Chapter 5 Synchrotron Radiation

Larmor’s equation is valid only for gyro radiation from a particle with charge $q$ moving with a small velocity $v\ll c$. The magnetic force $\overrightarrow{F}$ exerted on the particle by a magnetic field $\overrightarrow{B}$ is

$$\boxed{\overrightarrow{F}=\frac{q(\overrightarrow{v}\times \overrightarrow{B})}{c}.}$$

(5.1)

The magnetic force is perpendicular to the particle velocity, so $\overrightarrow{F}\cdot \overrightarrow{v}=0$. Consequently the magnetic force does no work on the particle, does not change the particle’s kinetic energy $m{v}^2/2$, and does not change the component of velocity $v_{\parallel }$ parallel to the magnetic field. Because both $|v|$ and $v_{\parallel }$ are constant, the magnitude of the velocity component $|v_{\perp }|$ perpendicular to the magnetic field must also be constant. In a uniform magnetic field, the particle moves along the magnetic field line on a helical path with constant linear and angular speeds. In the inertial frame moving with velocity $v_{\parallel }$, the particle orbits in a circle of radius $r$ perpendicular to the magnetic field with the angular velocity $\omega$ needed to balance the centripetal and magnetic forces:

$$m|\dot{v}|=m{\omega }^{2}r=\frac{q}{c}|\overrightarrow{v}\times \overrightarrow{B}|=\frac{q}{c}\omega rB;$$

(5.2)

the orbital angular frequency is

$$\omega =\frac{qB}{mc}.$$

(5.3)

Equation 5.3 implies that the orbital frequency is independent of the particle speed so long as $v\ll c$. The angular gyro frequency ${\omega }_{\mathrm{G}}$ is defined by

$$\boxed{{\omega }_{\mathrm{G}}\equiv \frac{qB}{mc}.}$$

(5.4)

This definition holds for any particle speed, so the gyro frequency equals the actual orbital frequency if and only if $v\ll c$.

The angular gyro frequency (rad s${}^{-1}$) of an electron is

	${\omega }_{\mathrm{G}}={\displaystyle \frac{eB}{m_{\mathrm{e}}c}}$	$={\displaystyle \frac{4.8\times {10}^{-10}\mathrm{statcoul}\cdot B}{9.1\times {10}^{-28}\mathrm{g}\cdot 3\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1}}}$		(5.5)
		$\approx 17.6\times {10}^6\mathrm{rad}{\mathrm{s}}^{-1}\cdot B(\mathrm{gauss}).$		(5.6)

In terms of ${\nu }_{\mathrm{G}}\equiv {\omega }_{\mathrm{G}}/(2\pi )$ the electron gyro frequency in MHz is

$$\boxed{\left(\frac{{\nu }_{\mathrm{G}}}{\mathrm{MHz}}\right)=2.8\left(\frac{B}{\mathrm{gauss}}\right).}$$

(5.7)

The typical interstellar magnetic field strength in a normal spiral galaxy like ours is $B\approx 10\mu$G, so the electron gyro frequency is only ${\nu }_{\mathrm{G}}=2.8\mathrm{MHz}\cdot 10\times {10}^{-6}\mathrm{gauss}\sim 28\mathrm{Hz}$. The associated gyro radiation cannot propagate through the ISM because its frequency $\nu \sim 28\mathrm{Hz}$ is less than the plasma frequency (Equation 6.40).

Gyro radiation from nonrelativistic electrons is observable in very strong magnetic fields such as the $B\sim {10}^{12}$ gauss magnetic field of a neutron star. For example, the binary X-ray source Hercules X-1 exhibits an X-ray absorption line at photon energy $E\approx 34$ keV (Figure 5.1).

This spectral feature is thought to be a gyro-resonance absorption, in which case the frequency of this absorption line directly measures the magnetic field strength near the Her X-1 neutron star. The observed photon energy corresponds to the frequency

$$\nu =\frac{E}{h}\approx \frac{34\times {10}^3\mathrm{eV}\cdot 1.60\times {10}^{-12}\mathrm{erg}{\mathrm{eV}}^{-1}}{6.63\times {10}^{-27}\mathrm{erg}\mathrm{s}}\approx 8.2\times {10}^{18}\mathrm{Hz}.$$

(5.8)

Equating this frequency to the gyro frequency yields the magnetic field:

	$B$	$={\displaystyle \frac{2\pi {\nu }_{\mathrm{G}}{m}_{\mathrm{e}}c}{e}}$		(5.9)
		$\approx {\displaystyle \frac{2\pi \cdot 8.2\times {10}^{18}\mathrm{Hz}\cdot 9.1\times {10}^{-28}\mathrm{g}\cdot 3\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1}}{4.8\times {10}^{-10}\mathrm{statcoul}}}$		(5.10)
		$\approx 2.9\times {10}^{12}\mathrm{gauss}.$		(5.11)

For any pointlike event, the Lorentz transform (see Appendix C for a derivation) relates the coordinates $(x,y,z,t)$ in the unprimed inertial frame and the coordinates $(x^{\prime },y^{\prime },z^{\prime },t^{\prime })$ in the primed frame moving with velocity $v$ in the $x$-direction (Figure 5.2). They are

$$\boxed{x=\gamma (x^{\prime }+v{t}^{\prime }),y=y^{\prime },z=z^{\prime },t=\gamma (t^{\prime }+\beta x^{\prime }/c),}$$

(5.12)

$$\boxed{x^{\prime }=\gamma (x-vt),y^{\prime }=y,z^{\prime }=z,t^{\prime }=\gamma (t-\beta x/c),}$$

(5.13)

where

$$\boxed{\beta \equiv v/c}$$

(5.14)

and

$$\boxed{\gamma \equiv {(1-{\beta }^2)}^{-1/2}}$$

(5.15)

is called the Lorentz factor. The Lorentz transform is linear, so even for finite coordinate differences $(\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z,\mathrm{\Delta }t)$ and $(\mathrm{\Delta }{x}^{\prime },\mathrm{\Delta }{y}^{\prime },\mathrm{\Delta }{z}^{\prime },\mathrm{\Delta }{t}^{\prime })$ between two events, the differential form of the Lorentz transform is

$$\boxed{\mathrm{\Delta }x=\gamma (\mathrm{\Delta }{x}^{\prime }+v\mathrm{\Delta }{t}^{\prime }),\mathrm{\Delta }y=\mathrm{\Delta }{y}^{\prime },\mathrm{\Delta }z=\mathrm{\Delta }{z}^{\prime },\mathrm{\Delta }t=\gamma (\mathrm{\Delta }{t}^{\prime }+\beta \mathrm{\Delta }{x}^{\prime }/c),}$$

