Distance Computation of Supernovae
Author: Sushil Jasper M.Sc., B.Ed., OCT
Address: Physics Teacher, 227 Bridge Avenue, Windsor, Ontario N9B 2M1, Canada.
Email: phys339@yahoo.com
Abstract: The main task in astronomy is the determination of the distance of cosmological objects. Measurement of distance is
quite important task and at the same instant, it is reasonably a challenged one too. Parallax method, Main-sequence fitting, RRLyrae variables Cepheid Variables are the popular primary distance indicator methods. Supernovae, Sunyaev-Zeldovich Effect
are the popular secondary distance indicators. Also here it is important to note that the secondary distance indicators depend on
primary indicators in distance calculation. The physical size and luminosity of the astronomical objects are highly related through
the distance to their angular dimension and flux respectively. Hence astronomers can estimate the distance by knowing the
physical size and luminosity in a distance –free method. Objects whose physical size and luminosity can be calculated in such a
manner are termed as standard rulers and standard candles respectively. In this report we will see the role of supernovae as
distance indicators. [1]
Keywords: Star Distance; Binary system; Accretion; Supernovae; White dwarf;
Definitions: :parsec; Mpc: Megaparsec; 1 pc = 3.08 × 1016m; 1 Mpc = 3.08 × 1022 m; Solar Radius =
6.9599 × 108 m (Equatorial); Solar Mass (Mʘ) = 1.989 × 1030 kg; 1 light year = 9.4605 × 1015 m.
Introduction: In this journal the role of supernovae as cosmological distance indicators is highlighted by
using a Maple software script. A supernova is the explosion of a star and it is the biggest explosion that
takes place in space. The major types of supernovae are explained here and also the criterion in which the
supernovae are categorized is also highlighted. Furthermore, the fact that the white dwarf stars are small but
massive was theoretically explained by using the radius-mass relationship. Here SN 1969L is taken as an
example to calculate its distance by Expanding Photosphere Method. EPM method is quite fit to calculate
the distance of supernovae type II at a range from 50 kpc to 200 Mpc.[10]
White dwarfs are the stars that have burned up the entire quantity of the hydrogen they once used as nuclear
fuel. The fusion in a star’s core generates heat and outward pressure but this pressure is maintained in
balance by the inward push of gravity due to the mass of the star.
Classification of supernovae
There are two main types of supernovae. Supernovae are classified by the characteristic of their spectra.
The main two types are referred as Type I and Type II.
Type I : There is no Hydrogen absorption lines in their spectrum.
Type II: Here astronomers found the existence of Hydrogen absorption lines in their spectrum.
Again type I is classified into 3 more sub categories and they are called as Type Ia, Ib, Ic.
Page 1 of 18
Sushil Jasper _ Distance Computation of Supernovae
SN Type Ia – In the spectra, there is No Hydrogen – No Helium – Strong Silicon lines.
SN Type Ib: In the spectra, there is No Hydrogen lines – Strong Helium lines.
SN Type Ic: In the spectra, there is No Hydrogen lines - No Helium lines – No Silicon lines. [2]
Formation of Supernovae Type Ia: Supernova Type Ia is formed by a sudden nuclear fusion in a
degenerate star. Here the transfer of matter has occurred in a close binary system. In Binary system, both
stars move in elliptical orbits around a common centre of mass in which one of the stars is a white dwarf
and the other may be a red giant star. Stars of mass less than 4 Mʘ normally will not explode. In most of
the binary systems, the stars are well separated. For example, the distance between Sirius A and Sirius B is
20 pc and this distance is approximately 4000 times the solar radius. But in some cases, the separation may
be small and hence the transfer of mass occurs easily which will lead the system to explode.
A degenerate white dwarf may gather sufficient material from its companion star by accretion.
Now the following thing happened to a white dwarf when its mass attains the Chandrasekhar‘s limit 1.44
Mʘ:
* cannot support its weight with electron degeneracy pressure, this means weight of the star overcomes
electron degeneracy pressure.
