Konrad Komorowski - Determination Of The Hubble Constant Based On Photometric Measurements Of Type Ia Supernova 2008ds Calibrated Using A Sample Of Nearby Type Ia Supernovae

Konrad Komorowski

Extended Essay in Physics

000704-005

An Extended Essay in Physics

Determination of the Hubble constant based on photometric measurements of Type Ia Supernova 2008ds calibrated using a sample of nearby Type Ia Supernovae

Candidate's name: Candidate's number: Session: Supervisor: Subject: Word count:

Konrad Komorowski 000704-005 May 2009 Ms. Elbieta Koziróg Physics 3897

Konrad Komorowski

Extended Essay in Physics

000704-005

Acknowledgements I would like to acknowledge and extend my heartfelt gratitude to the following persons who have made the completion of this Extended Essay possible: My supervisor, Ms. Elbieta Koziróg for motivating me to work systematically. Dr. Kevin Krisciunas of Texas A&M University for invaluable discussion of academic papers and providing me with essential data. Observation teams from the KAIT and Nickel telescopes at the Lick Observatory (University of California, Berkeley) for collecting data crucial to this Essay.

Konrad Komorowski

Extended Essay in Physics

000704-005

Abstract The aim of this paper was to experimentally determine the value of Hubble constant using a recent Supernova Type Ia explosion – SN 2008ds. The supernova's observations were refined with Milky Way dust extinction corrections, relativistic Doppler effect color shift corrections (K-corrections) and non-standard color filters adjustments (Scorrections). The brightness evolution rate was assessed using the m15 coefficient method (relative decline in brightness during 15 days after maximum) by writing an original light curve analysis program. The supernova's light curve was iteratively fitted to a set of discrete templates parametrized by m15, to yield a final value of m15 = 0.836 ± 0.071. The SN 2008ds measurements near the maximum were re-fitted to the templates to produce a nearcontinues representation of the supernova's brightness evolution near maximum. Hence the apparent magnitudes at maximum in different filters were found to be mB(tBmax) = 15.17 ± 0.04 [mag] and mV(tBmax) = 15.26 ± 0.03 [mag]. A set of 5 nearby supernovae (1989b, 1990n, 1991t, 1998aq and 1998bu) was chosen as a calibration sample. Their m15 coefficients were determined using the same method. Their absolute magnitudes at maximum were known from literature (thanks to the Cepheid-calculated distances to their host galaxies). In the sample, a clear correlations between m 15 and absolute brightness at maximum in different filters originated. The calibration functions (MB = -19.57 ± 0.19 [mag] and MV = -19.53 ± 0.17 [mag]) were used to estimate the absolute magnitudes of SN 2008ds at maximum. Compared to apparent magnitudes, they yielded the distance to SN 2008ds of DSN 2008ds = 92.05 ± 7.78 Mpc. The radial velocity between the Earth and SN 2008ds, uUGC 299 = 6238.65 ± 8.80 [km s-1], was calculated by applying the relativistic Doppler effect equations to the redshift of the supernova's host galaxy, UGC 299. The radial velocity and distance to SN 2008ds gave the final value of the Hubble constant of H0 = 68 ± 6 [km s-1 Mpc-1]. [283 words]

Konrad Komorowski

Extended Essay in Physics

000704-005

Table of Contents Introduction..........................................................................................................................................3 Type Ia Supernovae..............................................................................................................................4 Identification of Type Ia SNe...........................................................................................................4 Absolute magnitudes...................................................................................................................4 Spectroscopic features.................................................................................................................4 Brightness evolution...................................................................................................................4 Type Ia SNe as standardizable candles............................................................................................4 Standard and standardizable candles in astronomy.....................................................................4 Type Ia SNe as standardizable candles.......................................................................................4 Light curves..........................................................................................................................................5 UBVRI passbands............................................................................................................................5 Apparent magnitude scale................................................................................................................5 m15 – a numerical representation of SNe's decline rate.................................................................5 Set of templates parametrized by m15............................................................................................5 Light curve fitting program..................................................................................................................6 Algorithm for calculating the reduced 2 statistic between two light curves in one filter...............6 Algorithm for finding SN's m15 in a given filter............................................................................7 SN 2008ds............................................................................................................................................7 Spectrum..........................................................................................................................................7 Raw photometry...............................................................................................................................8 Analysis of SN 2008ds.........................................................................................................................8 Crude m15 assessment....................................................................................................................8 Galactic dust absorption corrections................................................................................................9 K-corrections...................................................................................................................................9 S-corrections....................................................................................................................................9 Application of the corrections and redetermination of m15.........................................................10 Apparent magnitudes at maximum................................................................................................11 Calibration of MX and m15 relationship............................................................................................11 Sample of nearby SNe...................................................................................................................11 Calibration function.......................................................................................................................12 Calculation of the Hubble constant....................................................................................................13 Distance to SN 2008ds...................................................................................................................13 Radial velocity of UGC 299..........................................................................................................13 The Hubble constant......................................................................................................................13 Evaluation and conclusion..................................................................................................................14 Comparison to the generally accepted value of the Hubble constant............................................14 Possible sources of error in the Hubble constant...........................................................................14 General comments about the method.............................................................................................14 References..........................................................................................................................................16 Appendices.........................................................................................................................................19 App. 1. – Source code of the light curve fitting Python script......................................................19 App. 2. – Spectrum of SN 2008ds.................................................................................................22 App. 3. – Uncorrected photometry of SN 2008ds from various sources.......................................22 App. 4. – SNe weighted for Jose Luis Prieto templates................................................................23 App. 5. – K-corrections for z = 0.021............................................................................................24 App. 6. – S-corrections for KAIT..................................................................................................24 1 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

App. 7. – S-corrections for Nickel.................................................................................................25 App. 8. – Corrected photometry of SN 2008ds.............................................................................25 App. 9. – Light curves fitted to the 5 nearby SNe.........................................................................26

2 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Introduction The aim of this Extended Essay is to determine the Hubble constant – a numerical representation of the rate of the expansion of the universe, as proposed by Edwin Hubble (1929). The Hubble constant is usually given in the units of km s -1 Mpc-1. It describes the relative velocity between any two objects across the universe divided by the distance between them (Equation 1). H 0

velocity km s distance Mpc

1

It is very important in astronomy because it allows to estimate distances to objects based only on their redshift. Precise orientation in spacetime of objects in the Universe is vital for analyzing the past and future of the Universe. The relative velocity between the Earth and any other object can be calculated using the Doppler effect for electromagnetic waves (Doppler 1846; Huggins 1868) corrected for the special relativity effects (Larmor 1897; Lorentz 1899, 1904; Einstein 1905). The wavelength of an identifiable emission/absorption line is measured in the laboratory (in the inertial frame of reference) and in the spectrum of the object (as it reaches the Earth). The two values, in conjunction with the speed of propagation of electromagnetic waves – c0, will make it possible to calculate the relative velocity between the Earth and the object. Since the Hubble's Law is a cosmological law – a description of large scale factors shaping the universe, objects at considerable distances need to be used for the assessment of the Hubble constant. This is to minimize the impact of local-scale forces, like direct gravitational attraction. However, determining reliable distances to points at cosmological scale poses difficulties. Small distances can be measured directly using purely geometrical methods, like the angle of parallax. Medium distances can be reached using common standard or standardizable candles, like tip of the red giant branch stars (Frogel et al. 1983) or Cepheid variables (Leavitt 1908). But distances at truly cosmological scales need other methods of measurement. Observations of Type Ia supernovae1, which can be calibrated as standardizable candles, are one possibility for this. In this essay I will analyze a recent explosion of SN 2008ds. I will calibrate its absolute magnitude using a sample of 5 nearby Type Ia SNe with known absolute magnitudes. Combining this data with the apparent magnitude of 2008ds, I will calculate distance to the object. The distance and relativistic Doppler effect-calculated speed of recession will give H0 – the Hubble constant value.