(5.16)

$$\boxed{\mathrm{\Delta }{x}^{\prime }=\gamma (\mathrm{\Delta }x-v\mathrm{\Delta }t),\mathrm{\Delta }{y}^{\prime }=\mathrm{\Delta }y,\mathrm{\Delta }{z}^{\prime }=\mathrm{\Delta }z,\mathrm{\Delta }{t}^{\prime }=\gamma (\mathrm{\Delta }t-\beta \mathrm{\Delta }x/c).}$$

(5.17)

The rest mass $m_{\mathrm{e}}$ of an electron can be converted to an energy by Einstein’s famous mass–energy equation

$$E=m{c}^2.$$

(5.18)

The energy corresponding to the electron’s rest mass $m_{\mathrm{e}}$ is

	$E_0=m_{\mathrm{e}}{c}^2$	$=9.1\times {10}^{-28}\mathrm{g}\cdot {(3\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1})}^2=8.2\times {10}^{-7}\mathrm{erg}$		(5.19)
		$={\displaystyle \frac{8.2\times {10}^{-7}\mathrm{erg}}{1.60\times {10}^{-12}\mathrm{erg}{(\mathrm{eV})}^{-1}}}=5.1\times {10}^5\mathrm{eV}=0.51\mathrm{MeV}.$		(5.20)

Cosmic-ray electrons having masses $m=\gamma m_{\mathrm{e}}\gg m_{\mathrm{e}}$ (Equation C.28) and total energies $E\gg E_0=0.51\mathrm{MeV}$ are called ultrarelativistic.

Ultrarelativistic electrons still move on spiral paths along magnetic field lines, but the angular frequencies ${\omega }_B$ of their orbits are lower because their inertial masses are multiplied by $\gamma$ (Appendix C):

$${\omega }_B=\frac{eB}{(\gamma m_{\mathrm{e}})c}=\frac{{\omega }_{\mathrm{G}}}{\gamma }.$$

(5.21)

The orbital frequency of a cosmic-ray electron with $\gamma ={10}^5$ in a $B\approx 10\mu$G interstellar magnetic field is only

$${\nu }_B\equiv \frac{{\omega }_B}{2\pi }\approx 28\times {10}^{-5}\mathrm{Hz}\approx 1\mathrm{cycle}\mathrm{per}\mathrm{hour}.$$

(5.22)

Because $v\approx c$ whenever $\gamma \gg 1$, the orbital radius $r$ of an ultrarelativistic electron is nearly independent of $\gamma$ and can be quite large:

$$r\approx \frac{c}{{\omega }_B}\approx \frac{3\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1}}{2\pi \cdot 28\times {10}^{-5}\mathrm{Hz}}\approx 1.7\times {10}^{13}\mathrm{cm}\approx 1\mathrm{AU}.$$

(5.23)

Equation 5.21 is not promising for the production of observable synchrotron radiation: the high observed masses $m=\gamma m_{\mathrm{e}}$ of relativistic electrons reduce their orbital frequencies and accelerations to extremely low values. However, two compensating relativistic effects can explain the strong synchrotron radiation observed at radio frequencies: (1) the total radiated power in the observer’s frame is proportional to ${\gamma }^2$ and (2) relativistic beaming turns the low-frequency sinusoidal radiation in the electron frame into a series of extremely sharp pulses containing power at much higher frequencies $\sim {\gamma }^3{\nu }_B={\gamma }^2{\nu }_{\mathrm{G}}$ in the observer’s frame. These relativistic corrections are derived in Sections 5.2.3 and 5.3.1.

Let primed coordinates describe the inertial frame in which the electron is (temporarily) nearly at rest. Larmor’s equation (Equation 2.143) gives the radiated power in the electron rest frame as

$$P^{\prime }=\frac{2{(e^{\prime })}^2{(a_{\perp }^{\prime })}^2}{3c^3}=\frac{2e^2{(a_{\perp }^{\prime })}^2}{3c^3}$$

(5.24)

because $e=e^{\prime }$ (electric charge is a relativistic invariant) follows directly from Maxwell’s relativistically correct equations.

The magnetic acceleration $a_{\perp }={(a_y^2+a_z^2)}^{1/2}$ of the electron in the frame of an observer at rest in the Galaxy can be derived by applying the chain rule for derivatives to the differential Lorentz coordinate transform (Equations 5.16 and 5.17).

$$a_y\equiv \frac{d{v}_y}{dt}=\frac{d{v}_y}{d{t}^{\prime }}\frac{d{t}^{\prime }}{dt}=\frac{1}{\gamma }\frac{d{v}_y^{\prime }}{d{t}^{\prime }}\frac{d{t}^{\prime }}{dt}=\frac{a_y^{\prime }}{{\gamma }^2}.$$

(5.25)

Similarly, $a_z=a_z^{\prime }/{\gamma }^2$ so

$$a_{\perp }=\frac{a_{\perp }^{\prime }}{{\gamma }^2}.$$

(5.26)

Thus

$$P^{\prime }=\frac{2e^2{(a_{\perp }^{\prime })}^2}{3c^3}=\frac{2e^2{a}_{\perp }^2{\gamma }^4}{3c^3}.$$

(5.27)

The next step is to transform from the radiated power $P^{\prime }=d{E}^{\prime }/d{t}^{\prime }$ in the electron frame to $P=dE/dt$, the power measured by an observer at rest in the Galaxy, by applying the chain rule to the mass–energy Equation 5.18:

$$P\equiv \frac{dE}{dt}=\frac{dE}{d{t}^{\prime }}\frac{d{t}^{\prime }}{dt}=\frac{dE}{d{E}^{\prime }}\frac{d{E}^{\prime }}{d{t}^{\prime }}\frac{d{t}^{\prime }}{dt}=\gamma P^{\prime }{\gamma }^{-1}=P^{\prime };$$

(5.28)

that is, the power is a relativistic invariant. Consequently,

$$P=P^{\prime }=\frac{2e^2{a}_{\perp }^2{\gamma }^4}{3c^3}\mathit{\hspace{1em}\hspace{1em}}(a_{\parallel }=0).$$

(5.29)

To calculate $a_{\perp }$, combine force balance in a circular orbit

$$a_{\perp }\equiv \frac{d{v}_{\perp }}{dt}={\omega }_B{v}_{\perp }$$

(5.30)

with Equation 5.21 for ${\omega }_B$ to get

$$a_{\perp }=\frac{eB{v}_{\perp }}{\gamma m_{\mathrm{e}}c}=\frac{eBv\sin \alpha }{\gamma m_{\mathrm{e}}c},$$

(5.31)

where the constant angle $\alpha$ between the electron velocity $\overrightarrow{v}$ and the magnetic field $\overrightarrow{B}$ is called the pitch angle.