* core temperature increases.
* radius decreases
* density increases and hence the mass increases
* in this high temperature and high density, the fusion starts
The gas put onto the white dwarf by its stellar companion is expected to be hydrogen and helium, but the
strong gravity at the white dwarf's surface compresses it to densities and temperatures high enough to fuse it
into carbon and oxygen.
Now the white dwarf acts as a fusion bomb and explodes. Now due to the sudden gravitational collapse in
its core, too much of gravitational potential energy will be released. The amount of energy was estimated by
the astronomers as 1×1044 Joules. This amount is equivalent to the total amount of energy which could be
radiated by Sun during its entire life time.
The light curve of supernova type Ia is a graph which represents luminosity as a function of time after the
explosion. The luminosity is due to the radioactive decay of Ni-56 to Co-56 to Fe-56 and some Silicon. The
peak of all type of supernova type Ia has a maximum absolute magnitude of -19.3 approximately. Supernova
Type Ia spectrum consists of silicon because silicon is one of the products of fusion. The main product of
the fusion is iron.
A type Ia supernova ejects about 1 solar mass of iron into interstellar medium. That is why iron is available
in abundance in earth.
Radioactive decay is the major source of energy in this process. [3] [10]
Page 2 of 18
Sushil Jasper _ Distance Computation of Supernovae
Ni :
56𝑁𝑖 + 𝑒 − → 56𝐶𝑜 + 𝜈𝑒 + 𝛾
Electron capture of 56Co :
56𝐶𝑜 + 𝑒 − → 56𝐹𝑒 + 𝜈𝑒 + 𝛾
Electron capture of
Beta Decay of
56
56
Co :
56𝐶𝑜 → 56𝐹𝑒 + 𝑒 + + 𝜈𝑒 + 𝛾
- Decay time is 6.1 days
- Decay time is 77.1 days
The interaction of electrons with the surrounding medium starts and consequently the surrounding medium
get heated and power the light curve. [10]
The figure: 1 is a light curve of supernova Type Ia
in which the luminosity relative to sun versus time.
[4]
The peak of the locus of the curve is due to the
decay of Nickel (Ni) and the tail part is due to
Cobalt (Co) and Iron (Fe).
Reasons for Type 1a supernova as standard
candle:




The collapse always occurs at 1.4 Mʘ.
At this juncture, the luminosity of the
explosion remains constant.
From the known luminosity, astronomers can estimate the distance.
They are very bright with an absolute magnitude of -19.5 with a peak magnitude variation of 0.3
The mass transfer process so called accretion from a red giant to white dwarf can be briefed as below: [5]





Two stars in the binary system – one is white dwarf and the other is red giant.
Matter starts to spill from red to white.
When white dwarf, reaches the Chandrasekhar limit 1.4Mʘ, it will explode.
At this stage, fusion starts.
Most of the carbon and Oxygen burnt into iron Ni, Co, Fe and some Si.
Just a teaspoon of substance from a white dwarf would weigh as approximately five tons. It is quite
incredible but it is the fact. The white dwarf stars are very dense and thick, their gravity is particularly
extreme. At the time of explosion, the white dwarfs are very bright and the resultant light is about 5 billion
times brighter than sun.
A usual white dwarf is half the mass of Sun, but slightly bigger than Earth. A white dwarf of equal size of
Earth has a density of 1 × 109 kg m -3. The density of a typical white dwarf star is between 107 and 1011
kgm-3. [6]
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Sushil Jasper _ Distance Computation of Supernovae
Actually, the mass of one cubic centimetre of white dwarf material weighs around 10000 kg !!
The density of earth is 5400 kg m-3. Hence a white dwarf is approximately 200,000 times the density of
earth.