1 SN, plural SNe

3 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Type Ia Supernovae Identification of Type Ia SNe Type Ia SNe are a class of objects which have been observationally found to share similar features. These include their absolute magnitudes, electromagnetic spectrum and brightness evolution over time. Absolute magnitudes Type Ia SNe are a subset of all SNe – very bright stellar outbursts. Their absolute magnitudes at peak brightness are usually about M = -19.5 [mag] (Phillips 1993). Spectroscopic features Type Ia SNe are characterized by some distinct spectroscopic features, namely:

“The spectra of Type Ia supernovae (SNe Ia) lack hydrogen and He I lines but unlike Type Ic events they do include among other distinguishing characteristics a deep absorption feature near 6100Å that is produced by blueshifted Si II  6347Å, 6371Å.” (Branch & Tammann 1992) Brightness evolution It has been first noticed by Pskovskii (1967) that these events share remarkably similar light curves, i.e. changes in apparent magnitude over time.

Type Ia SNe as standardizable candles Standard and standardizable candles in astronomy A standard candle or standardizable candle is an object which can be easily identified and whose absolute magnitude is known. Thanks to the absolute magnitude/apparent magnitude/distance relation, standard (or standardizable) candles are used as distance indicators to places where they can be identified.

The standard photometric equation to transform from apparent to absolute magnitude using the distance to the light source is as follows (MX – absolute magnitude in filter X, mX – apparent magnitude in filter X, D – distance in parsecs): M X m X 55log10  D

2

When MX and mX of an object are known, distance can be calculated using Equation 3 (derived from Equation 2): D10

M X m X 5 5

 pc

3

Note: Standard candles are objects with exactly the same absolute magnitudes (e.g. brightest red giant stars in a galaxy). Standardizable candles can have their absolute magnitude determined using a parameter, e.g. the rate of brightness evolution (SNe) or period of brightness fluctuations (Cepheids). Type Ia SNe as standardizable candles Rust (1974) and other astronomers pointed out a possible relation between the SNe's light curve2 shape and their absolute magnitude. Namely, fast-decliners (objects which quickly achieve maximum brightness but also fade quickly) tend to be less bright than the so called slow-decliners. Hence, Type Ia SNe can potentially act as standardizable candles – indicators of distance to the places where they appear. 2 LC, plural LCs

4 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Light curves UBVRI passbands Photometry on objects in the sky is usually done in different filters. I.e. the apparent magnitudes of an object are measured collecting only ultraviolet (U), blue (B), green (V), red (R) or infrared (I) light with the use of strictly defined photographic filters (Bessell 1990). Photometry of SNe is also conducted in different passbands to perform a more detailed analysis of their brightness evolution.

Apparent magnitude scale The apparent magnitude brightness scale is a logarithmic relative scale, as described by Equation 4, where L1 is the intensity of light received from the object, L2 – intensity of light received from some comparison source, m2 is chosen as the arbitrary point of reference and m1 is the resultant apparent magnitude (Pogson 1856): m1m22.5 log10 

L1  L2

4

As the apparent magnitude depends only on the L1/L2 ratio, a change in intensity of emitted light by a certain constant factor will always result in the same linear change of apparent magnitude – regardless of absolute magnitude or the total intensity of received/emitted light. As a result, the shapes of apparent magnitude LCs of SNe remain roughly the same regardless of the distance to the events.

m15 – a numerical representation of SNe's decline rate In order to perform statistical computations on Type Ia SNe it is vital to adopt some numerical representation of the properties of their light curves. One of the ways of assessing the rate of SNe's decline in brightness is the m15 parameter. It has been first proposed by Phillips (1993) and describes the difference between B-filter apparent magnitude of a SN during its maximum brightness and 15 days after it. Large values like m15 = 1.8 will describe fast-decliners – objects that faint quickly. Small values like m15 = 0.8 will describe slow-decliners.

Set of templates parametrized by m15 In reality, it is hard to accurately determine the apparent magnitude of a SN at its maximum and exactly 15 days after the maximum if one is relying only on discrete observational data. Without a continuous light curve (which would imply constant observations of the SN every minute – day and night) it is hard to determine the time and visual magnitude of the maximum brightness. In order to overcome these difficulties I decided to make use of a set of SNe light curve templates prepared by Jose Luis Prieto, ranging from m15 = 0.830 to m15 = 1.930 (Prieto 2006). The templates are parametrized by the m15 coefficient (calculated individually for each SN by its observational team) and had been weighted from the light curves of 14 well observed SNe (App. 4.) of different m15 coefficients. Although the templates contain light curves in BVRI filters, they are parametrized by the decline in B-band apparent magnitude – to simplify the calculations. The maximum brightness point of light curve templates has the coordinates: t3 = 0 [days], mX4 = 0 [mag]. The templates extend from 15 days before the maximum to 80 days after the maximum.

3 Time since B-band maximum 4 Magnitude with respect to the maximum magnitude.

5 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Light curve fitting program

In order to evaluate SN 2008ds (and then the nearby SNe for calibration) in terms of the m15 coefficient, their LCs need to be compared with the templates and it has to be determined template of what m15 fits the evaluated supernova best. To do so a Python program was written (App. 1). It utilizes the reduced chi-square5 method (Laub & Kuhn n.d.) to assess the accuracy of fit between the LCs of real SNe and the templates.

Algorithm for calculating the reduced 2 statistic between two light curves in one filter Note: Photometric observations of a SN will be referred to as LC SN. The template with which they are compared will be referred to as LCT. 1. LCT is given in its rest frame. LCSN is given in the frame of reference of the observer on Earth. In order to compare LCT with LCSN the frames of reference need to be agreed in the way described in the standard relativistic Equation 5 (t' is time in LCT agreed to the frame of reference of LCSN, t is time in LCT in its rest frame, z is the redshift of SN): t 't1 z 5 2. It is assumed that the brightest measurement in LC SN corresponds to the maximum of LC T. LCSN is shifted so that the two maxima overlap. 3. LCSN is shifted by every combination of the horizontal and vertical shifts from Fig. 1., where v6 = 0.1 [days] and h7 = 0.1 [mag].

+40v

-40h

...

-20h

...

0

...

21

...

221

...

241

...

261

...

281

...

21681

...

21701

...

23301

...

23321

...

24921

...

24941

...

26541

...

26561

... +20v

... 21621

...

21641

...

... 0

23241

...

23261

...

23281 ...

24861

...

24881

...

... -40v

21661 ...

... -20v

+20h ... +40h

24901 ...

26481

...

26501

...

26521

Fig. 1. – Columns represent different shifts in time, rows represent different shifts in magnitude, inside the table are the results of 6,561 2 tests 4. In each of the 6561 combinations of vertical and horizontal shifts a reduced 2 test is performed (Equation 6) to assess the goodness of fit between the two LCs. Apparent magnitude from any observation Pi is the observed value – Oi. LCT is linearly interpolated to find the expected value – Ei. Oi is the uncertainty at Pi, Ei is the interpolated uncertainty at 5 2 6 The length of one step in moving the supernova's light curve in the time (vertical) axis. 7 The magnitude of one step in moving the supernova's light curve in the magnitude (horizontal) axis.

