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:
| (4.1) |
Calculate the average density of electrons (nn) over all values of γ, where γ varies from 1 to 10^{6} in the ejecta by integrating the power-law energy distribution formula from 1 to 10^{6} (over all frequencies). (Use 'x' as the variable of integration for simplicity.)
| (4.2) |
Calculate the total synchrotron power of electrons in a magnetic field at frequency ν:
| (4.3) |
The total synchrotron power of the ejecta is approximately:
The power emitted at frequency ν is approximately 2.2 * 10^{34} watts.
| (4.4) |
To find the total luminosity of the remnant, integrate this power over the range of frequencies, from ν_{L} to ∞:
| (4.5) |
The power-law energy distribution of the electrons is described by:
#
The particle energy is
Change the variable of integration from γ to x and drop the Lorentz factor:
| (4.6) |
To find the lifetime of the radiation, divide the particle energy by the luminosity:
| (4.7) |
| (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.