Radius - Mass Equation of White Dwarf
White Dwarfs are high density celestial bodies. The mass of a white dwarf can be reasonably compared with
the mass of Sun, but its volume can be compared to Earth’s volume. The faint luminosity of white dwarves
is due to the emission of stored thermal energy. The nearest identified white dwarf is Sirius B - which is
located at 8.6 light years away from earth. As the mass increases, gravity compresses the star into small
volume. Hence they are small but massive and this face can be proved by Radius – Mass.
Energy of the white dwarf = its Gravitational Potential Energy + its Kinetic Energy.
The gravitational potential energy of a white dwarf [WD] can be represented as, 𝑬𝒈 =
Kinetic energy is the motion of electrons and it will be = 𝑬𝑲 =
−𝑮𝑴
𝑹
𝑵𝑷𝟐
𝟐𝒎
Where M – mass of WD star
R – radius of WD star
G = 6.67× 10-11 m3 kg-1 s-2
N – number of electrons in the unit mass of WD
P – average electron momentum
m – mass of electron [9.1 × 10-31 kg]
Since the electrons are degenerated, we can estimate the average electron momentum P to be on the order of
𝒉
uncertainty by ∆𝑷∆𝒙 ≥ 𝟐𝝅
Here ∆𝑥 represents the distance between the electrons. ∆𝒙 ≈ 𝒏
per unit volume.
−𝟏⁄
𝟑
where n – is the number of electrons
𝟐
Hence 𝑬𝑲 =
𝑵ħ𝟐 𝒏𝟑
𝟐𝒎
But we know that n =
Page 4 of 18
𝑵𝑴
𝑹𝟑
where NM – is the total number of electrons in the entire WD.
Sushil Jasper _ Distance Computation of Supernovae
𝟐
Hence we can write 𝑬𝑲 =
𝟐
𝑵ħ𝟐 𝑵𝟑 𝑴𝟑
𝟐𝒎𝑹𝟐
WD will be in equilibrium when the Ek + Eg is minimised. At this stage, Ek = Eg
𝟓
𝑮𝑴
𝑹
=
𝟐
𝑵𝟑 ħ𝟐 𝑴𝟑
𝟐𝒎𝑹𝟐
𝟓
Hence R =
𝟐
𝑵𝟑 ħ𝟐 𝑴𝟑
𝟐𝒎𝑮𝑴
Now we can see that R ∝
𝟓
𝑵 𝟑 ħ𝟐
=
𝟏
𝟐𝒎𝑮𝑴𝟑
𝟏
𝟏
𝑴𝟑
Here we can see the Radius of the white dwarf star is inversely proportional to the cube root of the mass.
Hence though the white dwarfs are small they are more massive. [6]
Supernovae Type II
Supernovae type II is massive supergiant stars. Normally a supergiant must be young – approximately 1
million years.
In terms of the source of energy, type Ia corresponds to Thermonuclear Fusion [C and O to Ni, Co Fe] and
type II corresponds to core collapse of iron core. Crab Nebula is the remnant of Type II supernova and it
consists of a neutron star in the centre. Tycho supernova remnant is the remnant of type Ia supernova and it
is very rich in iron.
Core-collapse Supernovae: In sun, in its core, H is fused into He and discharging thermal energy which
heats the sun’s core. This provides the adequate outward pressure. This helps the sun’s layers against the
collapse and maintains hydrostatic equilibrium. This is something good for the earth.
𝟒 𝟏𝟏𝑯 → 𝟒𝟐𝑯𝒆 + 𝟐 𝟎𝟏𝒆 .
Hence mass defect = 4 × 1.00813 – 4.00386 = 0.02866 amu.
To calculate the energy equivalent to 0.02866 amu,
1 amu = 1. 66 × 10-27 kg.
E =1. 66 × 10-27c2 = 1.494 × 10-10 Joules. 1 eV = 1.6 × 10-19Joules.
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Sushil Jasper _ Distance Computation of Supernovae
Hence 1.494 × 10-10 J =
1.494×10−10
1.6×10−19
= 933750000eV = 933 Mev. The standard theoretical value is 931
MeV
Energy released by this mass defect = 0.02866 × 931MeV = 26 MeV. This energy is not enough to fuse
Helium.
