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Duration of Synchrotron Radiation in a Typical Supernova Remnant

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Duration of Synchrotron Radiation in a Typical Supernova Remnant``NULL

Problem

Find an estimate for the length of time a typical supernova remnant can radiate, given its magnetic field strength.

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Hints:

 

Find the total number of electrons in the ejecta (N).

Integrate the power-law energy distribution formula from 1 to 106 (over all frequencies) to find particle density (nn).

Calculate the total synchrotron power of electrons in a magnetic field at frequency ν.

Find the total luminosity of the remnant by integrating the total synchrotron power over frequency range from νL to ∞.

Calculate the magnetic energy.

Calculate the particle energy.

Find the total energy content of the remnant by adding the magnetic energy (UB times V) and particle energy.

Divide the total energy by the luminosity to find the duration of the energy.

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Data

Number Density of Electrons

n := 10^6/Unit('m')^3

 

Volume of the Remnant

V := 2*(3.08*10^16)^3*Unit('m')^3

 

Speed of Light

c := 3*10^8*Unit('m')/Unit('s')

 

Thomson Cross-Section

sigma[T] := 6.65*10^(-25)*Unit('cm')^2

 

Magnetic Field Strength

B := 10^(-7)*Unit('T')

 

Permeability of Free Space``

mu[0] := 1.26*10^(-6)*Unit('H')/Unit('m')

 

 

Magnetic Energy Density

U[B] := B^2/(2*mu[0])``

 

Atomic Charge Unit

q := 1.6*10^(-19)*Unit('C')

 

 

Electron Mass

m := 9.0*10^(-31)*Unit('kg')

 

Critical Frequency

nu[L] := simplify(evalf(q*B/(2*Pi*m)))

 

Power Law Exponent for Synchotron Power of Electrons in Magnetic Field

p := 5

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Useful Equations

Total Number of Particle Species in Volume V

N = int(n, V) and int(n, V) = nV 

 

Particle Density of Supernova Remnant

nn = N*(int(1/x^5, x = 1 .. 10^6))    

 

Synchrotron Power of Electrons in Magnetic Field

P[nu] = (2/3)*c*sigma[T]*nn*U[B]*(nu/nu[L])^(-(p-1)*(1/2))/nu[L]   

 

Total Luminosity

L = int(P[nu], nu = nu[L] .. infinity)     

 

Magnetic Energy

U[B]*V*Unit('m')^3     

 

Particle Energy

nn*(int(gamma*m*c^2*(1/gamma^5), gamma = 1 .. 10^6))    

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Solution

 

First find the total number of electrons in the ejecta (N). N = the integral over the volume of the number density of electrons times the volume:NULL

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N := n*V

0.5843622400e56

(4.1)

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Calculate the average density of electrons (nn) over all values of γ, where γ varies from 1 to 106 in the ejecta by integrating the power-law energy distribution formula from 1 to 106 (over all frequencies). (Use 'x' as the variable of integration for simplicity.)

 

nn := evalf(N*(int(1/x^5, x = 1 .. 10^6)))

0.1460905600e56

(4.2)

 

Calculate the total synchrotron power of electrons in a magnetic field at frequency ν:

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P[nu] = (2/3)*c*sigma[T]*nn*U[B]*(nu/nu[L])^(-(p-1)*(1/2))/nu[L]

P[nu] = 0.2181578571e35*Units:-Unit('m')*Units:-Unit('cm')^2*Units:-Unit('Pa')*Units:-Unit(1/('s'))/(Units:-Unit('s')*nu^2)

(4.3)

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The total synchrotron power of the ejecta is approximately:

 

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The power emitted at frequency ν is approximately 2.2 * 1034 watts.NULL

 

P[nu] := 2.2*10^34*Unit('W')

0.2200000000e35*Units:-Unit('W')

(4.4)

 

To find the total luminosity of the remnant, integrate this power over the range of frequencies, from νL to ∞:

 

L := P[nu]*(int(1/nu^2, nu = 2829.421210 .. infinity))

0.7775441819e31*Units:-Unit('W')

(4.5)

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The power-law energy distribution of the electrons is described by:

 

#       nn(gamma)*`dγ` = N*gamma^(-p)*`dγ`NULL

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The particle energy is

 

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Change the variable of integration from γ to x and drop the Lorentz factor:

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pp := simplify(nn*m*c^2*(int(1/x^5, x = 1 .. 10^6)))

0.2958333840e42*Units:-Unit('J')

(4.6)

 

To find the lifetime of the radiation, divide the particle energy by the luminosity:

 

ls := simplify(pp/L)

0.3804714779e11*Units:-Unit('s')

(4.7)

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ls*Unit('yr')/((60*(365.25*24)*60)*Unit('s'))

1205.641360*Units:-Unit('yr')

(4.8)

 

Synchrotron energy of electrons in a magnetic field may enable a supernova remnant to radiate for more than a thousand years, compared with a relatively short period of time without such a magnetic field (See the worksheet "Crab"). Clearly, a synchrotron source is needed. This source is thought to be the pulsar that is found in the centre of a typical supernova remnant.

 

 

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