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Calculating the Distance to a Supernova by the Expanding Photosphere Method ***

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Problem: Use Doppler shifting and the "expanding photosphere method" to calculate the distance to the type II supernova remnant SN 1969L. (The calculations are based on extracts from Jasper (n.d.).)

 

Hints:

 

Calculate flux in janskys; convert to joules.

Calculate the frequency of the central wavelength of the B filter.

Calculate the Planck function for frequency.

Calculate the angular size of the supernova (θ) at 14 days.

Calculate the distance to the supernova in Mpc.

 

 

Data:

 

(Supernova data are from observations of the type II supernova SN 1969L reported in Kirshner and Kwan (1974).)

with(ScientificConstants)

 

t := 60*(14*24)*60

1209600

(1)

B := 13.4

13.4

(2)

(3)

V := 13.25

13.25

(4)

 

v[obs] := 8.5*10^6

8500000.0

(5)

NULL

`ς` := 1

1

(6)

R := v[obs]*t

0.1028160000e14

(7)

T := 10^4/(1.605*(B-V)+.67)

10979.96157

(8)

m := 13.4

13.4

(9)

z := 4260

4260

(10)

ja := 10^26

100000000000000000000000000

(11)

cw := .44

.44

(12)

c := evalf(Constant(c))

299792458.

(13)

h := evalf(Constant(h))

0.662606876e-33

(14)

k := evalf(Constant(k))

0.1380650277e-22

(15)

NULL

NULLNULL

NULL

Useful Equations:

NULL

Flux = 10^(-m/(2.5))*zeropoint*flux

B = 2*h*nu^3/(c^2*(exp(h*nu/(k*T))-1))

NULLNULL

NULL

  #  angular*size*on*the*sky

 

where

 

θ = angular size on the sky

D = distance to the supernova

fγ = observed flux density

ς = correction factor

Bν(T) = Planck function

R = radius of the supernova = vobs (t-t0) + R0  where

vobs = the observed velocity of expansion

t = time since t0

t0 = time of the explosion

R0 = the radius at t0

 

NULL

Solution:

NULL

unprotect(D)

NULLNULLNULL

NULL

Flux = 10^(-m/2.5)*zeropoint flux for the filter, where m is the visual magnitude (13.4 for SN 1969L). The zeropoint flux for the B filter is 4260 mJy, according to Bessel (1990).

NULL

4260*10^(-13.4/(2.5))

0.1859557445e-1

(16)

 

in janskys. Convert janskys to joules


f[nu] := 0.1859557445e-1*10^(-26)

0.1859557445e-27

(17)

NULL

NULL

The central wavelength of the B filter is 0.44 μm (Bessel, 1990). (See Table 2 in the text). Convert this to frequency.

 

nu := c/(.44*10^(-6))

0.6813464955e15

(18)

NULL

Planck's function for frequency:

 

(19)

B[nu] := proc (T, nu) options operator, arrow; 2*h*nu^3/(c^2*(exp(h*nu/(k*T))-1)) end procNULL

proc (T, nu) options operator, arrow; 2*h*nu^3/(c^2*(exp(h*nu/(k*T))-1)) end proc

(20)

evalf(B[nu](T, nu))

0.2500684058e-6

(21)

NULL

Calculate the angular size at 14 days.

 

theta := proc (T) options operator, arrow; sqrt(0.1859557445e-27/((`ς`^2*Pi*B[nu](T, nu)))) end proc

proc (T) options operator, arrow; sqrt(0.1859557445e-27/((`ς`^2*Pi*B[nu](T, nu)))) end proc

(22)

NULL

evalf(theta(T))

0.1538510450e-10

(23)

NULL

Distance in metres:

 

R/theta(T)

0.3770381684e24/(1/Pi)^(1/2)

(24)

NULL

In megaparsecs

 

evalf(0.3770381684e24/((1/Pi)^(1/2)*(3.08567758*10^6*10^16)))

21.65756908

(25)

NULL

This is close to the accepted value of 21.3 Mpc, given by Schmidt (2013).

 

---------------------------------------------------------------------------

References

 

Bessel, M. (1990). UBVRI Passbands. Publications of the Astronomical Society of the Pacific, 102, 1181-1199.

 

Jasper, S. (n.d.). Distance Computation of Supernovae. http://www.google.ca/search?hl=en-CA&source=hp&q=sushil+jasper+supernova&gbv=2&oq=sushil+jasper+supernova&gs_l=heirloom-hp.3...1662.6130.0.6427.23.12.0.10.10.0.425.1485.1j3j2j0j1.7.0....0...1ac.1.34.heirloom-hp..16.7.1485.rYZio6g6h1o (accessed: 2015-06-01).)

 

Kirshner, R. P., and Kwan, J. (1974). Distances to extragalactic supernovae. The Astrophysical Journal 193, 27-36.

 

Schmidt, B. (2013). The path to measuring an accelerating Universe. Physics-Uspekhi, 56(10).

NULL