* Hence H in the core get exhausted - fusion starts to slow down- gravity makes the core to contract.
* This contraction increases the temperature to start the He fusion.
* The electron degeneracy pressure and energy generated in the fusion can withstand the collapse
and maintain hydrostatic equilibrium.
In the case of stars of mass less than 8Mʘ, the carbon formed by Helium fusion does not fuse and the stars
started to cool down and finally turn into White Dwarf. If it gets a nearby companion star, there will be a
chance for supernova Ia.
But for more massive stars, the fusion starts from H, He and continues as per the order of periodic table. The
electron degeneracy pressure and energy generated during fusion are far enough to continue the fusion, to
prevent from collapse, to maintain hydrostatic equilibrium.[8] The steps of huge massive stars before their
collapse can be shown as in Table: 1
Core-Burning Nuclear Fusion Stages of a 25 Solar Mass Star
Process
Major Fuel
Main Products
Temperature
(Kelvin)
Duration
Hydrogen burning
Triple-Alpha process
Carbon burning
Neon burning
Oxygen burning
Silicon burning
Hydrogen
Helium
Carbon
Neon
Oxygen
Silicon
Helium
Carbon, Oxygen
Ne, Na, Mg, Al
O, Mg
Si, S, Ar, Ca
Nickel (decays into Iron)
Table: 1
7 × 107
2 ×108
8 × 108
1.6 × 109
1.8 × 109
2.5 × 109
1 × 107 years
1 ×106 years
1 × 103 years
3 years
0.3 years
5 days
This process lasts until a core of Fe, Ni is formed.


Then there is no fusion in Fe, Ni since there is no output energy as these elements have high Binding
Energy per nucleon.
Now the core is with huge gravitational pressure and hence the core becomes very dense.
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Sushil Jasper _ Distance Computation of Supernovae



When the mass of the core exceeds the threshold limit called Chandrasekhar limit 1.4Mʘ, the
degeneracy pressure only cannot withstand against gravity, and hence it will collapse.
At this juncture, the outer part of the core reaches a velocity up to 70000 km s-1 towards the centre.
Stars of mass greater than 50Mʘ, the star concerned will collapse directly into black hole without
forming supernova.
Type II, Type Ib and Type Ic supernovas are termed as core-collapse supernovas. Here it goes something
like the aforementioned process. When the nuclear power source at the center or core of a star is exhausted,
the core starts to collapse. Within a period of a second, a neutron star or a black hole, if the star is extremely
massive will be formed. The formation of a neutron star releases an enormous amount of energy in the form
of neutrinos and heat.
SN 1987A was discovered on 23 Feb 1987. This famous supernova star is very important to the world of
astronomy because of its close vicinity and this was the first Supernova star that could be seen by the bare
eye. Since it was quite near, it helped astronomers to enhance their observations over it. [9]
During this period, Hubble Telescope was not in orbit but it made very good observation of the expanding
gaseous remains of the star. Robert Kirshner of the Harvard-Smithsonian Center for Astrophysics in
Cambridge, USA has used the Hubble telescope to monitor/observe the supernova.
Supernova 1987A, blazed with tremendous power, which was equivalent to the power of 100 million suns
for several months since its discovery on Feb. 23, 1987.
Kirshner said "The sharp pictures from the Hubble telescope help us to ask and answer new questions about
Supernova 1987A," He further added "In fact, without Hubble we wouldn't even know what to ask”. The
iron in living body blood was manufactured in supernova explosions. Supernova 1987A is 163,000 lightyears away from us and located in the Large Magellanic Cloud and this has actually blown up about
161,000 B.C, but its light was able to observe only in 1987. [14]
Techniques to find the distance of supernovae
Astronomers are using the following mentioned three methods to find the distance of supernovae:
a) Expanding Photosphere method [EPM]
b) Expanding Shockfront method [ESM]
c) Standard Candles.