6 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

LCT. N is the number of observations: O iE i 2    2 i1 O E 2   N 1 N

i

i

6

5. The combination of vertical and horizontal shifts with the smallest 2 statistic is chosen to make the final shift of LCSN at this stage. 6. Procedures 3 to 5 are repeated for v = 0.01 days, h = 0.01 mag, then for v = 0.001 days and h = 0.001 mag, so that the fit is tightened and two light curves converge best. 7. The final shift of the last stage of point 6 is considered to be the best fit. Its 2 statistic is called the final goodness of fit between LCSN and LCT.

Algorithm for finding SN's m15 in a given filter

Note: Finding 2 statistic for a given m15 is defined as employing the previous algorithm to find the goodness of fit between LCSN and LCT (parametrized by an appropriate m15 value). 1. 2 is found for m15's from 0.83 to 1.93, with steps of 0.1. The value with best goodness of fit, 2, is chosen to be the temporary best m15. 2. Goodness of fit for m15's from (best m15 – ) to (best m15 + ) in steps of 0.1 is found. The amplitude  is 0.1 [mag or days]. The value with the lowest 2 goodness of fit is chosen as the new temporary best m15. 3. Step 2 is repeated for  = 0.01 [mag or days]. Temporary best m15 becomes the final m15.

SN 2008ds SN 2008ds, firstly known as SN 2008X3, was discovered by the Katzman Automatic Imaging Telescope (KAIT) at Lick Observatory on 28.47 June 2008 (UT) (Krisciunas 2008). It was identified to be a part of the UGC 299 galaxy (Leja et al. 2008) at redshift zUGC 299 = 0.021031 ± 0.000030 (Falco et al. 1999). It was observed by at least three telescopes: KAIT and Nickel telescopes at Lick Observatory (University of California, Berkeley) and the Takahashi refractor at the Etscorn Observatory (NMT – New Mexico Tech). Its photometry has not been published as of writing this essay so all the 2008ds data analyzed here comes from private communication with Dr. Kevin Krisciunas from Texas A&M University and has the permission to be published.

Spectrum A spectroscopic image of SN 2008ds (App. 2) was taken by the Lick observatory team (Silverman 2008). The shape of the spectrum closely resembles that of SN 1999aa – a Type Ia SN (Garavini et al. 2004). Distinct absorption feature is present at wavelength of about 6100Å. The outstanding similarity in shape is evident at first sight and serves as a proof that SN 2008ds is a Type Ia SN event too.

7 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Raw photometry Photometry in four filters has been prepared by each of the teams working on SN 2008ds. It is presented in App. 3. and Fig. 2.:

Fig. 2. – four graphs showing the variation in apparent magnitude of SN 2008ds. Horizontal axis is time represented by Julian Date8 – 2 454 000 (for readability purposes). Vertical axis is apparent magnitude in the corresponding filter.

Analysis of SN 2008ds Crude m15 assessment The Python program has been used to find the best fitting templates for the 2008ds data. The results are summarized in Table 1. and its comments. B

Vd

R

Ie

0.838

 8.300

0.838

 1.069

tBmax [days]

651.704

 650.889

652.010

 653.631

mX (tBmax)b [mag]

15.488

 15.471

15.399

 15.741

0.584

 0.767

0.894

 2.794

m15 a



2c

Table 1. – Properties of SN 2008ds as found by fitting the primary observational data to the templates. aTime of B-band maximum as derived from LCSN in particular filter in JD – 2,454,000; found from the horizontal shift of LC SN. bApparent magnitude in the appropriate band at B-band 8 Julian Date is defined as days since January 1, 4713 BC Greenwich noon

8 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

maximum; found from the vertical shift of LCSN. cReduced 2 statistic for fitting LCSN and LCT. d Could not be determined accurately because m15 was apparently smaller than 0.830 – outside the scope of templates. A rough estimate of 0.830 (slowest-declining template) was used to calculate the derived parameters. eThe template does not seem to be a good representation of the scattered Iband photometry at all (see Fig. 3., graph I and note 2 = 2.794). Although the machine calculated m15 and the derived parameters have been presented in the table, they should not be trusted. Since I-band photometry has been dropped later in the essay, any further analysis of SN 2008ds's I-band photometry would be redundant.

Galactic dust absorption corrections Light from SN 2008ds needs to pass through the interstellar matter in our galaxy – The Milky Way – before reaching telescopes on Earth. Galactic dust absorbs and scatters the light energy from the SN, decreasing its brightness. NASA/IPAC Extragalactic Database queried for an extragalactic object (like UGC 299 – the host galaxy of SN 2008ds) returns the absorption in the direction to the object using published Milky Way extinction maps (Schlegel et al. 1998). The values returned for UGC 299 are presented in Table 2. They are to be subtracted from the original photometry – to increase the apparent brightness to the real values, unaffected by dust extinction along the line of sight in the Milky Way.

AX [mag]

B

V

R

I

0.278

0.214

0.172

0.125

Table 2. – Galactic dust absorption towards UGC 299 in BVRI filters.

K-corrections If a source of light is moving away from the observer with a great velocity, its light is redshifted. In other words, its spectrum is shifted to the infrared. Intensities of light at particular wavelength that would be normally recorded in one filter (in the rest frame of the source of light) are sometimes recorded in the redder filter. As a result, the recorded photometry of the object is different in the observer frame than in the rest frame. K-corrections are mainly a function of redshift and spectrum of the object. Type Ia SNe share a lot of features, and among other have a similar spectrum evolution (Filippenko 1997). As a result, the K-corrections for them can be generalized (within the uncertainty of usual photometry) as a function of redshift and time since B-band maximum (arbitrary mean of accounting for spectral evolution in time). The appropriate generalized correction terms for z = 0.021, the value closest to zUGC 299, are listed in App. 5. (Jha 2007).

S-corrections UBVRI filters and their exact permittivities for different frequencies of light have been defined by Bessell (1990). In reality, not all filters behave like ideal Bessell filters. In order to account for this, S-corrections need to be employed. They are a function of the bandpass of the filter and spectrum of the object (Stritzinger et al. 2002). Since all Type Ia SNe share spectroscopic and color evolution features, it is possible to apply Scorrections of one SNe to the photometry of another SNe to get roughly corrected photometry. KAIT and Nickel telescopes have been used to observe SN 2005cf. It was identified by the 9 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

observing teams as another Type Ia SN and S-corrections have been calculated for different telescopes from 12 days before the maximum up to 87 days after the maximum of the SN (Wang et al. 2008). The figures are presented in App. 6. and App. 7. Since SN 2005cf and SN 2008ds are both very similar objects, S-corrections calculated for SN 2005cf may be used to roughly refine the photometry of SN 2008ds.

Application of the corrections and redetermination of m15 Whereas galactic dust extinction correction is a single figure, both K- and S-corrections are published as a function of days since B-band maximum. Rough estimate of tBmax was used to apply the corrections. Since S-corrections were not available for NMT photometry, this data was dropped. After applying the corrections, the Python program was run again in order to redetermine m15 (Table 3.). B

Vd

R

Ie

m15

0.838

 8.300

0.834

 1.025

tBmax a [days]

651.881

 651.188

651.949

 652.400

15.168

 15.263

15.314

 15.604

0.594

 0.785

0.919

 2.592

b

mX (tBmax) [mag] 

2c

Table 3. – Properties of SN 2008ds as found by fitting the corrected observational data to templates. See Table 1. for notes. See Fig. 3. for visualization.