Inserting $a_{\perp }$ from Equation 5.31 into Equation 5.29 gives the power radiated by a single electron moving with pitch angle $\alpha$:

$$P=\frac{2e^2}{3c^3}{\gamma }^2\frac{e^2{B}^2}{m_{\mathrm{e}}^2{c}^2}{v}^2{\sin }^2\alpha .$$

(5.32)

This power is usually written in terms of the Thomson cross section of an electron, ${\sigma }_{\mathrm{T}}$. The Thomson cross section is the classical radiation-scattering cross section of a charged particle. If a plane wave of electromagnetic radiation passes a free charged particle initially at rest, the electric field of that radiation will accelerate the particle, which in turn will radiate power in all directions according to Larmor’s equation. This process is called scattering rather than absorption because the total power in electromagnetic radiation is unchanged—all of the power extracted from the incident plane wave is reradiated at the same frequency but in other directions. It is a straightforward exercise to show that the geometric area that would intercept this amount of incident power from the plane wave is

$$\boxed{{\sigma }_{\mathrm{T}}\equiv \frac{8\pi }{3}{\left(\frac{e^2}{m_{\mathrm{e}}{c}^2}\right)}^2.}$$

(5.33)

Numerically,

$${\sigma }_{\mathrm{T}}=\frac{8\pi }{3}\left[\frac{{(4.80\times {10}^{-10}\mathrm{statcoul})}^2}{9.11\times {10}^{-28}\mathrm{g}{(3.00\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1})}^2}\right]\approx 6.65\times {10}^{-25}{\mathrm{cm}}^2.$$

(5.34)

The reason for using the Thomson cross section will become clear in the discussion of inverse-Compton scattering of radiation by the same cosmic rays that are producing synchrotron radiation (Section 5.5.1).

It is also conventional to eliminate the $B^2$ in Equation 5.32 in favor of the magnetic energy density

$$\boxed{U_B=\frac{B^2}{8\pi }.}$$

(5.35)

Then

$$P=\left[\frac{8\pi }{3}{\left(\frac{e^2}{m_{\mathrm{e}}{c}^2}\right)}^2\right]2\left(\frac{B^2}{8\pi }\right)c{\gamma }^2\frac{v^2}{c^2}{\sin }^2\alpha$$

(5.36)

simplifies to

$$\boxed{P=2{\sigma }_{\mathrm{T}}{\beta }^2{\gamma }^{2}c{U}_B{\sin }^2\alpha .}$$

(5.37)

The synchrotron power radiated by a single electron depends only on physical constants, the square of the electron kinetic energy (via ${\gamma }^2$), the magnetic energy density $U_B$, and the pitch angle $\alpha$.

The relativistic electrons in radio sources can have lifetimes of thousands to millions of years before losing their ultrarelativistic energies via synchrotron radiation or other processes. During their lifetimes they are scattered repeatedly by magnetic-field fluctuations and charged particles in their environment, and the distribution of their pitch angles gradually becomes random and isotropic. The average synchrotron power $⟨P⟩$ per electron in an ensemble of electrons having the same Lorentz factor $\gamma$ and isotropically distributed pitch angles $\alpha$ is

$$⟨P⟩=2{\sigma }_{\mathrm{T}}{\beta }^2{\gamma }^{2}c{U}_B⟨{\sin }^2\alpha ⟩,$$

(5.38)

where $⟨{\sin }^2\alpha ⟩$ is the average over all pitch angles:

	$⟨{\sin }^2\alpha ⟩$	$\equiv {\displaystyle \frac{\int {\sin }^2\alpha d\mathrm{\Omega }}{\int 𝑑\mathrm{\Omega }}}={\displaystyle \frac{1}{4\pi }}{\displaystyle \int {\sin }^2\alpha d\mathrm{\Omega }}$		(5.39)
		$={\displaystyle \frac{1}{4\pi }}{\displaystyle {\int }_{\varphi =0}^{2\pi }}{\displaystyle {\int }_{\alpha =0}^{\pi }}{\sin }^2\alpha \sin \alpha d\alpha d\varphi ={\displaystyle \frac{1}{4\pi }}2\pi {\displaystyle \frac{4}{3}}$		(5.40)
		$={\displaystyle \frac{2}{3}}.$		(5.41)

Thus the average synchrotron power per relativistic electron in a source with an isotropic pitch-angle distribution is

$$\boxed{⟨P⟩=\frac{4}{3}{\sigma }_{\mathrm{T}}{\beta }^2{\gamma }^{2}c{U}_B.}$$

(5.42)

For all $\gamma \gg 1$, the factor ${\beta }^2=1-{\gamma }^{-2}\approx 1$ can be ignored. Relativistic effects multiply the average radiated power by a factor ${\gamma }^2$ compared with the nonrelativistic ($\gamma =1$) Larmor equation.

Why does synchrotron radiation appear at frequencies much higher than ${\omega }_B={\omega }_{\mathrm{G}}/\gamma$? First, relativistic aberration beams the dipole pattern of Larmor radiation in the electron frame sharply along the direction of motion in the observer’s frame as $v$ approaches $c$ (Figure 5.3). Relativistic photon beaming follows directly from the relativistic velocity addition equations (Equations C.22 through C.26) that relate the photon velocity component $v_x$ in the unprimed frame of the observer to the photon velocity component $v_x^{\prime }$ in the primed frame and the velocity $v=\beta c$ of the primed frame in which the electron is instantaneously at rest:

	$v_x\equiv {\displaystyle \frac{dx}{dt}}$	$={\displaystyle \frac{dx}{d{t}^{\prime }}}{\displaystyle \frac{d{t}^{\prime }}{dt}}=\gamma \left({\displaystyle \frac{d{x}^{\prime }}{d{t}^{\prime }}}+v{\displaystyle \frac{d{t}^{\prime }}{d{t}^{\prime }}}\right){\left({\displaystyle \frac{dt}{d{t}^{\prime }}}\right)}^{-1}$		(5.43)
		$=\gamma (v_x^{\prime }+v){\left[\gamma \left(1+{\displaystyle \frac{\beta }{c}}{\displaystyle \frac{d{x}^{\prime }}{d{t}^{\prime }}}\right)\right]}^{-1},$		(5.44)

$$\boxed{v_x=(v_x^{\prime }+v){(1+\frac{\beta v_x^{\prime }}{c})}^{-1}.}$$

(5.45)

In the $y$-direction,

$$v_y\equiv \frac{dy}{dt}=\frac{dy}{d{t}^{\prime }}\frac{d{t}^{\prime }}{dt}=\frac{d{y}^{\prime }}{d{t}^{\prime }}{\left(\frac{dt}{d{t}^{\prime }}\right)}^{-1},$$

(5.46)

$$\boxed{v_y=\frac{v_y^{\prime }}{\gamma }{(1+\frac{\beta v_x^{\prime }}{c})}^{-1}.}$$