In the first two methods, the distance is measured by calculating the linear radius and angular radius. The
third one is by comparing with the star’s known absolute magnitude and then by the inverse square law to
calculate the distance.
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Sushil Jasper _ Distance Computation of Supernovae
Expanding Photosphere Method [EPM]:
Supernova type II works as a good distance indicator by Expanding Photosphere Method [EPM]. By using
this method, we can calculate the distance of the supernova directly from the linear radius and angular
radius. The angular radius will be calculated with the assumption that the supernova is optically thick in its
interior part and hence it can be treated as a blackbody. [10]
In 1974, Robert Kirshner and John Kwan firstly used the Badde-Wasselink method to calculate the distances
of supernovae.[11] The method has been already used by Halton Arp to compare the observed and predicted
expansion velocities in supernovae in the year 1961.[7]
Later Brian P. Schmidt , an American born Australian Astronomer [Australian National University –
Western Creek – Australia], Robert Kirshner [Harvard Smithsonian Centre for Astrophysics – USA]and
Ronald Eastman [University of California –USA] did the experiment to find the distance of 10 selected
Supernovae Type II [including the famous SN 1987A] by using the Expanding Photosphere Method.
Brian Schmidt got the Nobel Prize in Physics jointly with Adam Riess and Saul Pelmuter during 2011 for
their work on the ‘Accelerating Universe’ and the Nobel lecture dated 8 Dec 2011 can be seen through. [12]
For my calculation, I used the data of B, V magnitude and vused of the supernova SN 1969L from the journal
authored by Brian Schmidt, Robert Kirshner, and Ronald Eastman and published by American
Astronomical Society. [11]
I did calculate the
i)
ii)
iii)
iv)
temperature
angular size
radius of the photosphere at time (t), and finally
the distance of supernova star SN 1969L
I did my calculations by
i) Manual calculation method – time: 14 days
ii) Maple –version 10 student version program and
iii) Excel software program
My solutions have very good coincidence with the distance data provided by Brian Schmidt, Robert
Kirshner and Ronald Eastman in their Astrophysical journal, 395: 366-386, 20 Aug 1992. [11]
Supernova star SN 1969Lwas discovered by astronomers on 2 December 1969 in the outer zone of the
galaxy NGC 1058 as an apparent magnitude [𝑚𝐵 ] =13 object. The Photographic spectra and photometry
Page 8 of 18
Sushil Jasper _ Distance Computation of Supernovae
[BV] of this supernova were studied and the results were published by the astronomers Ciatti, Rossino and
Bertola in 1971. In the year 1974, Kirshner and Kwan published the spectra of this supernova. The data
are reasonably extensive and large with good quality and the results proved that SN 1969L was a Type II
supernova star. [11]
EPM assumes the principle that supernovae type II radiate as dilute blackbodies. So we can calculate the
angular size of the photosphere of the supernova by using the below mentioned formula:
𝜽=
𝑹
𝑫
𝒇
= √𝜻𝟐 𝝅𝑩𝜸 (𝑻) if redshift z<<1
𝝊
where
θ - angular dimension of the photosphere of the Supernova
R - radius of the photosphere
D - distance to the Supernova
fγ - observed flux density of the SN, and
Bυ (T) - the Planck function at any temperature T
Since the supernovae are not perfect blackbodies, Kirshner include a correction factor, , but in the
calculation, he gave the value for
=1 which is assigned for blackbody.
The radius of the photosphere R = vused (t – t0) + R0 where vused - observed velocity, t - elapsed time since
the time of explosion, t0.
Calculation of the distance of Type II SN 1969L by Expanding Photosphere Method [EPM]:
Data for time, B, V from Table 3A and vused from Table 3B
of the journal [11]
Calculation of Distance for a period of 14 days:
data can be seen in the second row of the table: 2
The
The value of distance of all the nine days is calculated by
using excel and the spreadsheet is attached in page: 17.