Fig. 3. – corrected photometry of SN 2008ds imposed over the best fit templates 10 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Once again, the m15 coefficient in V-filter was beyond the scope of templates. Also, although Iband photometry was converging better it still did not achieve 2 < 1. Therefore the mean of m15 derived from BR bands is considered the final result: m15 SN 2008ds = 0.836. Due to the small number of partial m15 estimates of m15 SN 2008ds (only BR bands), its measurement uncertainty cannot be calculated in a formal way, as it is done in Table 6. But as the m15 uncertainties in Table 6. are all very similar (0.071 ± 0.012), their mean has been adopted as m15 SN 2008ds's measurement uncertainty, yielding m15 SN 2008ds = 0.836 ± 0.071.

Apparent magnitudes at maximum 7 near maximum measurements (less than 5 days before and after the approximate B-band maximum) were fitted to templates again, in order to find the most accurate values of the apparent magnitudes at B-band maximum. In the templates zero point of B-band light curve is exactly the B-band maximum. So the SN's B-band light curve's shift in time axis is equal to tBmax. Its shift in magnitude axis is equal to mB(tBmax). To receive apparent magnitude in the VRI filters, their templates were shifted to the nearmaximum measurements and interpolated for mX at the time of B-band maximum. The mean uncertainty of the 7 near-maximum measurements was adopted as the uncertainty of mX(tBmax). B mX(tBmax) [mag]

V

R

I

15.17 ± 0.04 15.26 ± 0.03 15.31 ± 0.03 15.62 ± 0.04

Table 4. – Apparent magnitudes of SN 2008ds in different filters at B-band maximum

Calibration of MX and m15 relationship Sample of nearby SNe 5 SNe (1989b, 1990n, 1991t, 1998aq and 1998bu) have been chosen to be the calibration sample for SN 2008ds. Their m15 was determined using the same method as for SN 2008ds. Their absolute magnitudes at the time of B-band maximum9 (Table 5.) are known from literature, thanks to the Cepheid variables in their host galaxies. SN

MB (maxB) [mag] MV (maxB) [mag] MR (maxB) [mag] MI (maxB) [mag] Host galaxy Redshift Reference

1989b

−19.48 ± 0.07

−19.48 ± 0.07

...

...

NGC 3627 0.002425

(1) (6)

1990n

−19.52 ± 0.07

−19.48 ± 0.07

...

...

NGC 4639 0.003395

(2) (7)

1991t

−19.56 ± 0.23

−19.59 ± 0.19

...

−19.21 ± 0.13

NGC 4527 0.005791

(3) (8)

1998aq

−19.56 ± 0.21

−19.48 ± 0.20

...

...

NGC 3982 0.003699

(4) (6)

1998bu

−19.67 ± 0.20

−19.63 ± 0.19

−19.59 ± 0.18

−19.36 ± 0.18

NGC 3368 0.002992

(5) (6)

Table 5. – absolute magnitudes of 5 nearby SNe. References for absolute magnitudes calibration: (1) Saha et al. 1999 (2) Saha et al. 1997 (3) Gibson & Stetson 2001 (4) Saha et al. 2001 (5) Suntzeff et al. 1999. References for redshift of the host galaxy: (6) De Vaucouleurs et al. 1991 (7) Wong et al. 2006 (8) Strauss et al. 1992 Results of the reduction of the SNe's light curves using the Python script are presented in Table 6. and App. 9. 9 MX; as the SNe's absolute magnitudes in this paper are given only at the time of B-band maximum, there is no need to call the variable a more formal but longer name – MX(tBmax); the only times where MX(tBmax) [shortened to MX] will be given in a functional form, is when it described as an empirical function of m15, not time

11 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

SN

m15 from B

m15 from V

m15 from R

m15 from I

Mean m15

Reference

1989b

1.01.241

1.01.131

1.01.211

1.01.335

1.230 ± 0.084

(1)

1990n

1.01.1974

1.01.2005

0.925

1.01.119

1.031 ± 0.085

(2)

0.831

0.902

0.925

0.865 ± 0.058

(2)

1991t

a

0.802

b

1998aqa

1.01.198

1.01.146

1.01.1968

1.01.1967

1.120 ± 0.064

(3)

1998bu

1.01.1953

1.01.155

1.01.1992

1.01.199

1.125 ± 0.065

(4)

Table 6. – m15 coefficients of 5 nearby SNe, as derived from data in different filters (columns 2-5). References for the photometry used for fitting light curves: (1) Wells et al. 1994 (2) Lira et al. 1998 (3) Riess et al. 2005 (4) Suntzeff et al. 1999 aAs the rise rate did not match the decline rate only post-maximum data was fitted to the templates. bAs the 2 function changes quadratically when it approaches the minimum (Bevington & Robinson 1992), the value of this particular m15 has been extrapolated by fitting a quadratic function to the fitting results of templates from 0.830 to 0.840.

Calibration function Since all of the reference SNe had their absolute magnitude determined only in BV filters, relationships in these two filters will be investigated. Data analysis program Pro Fit has found the following reduced best fit lines between MX's and m15's: M B  m15 0.2078×  m15 – 19.7473  mag  M V   m15 0.1296 ×  m15 – 19.6394 mag 

7 8

A Python script has been written to generate 100,000 instances of each SNe. They have been randomized in MX and m15 with the Gaussian distribution (using the measurement uncertainty as the standard deviation). The standard deviation from the best fit lines for the total of 500,000 SNe was found to be 0.187 mag in B and 0.171 mag in V. Therefore, the final calibration functions, as shown in Fig. 4., are: M B  m15 0.2078×  m15 – 19.7473 ± 0.187 mag  M V   m15 0.1296 ×  m15 – 19.6394 ± 0.171  mag

9 10

Fig. 4. – relationship between MX and m15 in BV bands for 5 nearby SNe, dashed lines represent 1 standard deviation from the best fit line

12 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Calculation of the Hubble constant Distance to SN 2008ds Equations 9 & 10 were used to estimate the absolute magnitudes of SN 2008ds at B-band maximum using the value of m15 SN 2008ds: M B 0.836 ± 0.0710.2078 ×0.836 ± 0.071 – 19.7473 ± 0.187 mag  M V 0.836 ± 0.0710.1296 ×0.836 ± 0.071 – 19.6394 ± 0.171 mag 

11 12

A 5 million iterations Monte Carlo simulation returned: M B– 19.573 ± 0.187  mag  M V – 19.531 ± 0.171 mag  MX estimates from above and mX(tBmax) optical observations (Table 4.) were inserted into Equation 3 to give the distance to SN 2008ds: D B 10 DV 10

19.573 ±0.18715.17 ± 0.045 5 19.531 ± 0.171 15.26 ±0.03 5 5

 pc   pc 

13 14

A 5 million iterations Monte Carlo simulation run on Equations 13 & 14 gave: DB 92.98 ± 8.12 Mpc DV 91.12 ± 7.30 Mpc Weighted mean (5 million iterations Monte Carlo simulation) of which is: D SN 2008ds92.05 ± 7.78  Mpc

Radial velocity of UGC 299 The redshift of the light spectrum of UGC 299 can be used to calculate the velocity between the Milky Way and SN 2008ds. Redshift z is defined theoretically in Equation 15 (Taylor 2007), where u is the radial velocity between two objects, and c is the speed of light.