(5.47)

Consider the synchrotron photons emitted with speed $c$ at an angle ${\theta }^{\prime }$ from the $x^{\prime }$-axis. Let $v_x^{\prime }$ and $v_y^{\prime }$ be the projections of the photon speed onto the $x^{\prime }$- and $y^{\prime }$-axes. Then

$$\cos {\theta }^{\prime }=\frac{v_x^{\prime }}{c},\sin {\theta }^{\prime }=\frac{v_y^{\prime }}{c}.$$

(5.48)

In the observer’s frame the same photons have

$$\cos \theta =\frac{v_x}{c},\sin \theta =\frac{v_y}{c}.$$

(5.49)

Inserting the velocity Equations 5.45 and 5.47 into the angle Equations 5.48 and 5.49 yields relations connecting $\theta$ and ${\theta }^{\prime }$:

$$\cos \theta =\left(\frac{v_x^{\prime }+v}{1+\beta v_x^{\prime }/c}\right)\frac{1}{c}=\left(\frac{c\cos {\theta }^{\prime }+v}{1+\beta c\cos {\theta }^{\prime }/c}\right)\frac{1}{c}=\frac{\cos {\theta }^{\prime }+\beta }{1+\beta \cos {\theta }^{\prime }}$$

(5.50)

and

$$\sin \theta =\frac{v_y^{\prime }}{c\gamma (1+\beta v_x^{\prime }/c)}=\frac{\sin {\theta }^{\prime }}{\gamma (1+\beta \cos {\theta }^{\prime })}.$$

(5.51)

In the frame moving with the electron, the Larmor equation implies a power pattern proportional to ${\cos }^2{\theta }^{\prime }$ with nulls at ${\theta }^{\prime }=\pm \pi /2$. In the observer’s frame, these nulls are offset by much smaller angles

$$\theta =\pm \arcsin (1/\gamma ).$$

(5.52)

An observer at rest sees the radiation confined to a very narrow beam of width $2/\gamma$ between nulls, as shown in Figure 5.3. For example, a 10 GeV electron has $\gamma \approx 2\times {10}^4$ so $2/\gamma \approx {10}^{-4}\mathrm{rad}\approx 20$ arcsec! Although the electron is emitting continuously, the observer sees a short pulse of radiation only from the tiny fraction

$$\frac{2}{2\pi \gamma }=\frac{1}{\pi \gamma }$$

(5.53)

of the electron orbit where the electron is moving almost directly toward the observer.

The duration $\mathrm{\Delta }{t}_{\mathrm{p}}$ of the observed pulse is even shorter than the time $\mathrm{\Delta }t$ the electron needs to cover $1/(\pi \gamma )$ of its orbit because the electron is moving almost directly toward the observer with a speed approaching $c$ (Figure 5.4) when it is observable. As it travels toward the observer, the electron nearly keeps up with the radiation that it emits:

	$\mathrm{\Delta }{t}_{\mathrm{p}}$	$=t(\mathrm{end}\mathrm{of}\mathrm{observed}\mathrm{pulse})-t(\mathrm{start}\mathrm{of}\mathrm{observed}\mathrm{pulse})$		(5.54)
		$={\displaystyle \frac{\mathrm{\Delta }x}{v}}+{\displaystyle \frac{(x-\mathrm{\Delta }x)}{c}}-{\displaystyle \frac{x}{c}}.$		(5.55)

The first term in this equation represents the time taken by the electron to cover the distance $\mathrm{\Delta }x$, the second is the light travel time from the electron position at the end of the pulse seen by the observer, and the third is the light travel time from the electron position at the beginning of the pulse seen by the observer. Note that the observed pulse duration

$$\mathrm{\Delta }{t}_{\mathrm{p}}=\frac{\mathrm{\Delta }x}{v}-\frac{\mathrm{\Delta }x}{c}=\frac{\mathrm{\Delta }x}{v}\left(1-\frac{v}{c}\right)\mathit{\hspace{1em}}\ll \mathit{\hspace{1em}}\frac{\mathrm{\Delta }x}{v}=\mathrm{\Delta }t$$

(5.56)

is much less than the time $\mathrm{\Delta }t$ the electron needs to move a distance $\mathrm{\Delta }x$ because, in the observer’s frame, the electron nearly keeps up with its own radiation. In the limit $v\to c$,

$$\left(1-\frac{v}{c}\right)=\left(1-\frac{v}{c}\right)\frac{1+v/c}{1+v/c}=\frac{1-v^2/c^2}{1+v/c}\approx \frac{{\gamma }^{-2}}{2}=\frac{1}{2{\gamma }^2}$$

(5.57)

so

$$\mathrm{\Delta }{t}_{\mathrm{p}}=\frac{\mathrm{\Delta }t}{2{\gamma }^2}=\frac{\mathrm{\Delta }x}{v}\frac{1}{2{\gamma }^2}=\frac{\mathrm{\Delta }\theta }{{\omega }_B}\frac{1}{2{\gamma }^2}.$$

(5.58)

Recall that $\mathrm{\Delta }\theta \approx 2/\gamma$ (Figure  5.4) so

$$\mathrm{\Delta }{t}_{\mathrm{p}}=\frac{2}{\gamma {\omega }_{B}2{\gamma }^2}=\frac{1}{{\gamma }^3{\omega }_B}=\frac{1}{{\gamma }^2{\omega }_{\mathrm{G}}}$$

(5.59)

is the full observed duration of the pulse. Allowing for the motion of the electron parallel to the magnetic field replaces the total magnetic field by its perpendicular component $B_{\perp }=B\sin \alpha$, yielding

$$\mathrm{\Delta }{t}_{\mathrm{p}}=\frac{1}{{\gamma }^2{\omega }_{\mathrm{G}}\sin \alpha },$$

(5.60)

where $\alpha$ is the pitch angle of the electron. Thus a plot of the power received as a function of time is very spiky. If $\gamma \approx {10}^4$ and $B\sim 10\mu$G, the half width of each pulse is s and the spacing between pulses is $\gamma /{\nu }_{\mathrm{G}}>{10}^2$ s (Figure 5.5).

The observed synchrotron power spectrum is the Fourier transform of this time series of pulses. The pulse train is the convolution of the individual pulse profile with the shah function (see Figure A.1 or Bracewell [15], a valuable reference book):

$$\mathrm{III}(t/\mathrm{\Delta }t)\equiv \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\delta [(t/\mathrm{\Delta }t)]-n),$$

(5.61)

where each delta function $\delta$ is an infinitesimally narrow spike at integer $t/\mathrm{\Delta }t=n$ and whose integral is unity. The convolution theorem (Equation A.15) states that the Fourier transform of the pulse train is the product of the Fourier transform of one pulse and the Fourier transform of the shah function.