B = 13.40
V= 13.25
v used = 8500 km s-1 = 8.5×106 ms-1
t – t0 = 14 days
Page 9 of 18
Sushil Jasper _ Distance Computation of Supernovae
= 1.2 × 106s
i) The radius of the photosphere of the supernova [R]:
R = vused (t – t0) + R0 and R0 is very small and negligible.
Hence R = v [ t – t0]
R = 8.5×106 ×1.2 × 106 = 1.03 × 1013 m
Hence the radius of the photosphere of the supernova SN 1969L = 1.03 × 1013 m
ii) The temperature of the supernova [T]:
The colour index means the difference in magnitudes at two different wavelengths. Hence B-V colour index
means the difference between magnitudes in the blue and visual regions of spectrum.
𝟏𝟎𝟒 𝑲
= 𝟏. 𝟔𝟎𝟓 (𝑩 − 𝑽) + 𝟎. 𝟔𝟕
𝑻
104
T= [1.605 (0.15]+0.67] =
10980 K
iii) Angular size theta [𝜽] of the photosphere of the supernova:
𝜽=
𝒇𝜸
𝑹
= √ 𝟐
𝑫
𝜻 𝝅𝑩𝝊 (𝑻)
𝟐𝒉𝒄𝟐
Plank Curve formula tells , 𝑩λ (𝑻) =
𝒉𝒄
𝝀𝟓 𝒆𝝀𝒌𝑻 −𝟏
We need to convert 𝑩𝝀 (𝑻) into 𝑩υ (𝑻) by using c = 𝜆υ
We know 𝑩𝝀 (𝑻) = 𝑩υ (𝑻)
𝑩𝝀 =
𝑩υ 𝒅υ
𝒅𝜆
From c = 𝜆υ, υ=c𝜆−1
On differentiating υ with respect to 𝜆,
Page 10 of 18
Sushil Jasper _ Distance Computation of Supernovae
𝑑υ
𝑐
I got 𝑑 𝜆 = −
𝑑υ
|𝑑 𝜆 | =
𝜆2
𝑐
𝜆2
Hence
𝑩υ 𝒄
𝑩𝝀 = 𝟐
𝜆
𝑩υ = 𝑩𝝀
=
𝝀𝟐
𝒄
2ℎ𝑐 2
𝝀𝟐
ℎ𝑐
𝜆5 𝑒 𝜆𝑘𝑇 −1
𝒄
𝑩υ (𝑻) =
𝟐𝒉𝒄
𝝀𝟑
𝟏
𝒉υ
𝒆𝒌𝑻
Finally 𝑩υ (𝑻) =
=
−𝟏
𝟐𝒉υ𝟑
𝒄𝟐
𝟐𝒉𝒄
𝒄𝟑
υ𝟑
𝒉υ
𝒆𝒌𝑻 −𝟏
𝟏
𝒉υ
𝒆𝒌𝑻
w/m2/Hz/Sr
−𝟏
* A blackbody will absorbs all amount of light which hits it.
* This blackbody will emit thermal radiation – emission of photon is a good example of this phenomenon.
* The energy emitted per unit area depends on the blackbody’s temperature.
* The scientist Max Planck characterized the light coming from a blackbody.
* The aforementioned equation predicts the radiation of a blackbody at different temperatures and
frequency is known as ‘Planck’s law’.
I used the following online magnitude converter to get the equivalent Flux
http://ssc.spitzer.caltech.edu/warmmission/propkit/pet/magtojy/
The output will be in Milli-Jy and can be converted into MKS system by multiplying 10 - 29 w m-2 Hz-1
B = 13.4 has an effective wavelength of 0.43 µ m and its corresponding frequency can be calculated by
using c = λυ.