 

u 1  c z 1 u 1  c

15

The standard Equation 15 can be easily solved for u: uc

 z 121  z 121

16

The experimentally determined redshift of UGC 299 is zUGC 299 = 0.021031 ± 0.000030 (Falco et al. 1999). A five million iterations Monte Carlo simulation substituting zUGC 299 for z in Equation 16 yields: 1

u UGC 299 6238.65 ± 8.80 km s 

The Hubble constant The Hubble constant is defined as the ratio of velocity and distance between any two points in the universe. Assuming that the universe is homogenous and the relationship between our galaxy and UGC 299/SN 2008ds is typical of the universe's large scale structure, the Hubble constant can be 13 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

calculated from the velocity/distance ratio to this object. A Monte Carlo simulation for inputting uUGC 299 and DSN 2008ds into Equation 1 yields: 1

1

H 068 ± 6 km s Mpc 

Evaluation and conclusion Comparison to the generally accepted value of the Hubble constant The generally accepted values of H0 are for example 72 ± 8 [km s-1 Mpc-1] (Freeman et al. 2001) or more recently 70.5 ± 1.3 [km s-1 Mpc-1] (Komatsu et al. 2008). The result of this paper (68 ± 6 [km s-1 Mpc-1]) is therefore in perfect accordance with the common scientific knowledge.

Possible sources of error in the Hubble constant The small discrepancy between professional results and the ones received using this method could have resulted from many factors, but most probably from the following ones. The population of five SNe investigated as the calibration sample could have different properties from the far away supernovae, like SN 2008ds, which are unreachable with conventional astronomical measuring methods. In such case, the whole calibration procedure would be inherently flawed and give incorrect results. The only way of knowing that the distant SNe are truly similar to the nearby ones would be to investigate and fully understand the physical mechanism behind a Type Ia SN explosion, instead of just statistically analyzing their light curve properties and similarities. Peculiar motions affecting the velocity of our galaxy (Karachentsev & Makarov 1996), like the gravitational attraction to the Andromeda Galaxy, affect the radial velocity between the Earth and UGC 299. They have nothing to do with the large scale structure of the universe, therefore should be eliminated in any Hubble constant calculation. The way to do it would be to assess the velocity of Earth with respect to the Cosmic Microwave Background radiation (Fixsen et al. 1996) and remove this vector from the redshift-calculated velocity. Same peculiar motions probably affect UGC 299. To minimize the effects of the peculiar motions of the SN's host galaxy, many more objects than just SN 2008ds should be investigated and plotted on a velocity against distance graph. The tangent of the regression line would give a more universal and peculiar motion-free value of the Hubble constant. It has been proposed that the Hubble constant is changing over time (Riess et al. 1998). This implies a very scientifically advanced discussion of what exactly has been calculated by studying the relation between just the Earth and SN 2008ds. Again, observing more objects than just one would provide the necessary base for drawing any conclusions. The S- and K- corrections were either copied from the corrections for a similar object or generalized for the whole sample of SNe. The calculation of said corrections individually for SN 2008ds would give better results, but would also require the use of complicated procedures, which are beyond the scope of this project, and the frequent spectroscopic observations of the object, which are also unavailable.

General comments about the method The scientific method for determining the Hubble constant presented in this essay is not free from numerous potential sources of error enlisted above, nevertheless it allowed to calculate the Hubble constant accurately. The method is elegant (final step is taken directly from Hubble's equation), easily understandable and potentially extendable – bigger samples could be analyzed to reach more precise result. The error analysis method was very simple – each equation involving uncertainties was randomized 14 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

using the Gaussian distribution and this was repeated 5 million times. Although, it reduced the need for complicated error equations – simplified the method and made it possible to avoid the introduction of calculation errors, it was at the expense of computing power. This is why an extension of the method over a bigger sample could require either changes in the error analysis method or computing power bigger than that of a standard personal computer, as used in this essay.

15 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

References Bessell, M. S. (1990). UBVRI passbands. Publications of the Astronomical Society of the Pacific, 102, 1181-1199. Bevington, P. R., & Robinson, D. K. (1992). Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, Inc. Branch, D., & Tammann, G. A. (1992). Type Ia Supernovae as Standard Candles. Annual Review of Astronomy and Astrophysics, 30, 359-389. De Vaucouleurs, G., et al. (1991). Third Reference Catalogue of Bright Galaxies. Springer-Verlag Berlin Heidelberg New York, 1-3, 7. Doppler, C. A. (1846). Beitrage zur fixsternenkunde. Druck von G. Haase sohne. Einstein, A. (1905). On the Electrodynamics of Moving Bodies. Annalen der Physik, 17. Falco, E. E., et al. (1999). The Updated Zwicky Catalog (UZC). Publications of the Astronomical Society of the Pacific, 111(438). Filippenko, A. V. (1997). Optical Spectra of Supernovae. Annual Review of Astronomy and Astrophysics, 35, 309-355. Fixsen, D. J., et al. (1996). The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set. The Astrophysical Journal, 576. Freeman, W. L., et al. (2001). Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant. The Astrophysical Journal, 553, 47-72. Frogel, J. A., et al. (1983). Globular cluster giant branches and the metallicity scale. The Astrophysical Journal, 275, 773-789. Garavini, G., et al. (2004). Spectroscopic Observations and Analysis of the Peculiar SN 1999aa. The Astronomical Journal, 128, 387-404. Gibson, B. K., & Stetson, P. B. (2001). Supernova 1991T and the Value of the Hubble Constant. The Astrophysical Journal, 547(3), L103-L106. Hubble, E. (1929). A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae. Proceedings of the National Academy of Sciences of the United States of America, 15(3), 168-173. Huggins, W. (1868). Further Observations on the Spectra of Some of the Stars and Nebulae, with an Attempt to Determine Therefrom Whether These Bodies are Moving towards or from the Earth, Also Observations on the Spectra of the Sun and of Comet II., 1868. Philosophical Transactions of the Royal Society of London, 158, 529-564. Jha, S., et al. (2007). Improved Distances to Type Ia Supernovae with Multicolor Light-Curve Shapes: MLCS2k2. The Astrophysical Journal, 659(1), 122-148. Karachentsev, I. D., & Makarov, D. A. (1996). The Galaxy Motion Relative to Nearby Galaxies and the Local Velocity Field. The Astronomical Journal, 111, 794. Komatsu, E., et al. (2008). Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. The Astrophysical Journal Supplement Series. Krisciunas, K. (2008). Supernova 2008ds. In Newest home page of Kevin Krisciunas (Texas A & M). Retrieved December 22, 2008, from http://faculty.physics.tamu.edu/krisciunas/sn2008ds.html 16 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Krisciunas, K. (2009). Private communication. Larmor, J. (1897). A dynamical theory of the electric and luminiferous medium ? Part III: Relations with material media. Philosophical Transactions of the Royal Society of London, 190, 205-300. Laub, C., & Kuhl, T. L. (n.d.). A Critical Look at the Fitting of Reflectivity Models using the Reduced Chi-Square Statistic. Department of Chemical Engineering and Materials Science, University of California, Davis. Retrieved September 15, 2008, from http://neutrons.ornl.gov/workshops/sns_hfir_users/posters/Laub_Chi-Square_Data_Fitting.pdf Leavitt, H. S. (1908). 1777 Variables in the Magellanic Clouds. Annals of Harvard College Observatory, LX(IV), 87-110. Leja, J., et al. (2008). Central Bureau Electronic Telegrams, 1419(1). Lira, P., et al. (1998). Optical light curves of the Type IA supernovae SN 1990N and 1991T. The Astronomical Journal, 115, 234-246. Lorentz, H. A. (1899). Simplified theory of electrical and optical phenomena in moving systems. Proc. Acad. Science Amsterdam, I, 427-443. Lorentz, H. A. (1904). Electromagnetic phenomena in a system moving with any velocity less than that of light. Proc. Acad. Science Amsterdam, IV, 669-678. Phillips, M. M. (1993). The absolute magnitudes of Type IA supernovae. The Astrophysical Journal, 413, L105-L108. Pogson, N. R. (1856). Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857. Monthly Notices of the Royal Astronomical Society, 17, 12. Prieto, J. L. (2006). SN Ia distances using a new dm15 method. In Jose @ OSU. Retrieved July 7, 2008, from http://cassini.mps.ohio-state.edu/~prieto/paper_dm15/ Pskovskii, Y. P. (n.d.). The Photometric Properties of Supernovae. Astronomicheskii Zhurnal, 44, 82. Riess, A. G., et al. (1998). Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. The Astronomical Journal, 116(3). Riess, A. G., et al. (2005). Cepheid Calibrations from the Hubble Space Telescope of the Luminosity of Two Recent Type Ia Supernovae and a Redetermination of the Hubble Constant. The Astrophysical Journal, 627(2), 579-697. Rust, B. W. (1974). Use of supernovae light curves for testing the expansion hypothesis and other cosmological relations. Ph.D. Thesis Oak Ridge National Lab., TN. Saha, A., et al. (1999). Cepheid Calibration of the Peak Brightness of Type Ia Supernovae. IX. SN 1989B in NGC 3627. The Astrophysical Journal, 522, 802-838. Saha, A., et al. (1997). Cepheid Calibration of the Peak Brightness of Type Ia Supernovae VIII: SN 1990N in NGC 4639. The Astrophysical Journal, 486(1), 1-20. Saha, A., et al. (2001). Cepheid Calibration of the Peak Brightness of Type Ia Supernovae. XI. SN 1998aq in NGC 3982. The Astrophysical Journal, 562(1), 314-336. Schlegel, D. J., et al. (1998). Maps of Dust Infrared Emission for Use in Estimation of Reddening and Cosmic Microwave Background Radiation Foregrounds. Astrophysical Journal, 500, 525. Silverman, J. M., et al. (2008). Central Bureau Electronic Telegrams, 1419(2). 17 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Strauss, M. A., et al. (1992). A redshift survey of IRAS galaxies. VII - The infrared and redshift data for the 1.936 Jansky sample. Astrophysical Journal Supplement Series, 83(1), 29-63. Stritzinger, M. I., et al. (2002). Optical Photometry of the Type Ia Supernova 1999ee and the Type Ib/c Supernova 1999ex in IC 5179. The Astronomical Journal, 124, 2100-2117. Suntzeff, N. B., et al. (1999). Optical Light Curve of the Type IA Supernova 1998BU in M96 and the Supernova Calibration of the Hubble Constant. The Astronomical Journal, 117(3), 1175-1184. Taylor, S. M. (2007). General Doppler Shift Equation and the Possibility of Systematic Error in Calculation of Z for High Redshift Type Ia Supernovae. Retrieved December 28, 2008, from http://arxiv.org/pdf/0704.1303 Wang, X., et al. (2008). A Golden Standard Type Ia Supernova SN 2005cf: Observations from the Ultraviolet to the Near-Infrared Wavebands. Draft version. Retrieved November 10, 2008, from http://arxiv.org/pdf/0811.1205v2 Wells, L. A., et al. (1994). The Type IA supernova 1989B in NGC 3627 (M66). The Astronomical Journal, 108(6), 2233-2250. Wong, I., et al. (2006). HI Parkes All Sky Survey Final Catalog.

