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.