The Fourier transform of the shah function is also the shah function (Figure A.1), so the similarity theorem (Equation A.11) implies that the Fourier transform of

$$\mathrm{III}\left(\frac{t{\nu }_{\mathrm{G}}}{\gamma }\right)$$

(5.62)

in the time domain is proportional to

$$\mathrm{III}\left(\frac{\nu \gamma }{{\nu }_{\mathrm{G}}}\right),$$

(5.63)

which is a nearly continuous series of spikes in the frequency domain. Adjacent spikes are separated in frequency by only

(5.64)

Although this is not formally a continuous spectrum, the frequency shifts caused by even tiny fluctuations in electron energy, magnetic field strength, or pitch angle cause frequency shifts much larger than $\mathrm{\Delta }\nu$, so the spectrum of synchrotron radiation is effectively continuous.

Thus the synchrotron spectrum of a single electron is fairly flat at low frequencies and tapers off at frequencies above

$${\nu }_{\max }\approx \frac{1}{2\mathrm{\Delta }{t}_{\mathrm{p}}}\approx \pi {\gamma }^2{\nu }_{\mathrm{G}}\sin \alpha \propto {\gamma }^2{B}_{\perp }.$$

(5.65)

It isn’t necessary to know the Fourier transform of the pulse shape precisely to calculate the synchrotron spectra of celestial sources because real sources don’t contain electrons with just one energy and one pitch angle in a uniform magnetic field. The actual energy distribution of cosmic rays in a real radio source is a very broad power law, broad enough to smear out the details of the spectrum from each electron-energy range. Just for the record, the synchrotron power spectrum of a single electron is

$$\boxed{P(\nu )=\frac{\sqrt{3}{e}^{3}B\sin \alpha }{m_{\mathrm{e}}{c}^2}\left(\frac{\nu }{{\nu }_{\mathrm{c}}}\right){\int }_{\nu /{\nu }_{\mathrm{c}}}^{\mathrm{\infty }}{K}_{5/3}(\eta )d\eta ,}$$

(5.66)

where $K_{5/3}$ is a modified Bessel function and ${\nu }_{\mathrm{c}}$ is the critical frequency whose value is

$$\boxed{{\nu }_{\mathrm{c}}=\frac{3}{2}{\gamma }^2{\nu }_{\mathrm{G}}\sin \alpha .}$$

(5.67)

For the full mathematical derivation of Equations 5.66 and 5.67, see Pacholczyk [78], a valuable reference for details of radiation processes.

The synchrotron power spectrum of a single electron is plotted in Figure 5.6. It has a logarithmic slope

$$\frac{d\log P(\nu )}{d\log \nu }\approx \frac{1}{3}$$

(5.68)

at low frequencies, a broad peak near the critical frequency ${\nu }_{\mathrm{c}}$, and falls off sharply at higher frequencies. One way to look at ${\nu }_{\mathrm{c}}$ is

$${\nu }_{\mathrm{c}}=\left(\frac{3}{2}\sin \alpha \right){\left(\frac{E}{m{c}^2}\right)}^2\frac{eB}{2\pi m_{\mathrm{e}}c}\propto E^2{B}_{\perp }.$$

(5.69)

That is, the frequency at which each electron emits most strongly is proportional to the square of its energy multiplied by the strength of the perpendicular component of the magnetic field.

If a synchrotron source containing any arbitrary distribution of electron energies is optically thin ($\tau \ll 1$), then its spectrum is the superposition of the spectra from individual electrons and its flux density cannot rise faster than ${\nu }^{1/3}$ at any frequency $\nu$. In other words, the (negative) spectral index $\alpha \equiv -d\log P_{\nu }/d\log \nu$ (be careful not to confuse this spectral index $\alpha$ with the electron pitch angle $\alpha$) must always be greater than $-1/3$. Most astrophysical sources of synchrotron radiation have spectral indices near $\alpha \approx 0.75$ at frequencies where they are optically thin, and their high-frequency spectral indices reflect their electron energy distributions, not the spectra of individual electrons.

The energy distribution of cosmic-ray electrons in most synchrotron sources is roughly a power law:

$$n(E)dE\propto E^{-\delta }dE,$$

(5.70)

where $n(E)dE$ is the number of electrons per unit volume with energies $E$ to $E+dE$. The energy range around $\gamma \sim {10}^4$ is relevant to the production of radio radiation. Because $n(E)$ is nearly a power law over more than a decade of energy and the critical frequency ${\nu }_{\mathrm{c}}$ is proportional to $E^2$, the synchrotron spectrum will reflect this power law over a frequency range of at least ${10}^2=100$. Consequently the detailed spectra of individual electrons can be ignored because they are smeared out in the source spectrum by the broad power-law energy distribution. The source spectrum can be calculated with good accuracy from the approximation that each electron radiates all of its average power (Equation 5.42)

$$P=-\frac{dE}{dt}=\frac{4}{3}{\sigma }_{\mathrm{T}}{\beta }^2{\gamma }^{2}c{U}_B$$

(5.71)

at the single frequency

$$\nu \approx {\gamma }^2{\nu }_{\mathrm{G}},$$

(5.72)

which is very close to the critical frequency (Equation 5.67). Then the emission coefficient (Equation 2.26) of synchrotron radiation by an ensemble of electrons is

$$j_{\nu }d\nu =-\frac{dE}{dt}n(E)dE,$$

(5.73)

where

$$E=\gamma m_{\mathrm{e}}{c}^2\approx {\left(\frac{\nu }{{\nu }_{\mathrm{G}}}\right)}^{1/2}{m}_{\mathrm{e}}{c}^2.$$

(5.74)

Differentiating $E(\nu )$ in Equation 5.74 gives

$$dE\approx \frac{m_{\mathrm{e}}{c}^2{\nu }^{-1/2}}{2{\nu }_{\mathrm{G}}^{1/2}}d\nu$$

(5.75)

so

$$j_{\nu }\propto \left(\frac{4}{3}{\sigma }_{\mathrm{T}}{\beta }^2{\gamma }^{2}c{U}_B\right)(E^{-\delta })\left(\frac{m_{\mathrm{e}}{c}^2{\nu }^{-1/2}}{2{\nu }_{\mathrm{G}}^{1/2}}\right).$$

(5.76)

Eliminating $E$ in favor of $\nu /{\nu }_{\mathrm{G}}$ and then using ${\nu }_{\mathrm{G}}\propto B$ yields an expression for $j_{\nu }$ in terms of $\nu$ and $B$ only:

$$j_{\nu }\propto \left(\frac{\nu }{{\nu }_{\mathrm{G}}}\right)B^2{\left(\frac{\nu }{{\nu }_{\mathrm{G}}}\right)}^{-\delta /2}{(\nu {\nu }_{\mathrm{G}})}^{-1/2}\propto \left(\frac{\nu }{B}\right)B^2{\left(\frac{\nu }{B}\right)}^{-\delta /2}{(\nu B)}^{-1/2}.$$