18 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

Appendices App. 1. – Source code of the light curve fitting Python script #!/usr/bin/env python2.5 # -*- coding: utf-8 -*# discrete data fitting algorithm # by Konrad Komorowski # as a part of an Extended Essay in IB Physics # ------------------# IMPORT STATEMENTS # ------------------from __future__ import division import copy # ------------------# FUNCTIONS # ------------------def arange(start, stop=None, step=1): """if start is missing it defaults to zero, somewhat tricky""" if stop == None: stop = start start = 0 # allow for decrement if step < 0: while start > stop: yield start # makes this a generator for new start value start += step else: while start < stop: yield start start += step def interpolate(x_1, y_1, x_2, y_2, x): """Returns the value of y(x) for a given x between two points.""" return (y_1 - y_2) / (x_1 - x_2) * x + (x_1 * y_2 - x_2 * y_1) / (x_1 - x_2) def findChiSquare(observations, template, filters, reduced, shift, fittingParametersNumber): """Given some vertical and horizontal shift finds the minimum chi-square statistic in fitting data to the template.""" count = 0 chi2_sum = 0 adjustedObservations = copy.deepcopy(observations) for filter in filters: for datapoint in adjustedObservations[filter]: datapoint[0] = datapoint[0] + shift[0] datapoint[1] = datapoint[1] + shift[1] # find reference point in template (guess it is the first point) templateClosestDateID = 0 templateClosestDate = template[filter][0][0] maxDate = template[filter][0][0] minDate = template[filter][0][0] for i in arange(0, len(template[filter])): if abs(datapoint[0] - template[filter][i][0]) < abs(datapoint[0] - templateClosestDate): templateClosestDate = template[filter][i][0] templateClosestDateID = i if template[filter][i][0] > maxDate: maxDate = template[filter][i][0] if template[filter][i][0] < minDate: minDate = template[filter][i][0] if (datapoint[0] > maxDate) or (datapoint[0] < minDate): continue if (template[filter][i][0] - templateClosestDate) <= 0: templatePreviousDateID = templateClosestDateID else: templatePreviousDateID = templateClosestDateID - 1 try:

19 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

expectedValue = interpolate(template[filter][templatePreviousDateID][0], \ template[filter][templatePreviousDateID][1], \ template[filter][(templatePreviousDateID+1)][0], \ template[filter][(templatePreviousDateID+1)][1], datapoint[0]) if reduced: sdmTemplate = interpolate(template[filter][templatePreviousDateID][0], \ (template[filter][templatePreviousDateID][2])**0.5, \ template[filter][(templatePreviousDateID+1)][0], \ (template[filter][(templatePreviousDateID+1)][2])**0.5, datapoint[0]) sdmMeasurement = datapoint[2] sdm = sdmTemplate + sdmMeasurement else: sdm = 1 except IndexError: continue observedValue = datapoint[1] delta = observedValue - expectedValue chi2 = (delta / sdm)**2 chi2_sum = chi2_sum + chi2 count = count + 1 if count < 4: return -1 return (chi2_sum / (count - 1)) def loadData(template, name): """Loads a data file into the programs memory. Can be used to load either a light curve template or a light curve of an unknown object.""" global z global sdmz if template: dataFile = open("templates/light_curve_%.3f" % name, "r") else: dataFile = open("%s" % name, "r") data = {} for filter in allFilters: data[filter] = [] line = "line" while (line != ""): line = dataFile.readline() if (line[0:1] == "*") and not(template): lineSplit = line.split() z = float(lineSplit[1]) sdmz = float(lineSplit[2]) continue if (line[0:1] != "#") and (len(line) > 0): lineSplit = line.split() if template: time = float(lineSplit[0]) * (1 + z) else: time = float(lineSplit[0]) for filter in allFilters: try: data[filter].append([time, float(lineSplit[filtersPositions[filter][0]]), \ float(lineSplit[filtersPositions[filter][1]])]) except ValueError: continue return data def findClassicBestFit(observations, template, filters, reduced, degreesOfCheck): """Finds the minimal (reduced or not) chi-square statistic, minimum of the B band light curve, the best shift