Hence υ = c/λ = = 6.97 ×1014 Hz
𝐵υ (10980) =
and the
𝑓υ = 18 mJy = 18 ×10 - 29 w m-2 Hz-1
[2×6.634 ×10−34 ×[6.97×1014 ]3 [3×108 ]−2]
Page 11 of 18
6.634 ×10−34×6.97×10
𝑒 1.38×10−23×10980
14
−1
Sushil Jasper _ Distance Computation of Supernovae
2.48×10-7 w/m2/Hz/Sr
=
𝜽=
𝑹
𝑫
𝒇
= √𝜻𝟐 𝝅𝑩𝜸 (𝑻)
𝝊
where 𝜁 is the correction factor which accounts the effects of flux dilution. For a
blackbody, 𝜁 =1.
𝟏𝟖 ×𝟏𝟎𝟐𝟗
𝜽 = √𝟏×𝟑.𝟏𝟒×𝟐.𝟒𝟖×𝟏𝟎−𝟕 = 𝟏. 𝟓 × 𝟏𝟎−𝟏𝟏 𝒓𝒂𝒅𝒊𝒂𝒏
Hence the angular size of the photosphere of the Supernova 1969L is 1.5 ×10-11 radian
iv) Distance of the supernova [D]:
𝑹
D=𝜽 =
𝟏.𝟎𝟑×𝟏𝟎𝟏𝟑
𝟏.𝟓 × 𝟏𝟎−𝟏𝟏
=
𝟔. 𝟕𝟕 × 𝟏𝟎𝟐𝟑 𝐦
1 Mpc = 3.08 ×1022 m
Hence D =
6.77 × 1023
3.08 × 1022
= 21.98 Mpc
Hence the supernova SN 1969L was located at a distance of 21.98 Mpc from the earth.
This value is very close with Briyan P. Schmidt’s team’s calculated value in their research journal of the
distance of this supernova SN 1969L which is 21.3 Mpc
I computed this calculation by using Maple Computer Program and got the answer 21.97 Mpc and the Maple
worksheet is attached in the page: 12.
Also I have calculated the distance for all the 9 observed days by using Excel program and attached the
spreadsheet in the page: 16. My excel program calculated the distance value for t = 14 days as 22.5 Mpc.
Computation of the Distance of the supernova star SN1969L by Maple Script:
> `SN 1969L:
Calculations for t=14 days`;
SN 1969L: Calculations for t=14 days
> unprotect(D):
> `Universal constants:`;
c:=3E8;
`(light speed~m/s)`;
h:=6.634E-34;
Page 12 of 18
Sushil Jasper _ Distance Computation of Supernovae
`(Planck's constant)`;
k:=1.38E-23;
`(Boltzmann constant)`;
Universal constants:
c := 0.3 10 9
(light speed~m/s)
h := 0.6634 10 -33
(Planck's constant)
k := 0.138 10 -22
(Boltzmann constant)
> `Data for t=14 days:`;
zeta:=1;
`(correction factor)`;
t:=14*24*60*60;
`(elapsed time after explosion~s)`;
v_used:=8500E3;
`(expansion velocity~m/s)`;
R:=v_used*t;
`(radius of SN~m`;
B:=13.4;
`(apparent magnitude of blue light~non-dim)`;
V:=13.25;
`(apparent magnitude of visible light~non-dim)`;
T:=1E4/(1.605*(B-V)+0.67);
`(temperature of wave~deg K)`;
Data for t=14 days:
 := 1
(correction factor)
t := 1209600
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(elapsed time after explosion~s)
v_used := 0.8500 10 7
(expansion velocity~m/s)
R := 0.1028160000 10 14
(radius of SN~m
B := 13.4
(apparent magnitude of blue light~non-dim)
V := 13.25
(apparent magnitude of visible light~non-dim)
T := 10979.96157
(temperature of wave~deg K)
> `Planck's B_nu(T):`;
B_nu:=(T,nu)->(2*h*nu^3/c^2)*(exp((h*nu)/(k*T))-1)^(-1);
Planck's B_nu(T):
h 3
B_nu := ( T ,  ) 2
h
  k T 

c  e
 1 
> `Magnitude/Flux Converter:`;
2
`input:`;
`Filter:
Johnson B`;
B:=13.4;
`(magnitude for blue)`;
`output:`;
f_nu:=18E-3*1E-26;
`(flux density~mks)`;
lambda:=.43E-6;
`(effective wavelength~m)`;
nu:=c/lambda;
`(effective frequency~Hz)`;
Magnitude/Flux Converter:
input:
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Sushil Jasper _ Distance Computation of Supernovae
Filter: Johnson B
B := 13.4
(magnitude for blue)
output:
f_nu := 0.18 10 -27
(flux density~mks)