(5.77)

This simplifies to

$$\boxed{j_{\nu }\propto B^{(\delta +1)/2}{\nu }^{(1-\delta )/2}.}$$

(5.78)

Thus the spectrum of optically thin synchrotron radiation from a power-law distribution $n(E)\propto E^{-\delta }$ of electrons is also a power law, and the spectral index $\alpha =-d\ln S/d\ln \nu$ depends only on $\delta$:

$$\boxed{\alpha =\frac{\delta -1}{2}.}$$

(5.79)

In our Galaxy and in many other synchrotron sources, $\alpha \approx 0.75$ near $\nu \approx 1$ GHz, so the radio spectra imply $\delta \approx 2.5$. This value of $\delta$ reflects the initial power-law energy slope ${\delta }_0$ of cosmic rays accelerated in shocks, such as the shocks produced by supernova remnants expanding into the ambient interstellar medium, modified by loss processes that deplete the population of relativistic electrons. For example, the rate at which an electron loses energy to synchrotron radiation is proportional to $E^2$, so higher-energy electrons are depleted more rapidly. The critical frequency is also proportional to $E^2$, so synchrotron losses eventually steepen source spectra at higher frequencies. If relativistic electrons with initial power-law slope ${\delta }_0$ are continuously injected into a synchrotron source, synchrotron losses will eventually steepen that slope to $\delta ={\delta }_0+1$ at high energies and the high-frequency spectral index will steepen by $\mathrm{\Delta }\alpha =1/2$. See Pacholczyk [78] for a more detailed discussion of energy losses and their spectral consequences.

The brightness temperatures of synchrotron sources cannot become arbitrarily large at low frequencies because for every emission process there is an associated absorption process. If the emitting particles are in local thermodynamic equilibrium (LTE), they have a Maxwellian energy distribution and the source is thermal. No thermal source can have a brightness temperature greater than the kinetic temperature of the emitting particles. If the energy distribution of relativistic electrons in a synchrotron source were a (relativistic) Maxwellian, the electrons would have a well-defined kinetic temperature, and synchrotron self-absorption would prevent the brightness temperature of the synchrotron radiation from exceeding the kinetic temperature of the emitting electrons. Most astrophysical synchrotron sources are nonthermal sources because the energy distribution of the relativistic electrons is a power law and there is no well-defined electron temperature. However, synchrotron self-absorption occurs for any electron energy distribution, and the low-frequency spectrum of an optically thick synchrotron source is a power law whose slope is $\alpha =-d\ln S/d\ln \nu =-5/2$. That result is derived below.

Electrons with energy $E=\gamma m_{\mathrm{e}}{c}^2$ emit most of their synchrotron power near the critical frequency

$${\nu }_{\mathrm{c}}\sim \frac{{\gamma }^{2}eB}{2\pi m_{\mathrm{e}}c},$$

(5.80)

so the synchrotron emission at frequency $\nu$ comes primarily from electrons with Lorentz factors near

$$\gamma \approx {\left(\frac{2\pi m_{\mathrm{e}}c\nu }{eB}\right)}^{1/2}.$$

(5.81)

In this approximation that only those electrons having one particular energy $E$ contribute to the emission (and hence absorption) at each frequency $\nu$, all other electrons could have a relativistic Maxwellian energy distribution to match without changing the resulting emission and absorption at that frequency. Consequently, a sufficiently bright synchrotron source will be optically thick, and its brightness temperature at any frequency cannot exceed the effective temperature of the electrons emitting at that frequency.

In an ultrarelativistic gas, the ratio of specific heats at constant pressure and at constant volume is $c_{\mathrm{p}}/c_{\mathrm{v}}=4/3$, not the nonrelativistic $5/3$, so the relation between electron energy $E$ and temperature $T_{\mathrm{e}}$ is

$$E=3k{T}_{\mathrm{e}},\mathrm{not}\frac{3k{T}_{\mathrm{e}}}{2}.$$

(5.82)

Thus the effective temperature of relativistic electrons with energy $E$ can be defined as

$$T_{\mathrm{e}}\equiv \frac{E}{3k}=\frac{\gamma m_{\mathrm{e}}{c}^2}{3k},$$

(5.83)

even if the ensemble of electrons has a nonthermal energy distribution. Using Equation 5.81 to eliminate $\gamma$ in favor of $\nu$ gives the effective temperature of those electrons producing most of the synchrotron radiation at frequency $\nu$:

$$T_{\mathrm{e}}\approx {\left(\frac{2\pi m_{\mathrm{e}}c\nu }{eB}\right)}^{1/2}\frac{m_{\mathrm{e}}{c}^2}{3k}.$$

(5.84)

Numerically,

$$\boxed{\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)\approx 1.18\times {10}^6{\left(\frac{\nu }{\mathrm{Hz}}\right)}^{1/2}{\left(\frac{B}{\mathrm{gauss}}\right)}^{-1/2}.}$$

(5.85)

For example, the effective temperature of the relativistic electrons emitting synchrotron radiation at $\nu =0.1\mathrm{GHz}={10}^8\mathrm{Hz}$ in a $B=100\mu$gauss $={10}^{-4}$ gauss magnetic field is

$$\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)\approx 1.18\times {10}^6\cdot {({10}^8)}^{1/2}{({10}^{-4})}^{-1/2}\approx {10}^{12}.$$

(5.86)

At a sufficiently low frequency $\nu$, the brightness temperature $T_{\mathrm{b}}$ of any synchrotron source will approach the effective electron temperature $T_{\mathrm{e}}$ of electrons emitting at that frequency and the source will become opaque. Equation 2.33 defines brightness temperature in the Rayleigh–Jeans limit:

$$T_{\mathrm{b}}\equiv \frac{I_{\nu }{c}^2}{2k{\nu }^2}.$$

(5.87)

Setting $T_{\mathrm{b}}\approx T_{\mathrm{e}}$ and using Equation 5.85 to eliminate $T_{\mathrm{e}}$ in favor of $\nu$ and $B$ gives

$$I_{\nu }\approx \frac{2k{T}_{\mathrm{e}}{\nu }^2}{c^2}\propto {\nu }^{1/2}{\nu }^2{B}^{-1/2}={\nu }^{5/2}{B}^{-1/2}.$$

(5.88)

Thus at low frequencies the spectrum of a synchrotron self-absorbed and spatially homogeneous source is a power law of slope $5/2$:

$$\boxed{S(\nu )\propto {\nu }^{5/2},}$$

(5.89)

independent of the slope $\delta$ of the electron-energy spectrum. The flux density of an opaque but truly thermal source (e.g., an Hii region) is proportional to ${\nu }^2$; the extra ${\nu }^{1/2}$ for synchrotron radiation comes from the fact that $T_{\mathrm{e}}\propto {\nu }^{1/2}$ (Equation 5.85).

