Synchrotron energy constraint

Let us consider a sphere of hot plasma emitting synchrotron radiation. We assume that this plasma sphere is stationary, although the results below will also apply to a sphere expanding at a non relativistic speed. Suppose we measure the emitted spectrum from this sphere. This spectrum depends on four parameters of the plasma: the number of radiating electrons $N_e$, the radius $R$, the magnetic field $B$ and the average energy of the radiating electrons, expressed in terms of the thermal Lorentz factor $\gamma_e$. The observed spectrum can be described as a piecewise power - law, with two breaks, and whose indices are known in each region. One break is at the synchrotron frequency of the electrons, and another frequency when the sphere is marginally optically thick. Each such break provides two numbers: break frequency, and luminosity at the break. The two breaks therefore provide four numbers. However, only three are independent. In this entry we demonstrate how energy considerations can be used to provide a fourth constraint, with which the parameters of the plasma can be retrieved. For simplicity, we assume that both breaks coincide at frequency $\nu_b$, with spectral luminosity $L_{b}$. However, the same method can be used for cases when the breaks are separated. Also, we assume the distance to the plasma sphere is known, so the luminosity can be calculated from the measured flux.

We begin by writing the under - determined equations for the break parameters. The spectral luminosity from synchrotron is given by

$L_b \approx N_e \sqrt{B^2 r_e^3 m_e c^2}$

where $r_e$ is the classical electron radius, $m_e$ is the electron mass and $c$ is the speed of light. The synchrotron frequency of the electrons is given by

$\nu_b \approx \frac{c}{r_e} \sqrt{\frac{B^2 r_e^3}{m_e c^2}} \gamma_e^2$.

The spectral luminosity of a blackbody spectrum is given by

$L_b \approx R^2 \frac{\gamma_e m_e c^2 \nu_b^2}{c^2}$

We can express $B$, $N_e$ and $\gamma_e$ in terms of the radius $R$

$\gamma_e \approx \frac{L_b}{\nu_b^2 m_e R^2}$

$B \approx \frac{m_e^{5/2} R^4 \nu_b^5}{L_b^2 \sqrt{r_e}}$

$N_e \approx \frac{L_b^3}{c m_e^3 R^4 r_e \nu_b^5}$

Also, we can express the total energy, i.e. thermal energy of the electrons and magnetic energy, in terms of the radius

$U \approx B^2 R^3 + N_e me c^2 \gamma_e \approx \frac{c L_b^4}{m_e^3 R^6 r_e \nu_b^7} + \frac{m_e^5 R^{11} \nu_b^{10}}{L_b^4 r_e}$

Minimum energy is attained when

$R \approx \frac{c^{1/17} L_b^{8/17}}{m_e^{8/17} \nu_b}$

It is possible that the energy is larger than the minimum possible, and hence the true value of $R$ can be different from the value calculated above. However, the energy increases very steeply when the radius deviates from its optimum value. For values of $R$ larger than the optimum the energy increases as $R^{11}$, and for smaller values the energy increases as $R^{-6}$.

The radius $R$ can be substituted back to obtain the remaining parameters

$\gamma_e \approx \frac{\sqrt[17]{L_b}}{\sqrt[17]{m_e} c^{2/17}}$

$B \approx \frac{c^{4/17} \nu_b m_e^{21/34}}{L_b^{2/17} \sqrt{r_e}}$

$N_e \approx \frac{L_b^{19/17}}{c^{21/17} \nu_b m_e^{19/17} r_e}$

The Lorentz factor of the electrons is extremely close to unity $\gamma_e \approx 1$, also both the Lorentz factor and the temperature are very insensitive to the luminosity.