20 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

for a set of observational data against a given template.""" for filter in filters: M_max_aprox_time = observations[filter][0][0] M_max_aprox = observations[filter][0][1] for datapoint in observations[filter]: if datapoint[1] < M_max_aprox: M_max_aprox_time = datapoint[0] M_max_aprox = datapoint[1] bestShift = [-M_max_aprox_time, -M_max_aprox] bestShiftError = findChiSquare(observations, template, filters, reduced, bestShift, 1) centralBestShift = copy.deepcopy(bestShift) for step in [0.1, 0.01, 0.001]: for horizontalShift in arange (centralBestShift[0] – (step*degreesOfCheck), \ centralBestShift[0] + (step*degreesOfCheck), step): for filter in filters: for verticalShift in arange (centralBestShift[1] – (step*degreesOfCheck/10), \ centralBestShift[1] + (step*degreesOfCheck/10), step): temporaryShift = (horizontalShift, verticalShift) error = findChiSquare(observations, template, filter, reduced, temporaryShift, 1) if (error < bestShiftError) and error != -1: bestShift[0] = horizontalShift bestShift[1] = verticalShift bestShiftError = error centralBestShift = copy.deepcopy(bestShift) print "[%7.3f, %7.3f] %9.3f" % (bestShift[0], bestShift[1], bestShiftError) return (bestShift, bestShiftError) # ------------------# CONSTANTS AND VARIABLES # ------------------allFilters = ("B", "V", "R", "I") filtersPositions = {"B":(1,2), "V":(3,4), "R":(5,6), "I":(7,8)} z = 0 sdmz = 0 # ------------------# PROGRAM # ------------------def fit(name): for filter in allFilters: print filter + " filter" print errors = [] for resolution in (0.1, 0.01, 0.001): if resolution == 0.1: lowLimit = 0.83 highLimit = 1.3 else: bestDm15 = errors[0][0] smallestError = errors[0][1] for i in errors: if i[1] < smallestError: bestDm15 = i[0] smallestError = i[1] lowLimit = bestDm15 - (resolution * 10) highLimit = bestDm15 + (resolution * 10) if lowLimit < 0.83: lowLimit = 0.83 if highLimit > 1.8:

21 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

highLimit = 1.8 for i in arange(lowLimit, highLimit, resolution): print "%5.3f " % i, errors.append((i, findClassicBestFit(loadData(False, name), loadData(True, i), \ (filter), True, 40)[1])) print fit("sn2008ds_corrected_MAXIMUM")

App. 2. – Spectrum of SN 2008ds

App. 2. – Early spectrum of SN 2008ds, initially called SN 2008X3, (black line, Silverman et al. 2008) compared with two spectra of SN 1999aa from different times (where indicated – Garavini et al. 2004, “our database” – Silverman et al. 2008). The spectra of the two objects are remarkably similar, proving that the explosion discovered on 28.47 June 2008 (UT) was a Type Ia SN.

App. 3. – Uncorrected photometry of SN 2008ds from various sources ta

mB

mV

mR

mI

Source

647.89

15.714 ± 0.057

15.540 ± 0.044

15.514 ± 0.035

15.800 ± 0.074

NMT

651.91

15.458 ± 0.042

15.421 ± 0.028

15.389 ± 0.031

15.605 ± 0.050

NMT

654.87

15.607 ± 0.046

15.393 ± 0.031

15.406 ± 0.033

15.689 ± 0.056

NMT

663.97

...

15.924 ± 0.085

...

...

NMT

22 of 26

Konrad Komorowski

a

Extended Essay in Physics

000704-005

ta

mB

mV

mR

mI

Source

664.96

16.152 ± 0.055

15.859 ± 0.034

...

...

NMT

666.96

16.321 ± 0.145

15.968 ± 0.035

15.934 ± 0.040

...

NMT

670.94

16.592 ± 0.066

16.114 ± 0.038

16.158 ± 0.036

16.253 ± 0.068

NMT

653.63

15.462 ± 0.028

15.459 ± 0.015

15.400 ± 0.006

15.757 ± 0.024

Nickel

655.92

15.564 ± 0.029

15.499 ± 0.013

15.404 ± 0.011

15.813 ± 0.009

Nickel

658.93

15.664 ± 0.025

15.588 ± 0.009

15.539 ± 0.007

15.963 ± 0.010

Nickel

662.96

15.984 ± 0.033

15.787 ± 0.020

...

...

Nickel

670.90

16.739 ± 0.053

16.214 ± 0.027

16.209 ± 0.025

16.438 ± 0.031

Nickel

672.88

16.978 ± 0.048

16.285 ± 0.016

16.244 ± 0.015

16.367 ± 0.024

Nickel

694.89

18.437 ± 0.143

17.254 ± 0.039

16.892 ± 0.032

16.641 ± 0.050

Nickel

647.96

15.575 ± 0.033

15.603 ± 0.032

15.551 ± 0.037

15.658 ± 0.058

KAIT

649.97

15.547 ± 0.043

15.550 ± 0.033

15.510 ± 0.045

15.699 ± 0.039

KAIT

650.97

15.511 ± 0.050

15.476 ± 0.027

15.423 ± 0.037

15.725 ± 0.039

KAIT

653.94

15.529 ± 0.038

15.461 ± 0.030

15.442 ± 0.040

15.755 ± 0.039

KAIT

655.98

15.605 ± 0.059

15.566 ± 0.029

15.495 ± 0.048

15.954 ± 0.049

KAIT

662.93

16.026 ± 0.046

15.813 ± 0.032

15.838 ± 0.044

16.433 ± 0.055

KAIT

666.92

16.457 ± 0.061

16.022 ± 0.036

16.107 ± 0.049

16.560 ± 0.065

KAIT

674.97

17.234 ± 0.054

16.394 ± 0.046

16.263 ± 0.038

16.479 ± 0.046

KAIT

677.93

17.458 ± 0.055

16.524 ± 0.041

16.302 ± 0.044

16.340 ± 0.049

KAIT

680.98

17.602 ± 0.064

16.641 ± 0.059

16.374 ± 0.051

16.420 ± 0.060

KAIT

683.93

17.822 ± 0.108

16.821 ± 0.041

16.428 ± 0.057

16.448 ± 0.059

KAIT

686.93

18.065 ± 0.070

16.917 ± 0.039

16.552 ± 0.047

16.381 ± 0.056

KAIT

692.94

18.281 ± 0.103

17.411 ± 0.053

16.809 ± 0.054

16.608 ± 0.054

KAIT

697.92

...