 := 0.43 10 -6
(effective wavelength~m)
 := 0.6976744186 10 15
(effective frequency~Hz)
> `Angular distance theta(~rad):`;
theta:=(T)->sqrt((f_nu/(zeta^2*Pi*B_nu(T,nu))));
Angular distance theta(~rad):
 := T 
f_nu
  B_nu ( T ,  )
2
> `angular distance at t=14 days:`;
evalf(theta(T));
`(radians)`;
angular distance at t=14 days:
0.1520908879 10 -10
(radians)
> `linear distance at t=14 days:`;
D:=(3.25E-23)*evalf(R/theta(T));
`(linear distance~Mpc)`;
linear distance at t=14 days:
D := 21.97054700
(linear distance~Mpc)
>
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Conclusion
Supernovae play a dynamic role in calculating the distance of galaxies from the earth.
Supernova as distance indicator technique has been popular since the past few years. Supernova technology
is highly used by the astronomers in solving many problems in cosmology including Hubble Constant and
cosmological parameters.
We have seen the two major types of supernovae. Among these two, the type Ia has more attention by the
astronomers and relatively less attention for type II because they are considerably fainter. Anyhow one can
assume that type II will stand as an independent class of objects and the results of their studies will be useful
to confirm and compare the results of type Ia.
In the forthcoming years, supernovae studies will become more important for astronomers as the high
technology enhanced telescopes like 30MT and Great Magellan Telescope are likely to start functioning in
the nearest future. Hence it is very sure that many new theories for celestial objects with high redshift will
be discovered in the future.
References
1. Pearson, Chris P. "Evolutionary constraints from infrared source counts." Monthly Notices of the Royal
Astronomical Society 325.4 (2001): 1511-1526.
2. Harkness, Robert P., and J. Craig Wheeler. "Classification of supernovae." Supernovae. Springer New York,
1990. 1-29.
3. Niedermann, Florian. "Supernova Light Curves."
4. Branch, Asymptotic Giant. "From Wikipedia, the free encyclopedia." (2003).
5. Carroll, Bradley W., and Dale A. Ostlie. An introduction to modern astrophysics and cosmology. Vol. 1. 2006.
6. Regmi, Jeevan. "Dark Energy and Dark Matter." Himalayan Physics 4 (2013): 91-94.
7. Arp, Halton. "The Globular Cluster M5." The Astrophysical Journal 135 (1962): 311.
8. Murdoch, Lachlan, et al. "Encyclopedia> Star."
9. Shapiro, Stuart L., and Saul A. Teukolsky. Black holes, white dwarfs and neutron stars: the physics of
compact objects. John Wiley & Sons, 2008.
10. Webster, Josh. "Galaxies and Supermassive Black Holes."
11. Kirshner, R. P., and J. Kwan. "Distances to extragalactic supernovae." The Astrophysical Journal 193 (1974):
27-36.
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12. Schmidt, B. P. (2013). The path to measuring an accelerating Universe. Physics-Uspekhi, 56(10).
13. Kirshner, R. P., and J. Kwan. "Distances to extragalactic supernovae." The Astrophysical Journal 193 (1974):
27-36.
14. Arnett, W. David, et al. "Supernova 1987A." Annual review of astronomy and astrophysics 27 (1989): 629700.
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