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• Outline
• Glossary
• 1. Radio Science using Interferometric Arrays
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  • Next: 1.7 Line emission

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1.6.1 Synchrotron Emission:

Sychrotron emission is one of the most commonly encountered forms of radiation found from astronomical radio sources. This type of radiation originates from relativistic particles get accelerated in a magnetic field.

The mechanism by which synchrotron emission occurs depends fundamentally on special relativistic effects. We won't delve into the details here. Instead we will try to explain (in a rather hand wavy way) some of the underlying physics. As we have seen in $\S$ 1.2.1 ➞,

LB:RF:this is the original link but I don't think it points to the right place. Add a reference to where this is discussed and link to that. See also comment in previous section about where the Larmor formula is first introduced

an accelerating charge emitts radiation. The acceleration is a result of the charge moving through an ambient magnetic field. The non-relativistic Larmor formula for the radiated power is:

$$P= \frac{2}{3}\frac{q^{2}a^{2}}{c^{3}}$$

If the acceleration is a result of a magnetic field $B$, we get:

$$P=\frac{2}{3}\frac{q^{2}}{c^{3}}\frac{v_{\perp}^{2}B^{2}q^{2}}{m^{2}} $$

where $v_{\perp}$ is the component of velocity of the particle perpendicular to the magnetic field, $m$ is the mass of the charged particle, $q$ is it's charge and $a$ is its acceleration. This is essentially the cyclotron radiation. Relativistic effects (i.e. as $v_\perp \rightarrow c$) modifies this to:

$$P = \gamma^{2} \frac{2}{3}\frac{q^{2}}{c^{3}}\frac{v_{\perp}^{2}B^{2}q^{2}}{m^{2}c^{2}} = \gamma^{2} \frac{2}{3}\frac{q^{4}}{c^{3}}\frac{v_{\perp}^{2}B^{2}}{m^{2}c^{2}} $$

where $$\gamma = \frac{1}{\sqrt{1+v^{2}/c^{2}}} = \frac{E}{mc^{2}} $$

LB:IC: This is a very unusual form for the relativistic version of Larmor's formula. I suggest clarifying the derivation.

is a measure of the energy of the particle. Non-relativistic particles have $\gamma \sim 1$ whereas relativistic and ultra-relativistic particles typically have $\gamma \sim 100$ and $\gamma \geq 1000$ respectively. Since $v_{\perp}= v \sin\alpha$, with $\alpha$ being the angle between the magnetic field and the velocity of the particle, the radiated power can be written as:

$$P=\gamma^{2} \frac{2}{3}\frac{q^{4}}{c^{3}}\frac{v^{2}B^{2}\sin\alpha^{2}}{m^{2}c^{2}} $$

From this equation it can be seen that the total power radiated by the particle depends on the strength of the magnetic field and that the higher the energy of the particle, the more power it radiates.

In analogy with the non-relativistic case, there is a frequency of gyration. This refers to the path the charged particle follows while being accelerated in a magnetic field. The figure below illustrates the idea.

Figure 1.6.1 Example path of a charged particle accelerated in a magnetic field

The frequency of gyration in the non-relativistic case is simply

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

For synchrotron radiation, this gets modified to

$$\omega_{G}= \frac{qB}{\gamma mc} $$

since, in the relativistic case, the mass is modified to $m \rightarrow \gamma m$.

In the non-relativistic case (i.e. cyclotron radiation) the frequency of gyration corresponds to the frequency of the emitted radiation. If this was also the case for the synchrotron radiation then, for magnetic fields typically found in galaxies (a few micro-Gauss or so), the resultant frequency would be less than one Hertz! Fortunately the relativistic beaming and Doppler effects come into play increasing the frequency of the observed radiation by a factor of about $\gamma^{3}$. This brings the radiation into the radio regime. This frequency, known also as the 'critical frequency' is at most of the emission takes place. It is given by

$$\nu_{c} \propto \gamma^{3}\nu_{G} \propto E^{2}$$

LB:IC: The last sentence is not clear. Why is it called the critical frequency? How does it come about?

So far we have discussed a single particle emitting synchrotron radiation. However, what we really want to know is what happens in the case of an ensemble of radiating particles. Since, in an (approximately) uniform magnetic field, the synchrotron emission depends only on the magnetic field and the energy of the particle, all we need is the distribution function of the particles. Denoting the distribution function of the particles as $N(E)$ (i.e. the number of particles at energy $E$ per unit volume per solid angle), the spectrum resulting from an ensemble of particles is:

$$ \epsilon(E) dE = N(E) P(E) dE $$

LB:IC: Clarify what is $P(E)$. How does the spectrum come about?

The usual assumption made about the distribution $N(E)$ (based also on the observed cosmic ray distribution) is that of a power law, i.e.

$$N(E)dE=E^{-\alpha}dE $$

Plugging in this and remembering that $P(E) \propto \gamma^{2} \propto E^{2}$, we get

$$ \epsilon(E) dE \propto E^{2-\alpha} dE $$

Shifting to the frequency domain

$$\epsilon(\nu) \propto \nu^{(1-\alpha)/2} $$

The usual value for $\alpha$ is 5/2 and since flux $S_{\nu} \propto \epsilon_{\nu}$

$$S_{\nu} \propto \nu^{-0.75} $$

This shows that the synchrotron flux is also a power law, if the underlying distribution of particles is a power law.

LB:IC: The term spectral index is used below without being properly introduced. Introduce the notion of a spectral index here.