The full spectrum of a homogeneous cylindrical synchrotron source (Figure 5.7) is [78]

$$S\propto {\left(\frac{\nu }{{\nu }_1}\right)}^{5/2}\left\{1-\exp \left[-{\left(\frac{\nu }{{\nu }_1}\right)}^{-(\delta +4)/2}\right]\right\},$$

(5.90)

where ${\nu }_1$ is the frequency at which $\tau =1$. Real astrophysical sources are inhomogeneous, so synchrotron self-absorption always yields slopes much lower than $5/2$, as illustrated by Figure 5.8.

Substituting $T_{\mathrm{b}}\approx T_{\mathrm{e}}$ into Equation 5.85 yields an estimate of the magnetic field strength in a self-absorbed source whose brightness temperature has been measured at frequency $\nu$:

$$\boxed{\left(\frac{B}{\mathrm{gauss}}\right)\approx 1.4\times {10}^{12}\left(\frac{\nu }{\mathrm{Hz}}\right){\left(\frac{T_{\mathrm{b}}}{\mathrm{K}}\right)}^{-2}.}$$

(5.91)

For example, a self-absorbed radio source with observed brightness temperature $T_{\mathrm{b}}\approx {10}^{11}$ K at $\nu =1$ GHz has a magnetic field strength

$$\left(\frac{B}{\mathrm{gauss}}\right)\approx 1.4\times {10}^{12}\cdot {10}^9\cdot {({10}^{11})}^{-2}\approx 0.1.$$

The spectra of celestial radio sources are more complex because real sources have nonuniform magnetic fields and electron energy distributions in geometrically complex structures. Representative spectra of powerful radio galaxies and quasars are illustrated in Figure 5.8.

Synchrotron radiation from cosmic-ray electrons accelerated by the supernova remnants of relatively massive ($M>8M_{\odot }$) and short-lived ( yr) stars dominates the radio continuum emission of the nearby starburst galaxy M82 (Figure 8.13) at frequencies GHz (see the dot-dash line in Figure 2.24). Thermal emission (dashed line) from Hii regions ionized primarily by even more massive ($M>15M_{\odot }$) and shorter-lived stars is strongest between about 30 and 200 GHz. At frequencies well below 1 GHz, free–free absorption flattens the overall spectrum.

The existence of a synchrotron source implies the presence of relativistic electrons with some energy density $U_{\mathrm{e}}$ and a magnetic field whose energy density is $U_B=B^2/(8\pi )$. What is the minimum total energy in relativistic particles and magnetic fields required to produce a synchrotron source of a given radio luminosity

$$L={\int }_{{\nu }_{\min }}^{{\nu }_{\max }}{L}_{\nu }𝑑\nu$$

(5.92)

over the frequency range conventionally bounded by ${\nu }_{\min }={10}^7$ Hz and ${\nu }_{\max }={10}^{11}$ Hz?

The energy density of relativistic electrons in the energy range $E_{\min }$ to $E_{\max }$ is

$$U_{\mathrm{e}}={\int }_{E_{\min }}^{E_{\max }}En(E)𝑑E,$$

(5.93)

where $n(E)dE$ is the number density of electrons in the energy range $E$ to $E+dE$. Electrons with energy $E$ emit most of the radiation seen at frequency $\nu \propto E^{2}B$, so the electron energy corresponding to radiation at frequency $\nu$ satisfies

$$E\propto B^{-1/2}.$$

(5.94)

Thus the ratio of $U_{\mathrm{e}}$ to $L$ can be written in terms of the energy limits:

$$\frac{U_{\mathrm{e}}}{L}\propto \frac{{\int }_{E_{\min }}^{E_{\max }}En(E)𝑑E}{-{\int }_{E_{\min }}^{E_{\max }}(dE/dt)n(E)𝑑E},$$

(5.95)

where the synchrotron power emitted per electron is $(-dE/dt)\propto B^2{E}^2$. For a power-law electron energy distribution $n(E)\propto E^{-\delta }$,

$$\frac{U_{\mathrm{e}}}{L}\propto \frac{{\int }_{E_{\min }}^{E_{\max }}{E}^{1-\delta }𝑑E}{B^2{\int }_{E_{\min }}^{E_{\max }}{E}^{2-\delta }𝑑E}\propto \frac{{E^{2-\delta }|}_{E_{\min }}^{E_{\max }}}{{B^2{E}^{3-\delta }|}_{E_{\min }}^{E_{\max }}}.$$

(5.96)

The energy limits $E_{\min }$ and $E_{\max }$ are both proportional to $B^{-1/2}$ (Equation 5.94) so

$$\frac{U_{\mathrm{e}}}{L}\propto \frac{{(B^{-1/2})}^{2-\delta }}{B^2{(B^{-1/2})}^{3-\delta }}=\frac{B^{-1+\delta /2}}{B^2{B}^{-3/2+\delta /2}}=B^{-3/2}$$

(5.97)

Thus the electron energy density needed to produce a given synchrotron luminosity scales as

$$\boxed{U_{\mathrm{e}}\propto B^{-3/2},}$$

(5.98)

while the magnetic energy density is

$$U_B\propto B^2.$$

(5.99)

The “invisible” cosmic-ray protons and heavier ions emit negligible synchrotron power but they still contribute to the total cosmic-ray particle energy. If the ion/electron energy ratio is $\eta$, then the total energy density in cosmic rays is $U_{\mathrm{E}}=(1+\eta )U_{\mathrm{e}}$ and the total energy density $U$ of both cosmic rays and magnetic fields is

$$\boxed{U=(1+\eta )U_{\mathrm{e}}+U_B.}$$

(5.100)

Cosmic rays collected near the Earth have $\eta \approx 40$, but the value of $\eta$ in radio galaxies and quasars has not been measured.

The greatly differing dependences of $U_{\mathrm{e}}$ and $U_B$ on $B$ means that the total energy density $U$ has a fairly sharp minimum near equipartition, the point at which $(1+\eta )U_{\mathrm{e}}\approx U_B$ (Figure 5.9).