17.636 ± 0.140

16.909 ± 0.097

16.812 ± 0.104

KAIT

Julian Date minus 2,454,000

App. 4. – SNe weighted for Jose Luis Prieto templates SN

m15

SN

m15

1992al

1.110

1994ae

0.938

1992bc

0.870

1995d

1.052

1991bg

1.930

1995al

0.894

1992bo

1.690

1996x

1.251

1991t

0.940

1998bu

1.047

1992a

1.470

1999aa

0.827

1994d

1.343

2001el

1.120

23 of 26

Konrad Komorowski

Extended Essay in Physics

000704-005

App. 5. – K-corrections for z = 0.021 ta

a

KB

KV

KR

KI

ta

KB

KV

KR

KI

ta

KB

KV

KR

-7

-0.016 -0.023 -0.034 -0.060

14

0.039

0.014 -0.051 0.046

35 0.065 0.068 -0.055 0.044

-6

-0.014 -0.022 -0.038 -0.054

15

0.042

0.017 -0.050 0.046

36 0.064 0.068 -0.055 0.043

-5

-0.013 -0.020 -0.042 -0.048

16

0.045

0.021 -0.049 0.045

37 0.064 0.067 -0.055 0.042

-4

-0.011 -0.019 -0.046 -0.042

17

0.047

0.024 -0.049 0.045

38 0.063 0.067 -0.055 0.040

-3

-0.009 -0.017 -0.049 -0.035

18

0.050

0.027 -0.048 0.044

39 0.062 0.066 -0.055 0.039

-2

-0.007 -0.016 -0.052 -0.028

19

0.053

0.031 -0.048 0.043

40 0.062 0.065 -0.055 0.037

-1

-0.005 -0.014 -0.055 -0.021

20

0.055

0.035 -0.048 0.042

41 0.062 0.064 -0.054 0.035

0

-0.003 -0.013 -0.057 -0.014

21

0.057

0.038 -0.048 0.041

42 0.061 0.062 -0.054 0.032

1

-0.001 -0.011 -0.059 -0.007

22

0.059

0.042 -0.048 0.041

43 0.061 0.060 -0.053 0.030

2

0.002 -0.010 -0.060 -0.001

23

0.061

0.045 -0.048 0.040

44 0.060 0.059 -0.052 0.027

3

0.004 -0.008 -0.061 0.006

24

0.063

0.048 -0.049 0.040

45 0.060 0.057 -0.052 0.024

4

0.007 -0.007 -0.061 0.012

25

0.064

0.052 -0.049 0.041

46 0.060 0.055 -0.051 0.021

5

0.010 -0.005 -0.061 0.018

26

0.065

0.054 -0.050 0.041

47 0.060 0.053 -0.050 0.018

6

0.013 -0.004 -0.061 0.024

27

0.066

0.057 -0.050 0.042

48 0.059 0.051 -0.050 0.015

7

0.016 -0.002 -0.060 0.029

28

0.067

0.059 -0.051 0.042

49 0.059 0.050 -0.050 0.012

8

0.019

0.000 -0.059 0.033

29

0.067

0.061 -0.051 0.043

50 0.059 0.048 -0.050 0.008

9

0.023

0.002 -0.058 0.037

30

0.067

0.063 -0.052 0.044

10

0.026

0.004 -0.056 0.040

31

0.067

0.065 -0.053 0.044

11

0.029

0.006 -0.055 0.043

32

0.067

0.066 -0.053 0.044

12

0.032

0.009 -0.054 0.044

33

0.066

0.067 -0.054 0.044

13

0.035

0.011 -0.052 0.045

34

0.066

0.067 -0.054 0.044

Time since B-band maximum

App. 6. – S-corrections for KAIT

a

KI

ta

SB

SV

SR

SI

ta

-10

-0.033

0.003

0.002

-0.012

20

-5

-0.028 -0.004

0.008

-0.002

0

-0.027 -0.004

0.018

5

-0.027 -0.004

10 15

SR

SI

-0.011 -0.004

0.013

-0.012

25

-0.021 -0.001

0.012

-0.007

-0.008

30

-0.025

0.003

0.011

-0.001

0.025

-0.013

35

-0.025

0.003

0.011

0.004

-0.012 -0.004

0.019

-0.018

40

-0.026

0.004

0.012

0.005

-0.010 -0.005

0.014

-0.017

45

-0.028

0.004

0.015

-0.006

Time since B-band maximum

24 of 26

SB

SV

Konrad Komorowski

Extended Essay in Physics

000704-005

App. 7. – S-corrections for Nickel

a

ta

SB

SV

SR

SI

ta

SB

SV

SR

SI

-10

-0.039

0.003

0.002

0.017

20

-0.023

0.006

0.027

0.048

-5

-0.033 -0.010

0.014

0.012

25

-0.041

0.007

0.024

0.048

0

-0.024 –0.009

0.026

0.012

30

-0.048

0.008

0.018

0.042

5

-0.026 -0.003

0.035

0.026

35

-0.050

0.009

0.020

0.007

10

-0.022

0.005

0.034

0.050

40

-0.054

0.009

0.023

0.013

15

-0.023

0.006

0.030

0.057

45

-0.056

0.007

0.028

0.029

Time since B-band maximum

App. 8. – Corrected photometry of SN 2008ds

a

ta

t(tBmax)b

mB

mV

mR

mI

Source

647.96

-3.92

15.280 ± 0.033

15.404 ± 0.032

15.433 ± 0.037

15.573 ± 0.058

KAIT

649.97

-1.91

15.249 ± 0.043

15.348 ± 0.033

15.408 ± 0.045

15.594 ± 0.039

KAIT

650.97

-0.91

15.211 ± 0.050

15.272 ± 0.027

15.324 ± 0.037

15.613 ± 0.039

KAIT

653.94

2.06

15.222 ± 0.038

15.253 ± 0.030

15.348 ± 0.040

15.623 ± 0.039

KAIT

655.98

4.10

15.293 ± 0.059

15.355 ± 0.029

15.409 ± 0.048

15.804 ± 0.049

KAIT

662.93

11.05

15.707 ± 0.046

15.589 ± 0.032

15.740 ± 0.044

16.247 ± 0.055

KAIT

666.92

15.04

16.127 ± 0.061

15.786 ± 0.036

15.999 ± 0.049

16.372 ± 0.065

KAIT

674.97

23.09

16.874 ± 0.054

16.134 ± 0.046

16.151 ± 0.038

16.307 ± 0.046

KAIT

677.93

26.05

17.094 ± 0.055

16.255 ± 0.041

16.192 ± 0.044

16.167 ± 0.049

KAIT

680.98

29.10

17.232 ± 0.064

16.369 ± 0.059

16.264 ± 0.051

16.251 ± 0.060

KAIT

683.93

32.05

17.452 ± 0.108

16.544 ± 0.041

16.320 ± 0.057

16.278 ± 0.059

KAIT

686.93

35.05

17.697 ± 0.070

16.638 ± 0.039

16.446 ± 0.047

16.216 ± 0.056

KAIT

692.94

41.06

17.915 ± 0.103

17.137 ± 0.053

16.703 ± 0.054

16.453 ± 0.054

KAIT

697.92

46.04

...

17.37 ± 0.140

16.803 ± 0.097

16.67 ± 0.100

KAIT

653.63

1.75

15.158 ± 0.028

15.245 ± 0.015

15.314 ± 0.006

15.645 ± 0.024

Nickel

655.92

4.04

15.253 ± 0.029

15.283 ± 0.013

15.328 ± 0.011

15.702 ± 0.009

Nickel

658.93

7.05

15.344 ± 0.025

15.367 ± 0.009

15.462 ± 0.007

15.835 ± 0.010

Nickel

662.96

11.08

15.655 ± 0.033

15.564 ± 0.020

...

...

Nickel

670.90

19.02

16.385 ± 0.053

15.975 ± 0.027

16.112 ± 0.025

16.318 ± 0.031

Nickel

672.88

21.00

16.620 ± 0.048

16.039 ± 0.016

16.147 ± 0.015

16.249 ± 0.024

Nickel

694.89

43.01

18.042 ± 0.143

16.987 ± 0.039

16.801 ± 0.032

16.515 ± 0.050

Nickel

Julian Date minus 2,454,000 bDays since B-band maximum

25 of 26

Konrad Komorowski

Extended Essay in Physics

App. 9. – Light curves fitted to the 5 nearby SNe

26 of 26

000704-005