This is approximately valid for 'fresh' collection of radiating particles. However, as mentioned above, the higher energy particles lose energy through radiation much faster than the lower energy particles. This means that the distribution of particles over time gets steeper at higher frequencies (which is where the contribution from the high energy particles comes in). As we will see below, this steepening of the spectral index is a typical feature of older plasma in astrophysical scenarios.

1.6.2 Sources of Synchrotron Emission:

So where do we actually see synchrotron emission? As mentioned above, the prerequisites are magnetic fields and relativistic particles. These conditions are satisfied in a variety of situations. Prime examples are the lobes of radio galaxies. The lobes contain relativistic plasma in magnetic fields of strength ~ $\mu$G. It is believed that these plasmas and magnetic fields ultimately originate from the activity in the center of radio galaxies where a supermassive black hole resides. The figure below shows a radio image of the radio galaxy nearest to us, Cygnus A.

Figure 1.6.2 Cygnus A: Example of Synchroton Emission

The jets, which carry relativistic charged particles or plasma originating from the centre of the host galaxy (marked as 'core' in the figure), collide with the surrounding medium at the places labelled as "hotspots" in the figure. The plasma responsible for the radio emission (the lobes) tends to stream backward from the hotspots. As a result we can expect the youngest plasma to reside in and around the hotspots. On the other hand, we can expect the plasma closest to the core to be the oldest. But is there a way to verify this?

Well, the non-thermal nature of the emission can be verified by measuring the spectrum of the radio emission. A value close to -0.7 suggests, by the reasoning given above, that the radiation results from a synchroton emission mechanism. The plots below show the spectrum of the lobes of Cygnus A within a frequency range of 150 MHz to 14.65 GHz. LB:RF: Add proper citation.

Figure 1.6.3 Lobe flux vs. hotspot flux

LB:IC: Give a descriptive caption for this figure. Also maybe overlay a power law plot with spectral index of -0.7.

Since these curves are consistent with the expected power law behaviour, we should expect that the radiation is emitted by the synchroton mechanism rather than by some thermal mechanism. Note that the spectral index measured in the lobes is steeper than that from the hotspot. Now, since higher energy particles are expected to lose energy faster than low energy particles, fewer of them make it to the region of the lobes closest to the core. As a result there is less flux emitted at high frequencies in these regions. This is consistent with the idea that the youngest plasma resides at the hotspots.

As mentioned before, Synchrotron emission is an example of non-thermal continuum emission (i.e. emission occurring over a wide range of frequencies). In the next section we'll see an example of spectral line emission (i.e. emission which is confined to a very narrow range in frequencies). We will look at the particular example of the famous 21 cm line given off by neutral hydrogen.

1.6.3 Inverse Compton:

Another important mechanism by which astronomical objects emit at radio frequencies is inverse Compton scattering. In Compton scattering a high energy photon loses energy through inelastic scattering by a non-relativistic charged particle (mostly electrons). Conversely, inverse Compton scattering is the process through which a low energy photon gets up scattered to higher energies (and consequently, higher frequencies) by colliding with a relativistic particle (again usually an electron). This process is important to understand several astronomical scenarios.

Once again we will sacrifice rigour for simplicity in order to try and convey the important underlying physics. We start with a situation, analogous to the derivation of Bremsstrahlung radiation, in which a photon is interacting with an electron.

The 'acceleration' of the electron in this case is:

$$\mathbf{a} = \frac{e}{m_{e}}\mathbf{E'} $$

with $\mathbf{E'}$ being the electric field of the incident photon (Note that this is in the rest frame of the electron).

LB:IC: Put this way it sounds like the electric field is produced by the photon which is absurd. This needs to be re-written.

Then the Larmor formula gives the total radiated power as:

$$P = \frac{2\,e^{4}}{3m_{e}^{2}c^{3}} \mathbf{E'}^{2}$$

In the lab frame, this turns out to be:

$$P=2\frac{8\pi}{3}\frac{e^{4}}{m_{e0}^{2}c^{4}}\gamma^{2}(\mathbf{E}+\frac{\mathbf{v}}{\mathbf{c}}\times\,\mathbf{B})^{2}\frac{c}{8\pi}$$

where we apply the Lorentz transform for the electric field. The expression for the radiated power can be put in a more elegant form:

$$P = \sigma_{T}\gamma^{2}c\,u_{rad}$$

where $u_{rad}$ is the energy density of the photon and $\sigma_{T}$ is the Thomson cross section of the electron. Assuming that this energy (i.e. the energy lost by the electron) is the same as the energy gained by the photon during the scattering process, we have:

$$\frac{\Delta E}{E} = (\gamma^{2}-1)\sim \frac{v^{2}}{c^{2}}$$

This process takes place with a low energy photon and high energy plasma. One of the main examples is photons from the cosmic microwave background interacting with hot intra-cluster gas (i.e. the medium between galaxies within a galaxy cluster). As we learned from $\S$ 1.5 ⤵, the CMB has a temperature $T = 2.725$ K. Hot intra-cluster gas, on the other hand, has a temperature of $\sim 10^7 - 10^8$ K. The photons from the CMB interact with this medium resulting in in increase of the overall energy of the photons. This effectively marks a change in the CMB temperature in the direction of the cluster and is known as the Sunyaev-Zeldovich effect (see here for example). The change in the temperature can be written as:

$$\frac{\Delta T}{T} = \frac{2kT}{m_{e}c^{2}}\sigma_{T}\,N_{e}L$$

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