noise is greater, and image subtraction artifacts are more of a problem.

C.2. Supernova Light Contamination Bias
The Annis et al. stacks were made using some epochs that included SN light.Is it possible that this SN light bled through into the stacks, and biased our galaxy fitting?To test this, we repeated the imfit fitting procedure with the SN position masked for each host, using a circular mask of radius 0.8 arcsec.These results show that the effect of SN contamination on the imfit fits is quite small for most objects (Fig. 10).L inc /L acc values agree to within ±0.1 (±0.05) for 90 (80) per cent of the primary sample; none of the conclusions of the paper are affected.Most of the deviant points are centrally located supernovae, as is also the case in a comparison of masked and unmasked B/T values.

C.3. SN Positional Errors
As mentioned in §3.1, the centroids of SNe and their hosts are not strongly affected by seeing.For single SDSS scans, the centroiding error is σ xy ≃ 0.1 arcsec, both for stars at r = 22 mag (the brightness of our faintest SNe), and also for galaxies at r = 20 mag (our faintest hosts) (Pier et al. 2003).These errors will become smaller in the stacked images.
We have simulated the effects of this positional error on L inc /L acc for both disk and bulge components.The size of the effect depends on σ xy /r ef f and on Sérsic index n.For small disk galaxies (r ef f = 0.8 arcsec), the effect is negligible; for bulges with the same r ef f , there is a deficit of objects at L inc /L acc ≲ 0.1, which are scattered outwards from the cores of bulges.We therefore try restricting the bulge analysis to L inc /L acc ≥ 0.1 in §6.
From the above literature several things are clear.Galaxies with prominent bulges (B/T ≳ 0.2, roughly Sb or earlier) tend to have larger n bulge (though note the different n distributions for classical bulges in Fisher &Drory 2008 andGao et al. 2020), and there is a clear trend of increasing n bulge with B/T .Galaxies with B/T > 0.2 are the systems discussed in §6.Luminous galaxies (such as in magnitude limited surveys of SN hosts -e.g.§3.4) also tend to have large n bulge .Given that we cannot fit n bulge from our data, n bulge = 4 is a reasonable starting point.Our approach in §6 will be to see which value of n bulge best fits deviations from a straight line in the L inc /L acc plot calculated with n bulge = 4.
What is the effect of using an incorrect value of n bulge on our estimated value of B/T ?Surprisingly, not much.We have used imfit to fit seeing degraded models of M31 (n true bulge = 2.2, n f it bulge = 4) redshifted to z = 0.1 − 0.3, and find that B/T is recovered to ±0.03.This is because, at these redshifts, bulges are at best poorly resolved.Errors become larger for z < 0.1, seeing FWHM < 0.5 arcsec, or for large bulges.
Finally we note that, because of resolution effects, nuclei and bulges are difficult to distinguish in our data.Galaxies with bright nuclei will be fitted with an overly bright bulge and an r ef f value that is too small.

Figure 1 .
Figure1.Definition of Linc and Lacc.The upper panel shows a SN (in red) superimposed on its host galaxy.The red shaded area is the area used to calculate Linc; its boundary is defined by an isophote passing through the SN; this isophote is not necessarily elliptical.Lacc is computed within an elliptical boundary with semi-major axis a = 1.959a25 and semi-minor axis b = 1.959b25.This is the area within which a SN can be matched to its host; see §2.2.Note that Linc and Lacc are computed using the fitted model light distribution.The middle panel shows one simulation of 100 SNe in which SNe trace light (bin size 0.01).The lower panel shows 30 simulations of 100 SNe, plotted as cumulative distributions.

Figure 3 .
Figure3.Cumulative distribution of the radial offset of SNe Ia divided by the fitted half light radius of their host galaxies.Red solid line: bulge dominated objects (B/T > 0.75); blue solid line: disk dominated objects (B/T < 0.25).This is for the primary sample of objects, and the GIM2D analysis.The dot-dash lines are Sérsic models: blue: n = 1 (disk); green: n = 2.5; red: n = 4 (bulge).The Sérsic profiles have been corrected for missing light at ρ25 > 1.959, and hence do not pass through (R/HLR = 1, f = 0.5).

Figure 9 .
Figure 9.Comparison of Linc/Lacc from GIM2D and imfit fits of the host galaxy light distribution.The first three panels are from two component (bulge plus disk) models, with Linc/Lacc calculated from (left to right) fitted bulge light only, fitted disk light only, and fitted total (i.e.bulge plus disk) light.The right panel is calculated using a single component model (i.e.pure Sérsic or pS model).The plotted points are from the Primary Sample of objects ( §3.4): black -r host < 20, red -r host < 19, blue -r host < 18.

Table 1 .
Definition of the Primary Sample Sample GIM2D a imfit a d 6 objects do not have redshift information, and have been excluded.e After a visual inspection of the residuals.