The minimum of the total energy density $U$ occurs at

$$\frac{dU}{dB}=\frac{d[(1+\eta )U_{\mathrm{e}}+U_B]}{dB}=0.$$

(5.101)

The logarithmic derivative of the electron energy density $U_{\mathrm{e}}\propto B^{-3/2}$ is

$$\frac{d{U}_{\mathrm{e}}}{dB}\cdot U_{\mathrm{e}}^{-1}=-\left(\frac{3}{2}\right)B^{-5/2}{B}^{3/2}=-\frac{3}{2B},$$

(5.102)

so

$$\frac{d{U}_{\mathrm{e}}}{dB}=-\frac{3U_{\mathrm{e}}}{2B}.$$

(5.103)

The logarithmic derivative of the magnetic-field energy density $U_B\propto B^2$ is

$$\frac{d{U}_B}{dB}\cdot U_B^{-1}=\frac{2B}{B^2}=\frac{2}{B},$$

(5.104)

so

$$\frac{d{U}_B}{dB}=\frac{2U_B}{B}.$$

(5.105)

Inserting Equations 5.103 and 5.105 into the minimum-energy Equation 5.101 gives

$$-\frac{3(1+\eta )U_{\mathrm{e}}}{2B}+\frac{2U_B}{B}=0.$$

(5.106)

The ratio of cosmic-ray particle energy density to magnetic field energy that minimizes the total energy is

$$\boxed{\frac{\mathrm{particle}\mathrm{energy}\mathrm{density}}{\mathrm{magnetic}\mathrm{field}\mathrm{energy}\mathrm{density}}=\frac{(1+\eta )U_{\mathrm{e}}}{U_B}=\frac{4}{3}.}$$

(5.107)

This ratio is nearly unity, so minimum energy implies (near) equipartition of energy: the total cosmic-ray energy density (including the energy of the nonradiating ions) $(1+\eta )U_{\mathrm{e}}$ is nearly equal to the total magnetic energy density $U_B$. It is not known whether most synchrotron sources are in equipartition, but radio astronomers often assume so because

1. 1.

   it is physically plausible—systems with interacting components often tend toward equipartition;

2. 2.

   extragalactic radio sources with high luminosities $L$ and large volumes $V$ such as Cyg A have enormous total energy $E=UV$ requirements even near equipartition; the “energy problem” is even worse otherwise;

3. 3.

   it eliminates an unknown parameter and permits estimates of the relativistic particle energies and the magnetic field strengths of radio sources with measured luminosities and sizes.

Getting the actual numerical values of the particle and magnetic-field energy densities from the synchrotron emission coefficient is a straightforward but tedious algebraic chore (Wilson et al. [116, Section 10.10]). The results (from Pacholczyk [78, p. 171]) are summarized in Equations 5.109 and 5.110. The functions $c_{12}$ and $c_{13}$ in these equations absorb the integrations from frequency ${\nu }_{\min }$ to ${\nu }_{\max }$ and the physical constants in Gaussian CGS units. The values of $c_{12}$ and $c_{13}$ are plotted in Figures 5.10 and 5.11.

For a spherical radio source with radius $R$ and magnetic field strength $B$, the total magnetic energy is

$$E_B=U_{B}V=\frac{B^2}{8\pi }\frac{4\pi R^3}{3}=\frac{B^2{R}^3}{6}.$$

(5.108)

The minimum-energy magnetic field strength for a source of radio luminosity $L$ is

$$\boxed{B_{\min }={[4.5(1+\eta )c_{12}L]}^{2/7}{R}^{-6/7}\mathrm{gauss}}$$

(5.109)

and the corresponding total energy is

$$\boxed{E_{\min }(\mathrm{total})=c_{13}{[(1+\eta )L]}^{4/7}{R}^{9/7}\mathrm{ergs}.}$$

(5.110)

The synchrotron lifetime of a source is defined as the ratio of total electron energy $E_{\mathrm{e}}$ to the energy loss rate $L$ from synchrotron radiation:

$${\tau }_{\mathrm{s}}\equiv \frac{E_{\mathrm{e}}}{L}.$$

(5.111)

It approximates the lifetime of a synchrotron source if the primary loss mechanism is synchrotron radiation; if other loss mechanisms (e.g., inverse-Compton scattering) are significant, the actual source lifetime will be shortened. The synchrotron lifetime can be written as

$$\boxed{{\tau }_{\mathrm{s}}\approx c_{12}{B}_{\perp }^{-3/2}.}$$

(5.112)

The steady-state luminosity of an astronomical object of total mass $M$ is limited by the requirement that the outward radiation pressure cannot exceed the pull of gravity. Otherwise, radiation pressure would expel the outer layers of a star or disrupt accretion onto a compact object such as a black hole or neutron star. If the atmosphere of the star or the infalling material is primarily ionized hydrogen, the free electrons will Thomson-scatter outflowing radiation. The Thomson-scattering cross section ${\sigma }_{\mathrm{T}}$ is given by Equation 5.33. Each electron being pushed away by radiation pressure will drag along one proton ($m_{\mathrm{p}}\gg m_{\mathrm{e}}$) with it to maintain charge neutrality. Balancing the forces from radiation and gravity on each electron/proton pair at distance $r$ from the accreting object defines the Eddington luminosity:

$$\frac{L_{\mathrm{E}}}{4\pi r^2}\frac{{\sigma }_{\mathrm{T}}}{c}=\frac{GM(m_{\mathrm{p}}+m_{\mathrm{e}})}{r^2}\approx \frac{GM{m}_{\mathrm{p}}}{r^2}.$$

(5.113)

Both forces are proportional to $r^{-2}$ so

$$L_{\mathrm{E}}\approx \frac{4\pi GM{m}_{\mathrm{p}}c}{{\sigma }_{\mathrm{T}}}$$

(5.114)

is proportional to the mass $M$ and independent of distance. In CGS units,

	$L_{\mathrm{E}}={\displaystyle \frac{4\pi \cdot 6.67\times {10}^{-8}\mathrm{dyne}{\mathrm{cm}}^2{\mathrm{g}}^{-2}\cdot M\cdot 1.66\times {10}^{-24}\mathrm{g}\cdot 3\times {10}^{10}\mathrm{cm}{\mathrm{s}}^{-1}}{6.65\times {10}^{-25}{\mathrm{cm}}^2}},$
	$L_{\mathrm{E}}(\mathrm{erg}{\mathrm{s}}^{-1})=6.28\times {10}^{4}M(\mathrm{g}).$		(5.115)

Normalized to “solar” units $L_{\odot }\approx 3.83\times {10}^{33}$ erg s${}^{-1}$ and $M_{\odot }\approx 1.99\times {10}^{33}$ g,

$$\left(\frac{L_{\mathrm{E}}}{L_{\odot }}\right)\approx \frac{6.28\times {10}^4\cdot 1.99\times {10}^{33}\mathrm{g}}{3.83\times {10}^{33}\mathrm{erg}{\mathrm{s}}^{-1}}\left(\frac{M}{M_{\odot }}\right),$$

(5.116)

$$\boxed{\left(\frac{L_{\mathrm{E}}}{L_{\odot }}\right)\approx 3.3\times {10}^4\left(\frac{M}{M_{\odot }}\right).}$$

(5.117)