Question

The synchrotron power P radiated by an ultrarelativistic (1) electron is extracted from its kinetic energy E mec2. The synchrotron lifetime of such an electron is:

Estimate the synchrotron lifetime of electrons responsible for the 1 GHz radiation from our Galaxy. You may make the approximation that Galactic synchrotron radiation at frequency v = 1 GHz is produced primarily by electrons whose critical frequency in the B = 5 × 10−6 G magnetic field of our Galaxy is vc = 1 GHz.

Answer 1

Answer on Question 55061, Physics / Astronomy | Astrophysics:

Question:

The synchrotron power P radiated by an ultrarelativistic (1) electron is extracted from its kinetic energy E mec2. The synchrotron lifetime of such an electron is?

Estimate the synchrotron lifetime of electrons responsible for the 1 GHz radiation from our Galaxy. You may make the approximation that Galactic synchrotron radiation at frequency $v = 1$ GHz is produced primarily by electrons whose critical frequency in the $B = 5 \times 10^{-6}$ G magnetic field of our Galaxy is $v_c = 1$ GHz.

Solution:

Synchrotron power is given by:

$$P = \frac {4}{3} \sigma_ {T} \beta^ {2} \gamma^ {2} c U _ {B}, \text{ where}$$ $$U _ {B} = \frac {B ^ {2}}{8 \pi}, \text{ and}$$ $$\sigma_ {T} = \frac {8 \pi}{3} \left(\frac {e ^ {2}}{m _ {e} c ^ {2}}\right) ^ {2}$$

Therefore:

$$P = \frac {4}{3} \frac {8 \pi}{3} \left(\frac {e ^ {2}}{m _ {e} c ^ {2}}\right) \beta^ {2} \gamma^ {2} c \times \frac {B ^ {2}}{8 \pi} = \frac {4}{9} \frac {\varepsilon^ {4} \beta^ {2} \gamma^ {2} B ^ {2}}{m _ {e} ^ {2} c ^ {2}}$$

The critical frequency is:

$$\nu_ {c} = \frac {3}{2} \gamma^ {2} \nu_ {G} \sin a = \frac {3}{2} \gamma^ {2} \frac {\omega_ {G}}{2 \pi} \sin a = \frac {3 \gamma^ {2}}{4 \pi} \frac {e B}{m _ {e} c} \sin a \Rightarrow \lambda = \sqrt {\frac {4 \pi m _ {e} c \nu_ {c}}{3 e B \sin a}}$$

Assume $\sin \sim 0.5$. Then we get $\gamma \sim 10^4$, and $\beta \sim 1$.

$$\tau \approx \frac {E}{P} = \frac {\lambda m _ {e} c ^ {2}}{\frac {4}{9} \frac {\varepsilon^ {4} \beta^ {2} \gamma^ {2} B ^ {2}}{m _ {e} ^ {2} c ^ {2}}} = 3 \times 10^{15} \text{sec}$$

Answer: $t \sim 10^{8}$ years

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