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Notice that the factor in the critical frequency makes synchrotron radiation “harder” than cyclotron radiation. In a cyclotron, the power radiated into all solid angles is given by the Larmor Forumla:
Let’s derive this for the synchrotron. In the electron frame:
It turns out that power is a relativistic invariant. To see this, note , and we know the following:
and in the prime frame (), .
So we’ve shown that once we calculate , we know in all frames. What we need to do now is calculate . We’ll do the following:
- Get , the Lorentz transform of acceleration.
- Get due to .
Since is at rest in the primed frame:
Therefore, the magnitude of the acceleration is:
(See below for derivations up to this point.)
And so the power radiated is:
Note that as ,
Thus we get way more power () out of the synchrotron. How long can an hold up radiating this kind of power?
The time it takes an to go in the circle is just:
Taking the ratio of these, we find that the critical required to make these timescales comparable is:
Getting back to P, there is a prettier way of writing it:
where is the Thomson cross-section ( being defined by ), and is the magnetic field Energy Density .
Synchrotron Cooling Time
To estimate the synchrotron cooling time, we’ll set up our standard expression of self-energy over power radiated:
Instead of doing anything fancy with , we’ll just use that . we can estimate as , giving us:
- Let’s examine the cooling time for radio jets, where , and . Plugging this in, we get . Compare this to .
- We’ll also estimate the cooling time of the Crab Nebula. To set an upper bound on , we’ll use the most energetic X-rays. We can do this because uniquely determines the electron energy and as a result, it uniquely determines a photon energy. For the Crab Nebula, we’ll pick a photon energy: . Then:
mG gives us that . Clearly, since the Crab Nebula was created some 1000 years ago, there must be a source of fresh electrons. This source is the pulsar, sitting in the middle of the nebula.
Single Electron Power Distribution
For a non-relativistic cyclotron emission, an observer will see a power spectrum with time that oscillates sinusoidally with a single characteristic frequency. To obtain the power spectrum with frequency, P(), the Fourier Transform is taken of the sine wave which gives a delta function at the characteristic frequency, .
For synchrotron radiation, however, the radiation will be emitted in a narrow beam of angular width , so P(t) will be a series of sharp peaks. Taking the Fourier transform of this distribution is not so straightforward. Jumping to the result, the spectrum for a single electron can be found:
Where is the critical photon frequency found for synchrotron radiation and is a modified Bessel function. In general this function is hard to work with, however it has some very nice properties. Note in the plot how:
for small and
for large . Furthermore, it is highly peaked around .3 , which allows us to approximate the power as being radiated at that single frequency when doing further calculations.
To find the total power radiated we can then integrate with respect to , giving
which, making the substitution , gives
The integral over F is just some constant factor, so we see that the important scalings are P , which is what we found earlier using the Larmor Formula.
Spectra of Synchrotron Radiation
The power spectrum of a single undergoing synchrotron radiation peaks at . For small , goes as , and for , goes as . In general, recall that .
We would like to calculate the power spectrum of an ensemble of . To do this, we need to describe how many electrons there are per energy. We’ll assume a power law distribution of energies, i.e.
where is the differential energy spectrum index. We make this assumption simply because this coincides with our observations (see Nilsen and Zager). Let’s consider the power radiated by electrons with energies between and :
Thus we have:
Now, we want to relate this electron energy, E, to a photon frequency . For cyclotron emission this was straightforward, since an electron traveling with a given energy would only radiate at one frequency, a one-to-one correspondence. However, as we found in the last section, synchrotron electrons with a specific energy radiate at a whole continuum of frequencies, represented by the function F(). Luckily, we saw that this function is sharply peaked around , so to reasonable approximation we can use
Which means that we have recovered the one-to-one relationship between and . That is, we can say that electrons of a specific energy are responsible for generating photons of frequency . Note also that , so .
Usually, . Note that our assumption of one-to-one correspondence was not necessary to get this power law dependence on . If hadn’t made that assumption, we would have found that the sum of the contributions of electrons in nearby energies would have yielded the same result we got, except near the edges of . Also note that this expression relies on an influx of to replace old ones which cooled down. If we cut off this influx, we’ll see that since , the emitting higher photons decay first, and so we see turn-offs from a power law distribution for increasingly low as time goes by.
For measuring , let’s define . Observations of extended radio sources have measured . In general, we find that , or . We’ve made an assumption of a constant magnetic field. Suppose we have an optically thin synchrotron emitting gas with a power law emissivity for (where we can define energies and that directly correspond to and ). We will use observations through this thin gas to infer a “minimum pressure” or “minimum energy density”. The electron pressure (or electric energy density) is given by:
In reality, spectra might cut off outside of the range of our observations. To correct for this, we’ll carry around a correction factor of . Measuring , we get:
where is the magnetic energy density. Now since :
This expression has a minimum for a unique . This tells us the way energy is partitioned in the system between and . The minimum should occur when .
Recall that last time we derived for an optically thin synchrotron gas that:
Thus, the minimum total power occurs near the equipartition point between and . This gives us:
Supernova Remnants Effective Source Sizes
Imagine that we had a continuous, power law source of highly relativistic electrons that are subsequently carried away by a non-relativistic magnetized flow at speed v. An example of this situation would be the shock of a supernova remnant. Typical speeds are of order v 1250 . We want to see if the finite cooling time for synchrotron radiation will effect what we observe.
The distance such an electron will travel will before cooling will be simply:
where 40 G is a more appropriate magnetic field for the post shock environment in a supernova remnant. We can convert the electron energy, to a photon energy using the critical frequency found in past lectures:
Which is smaller than the typical supernova remnant radius around keV energies. As a result, when viewing supernova remnants at higher energies ( keV and above), we see synchrotron emission concentrated around very thin nonthermal filaments that should decrease in size with energy. However, electrons also have the ability to diffuse through media, which is characterized by a diffusion coefficient, with dimensions of [length][velocity] and can be written as :
For very relativistic electrons, and is taken as some constant multiple of the gyroradius , so that
Quick dimensional analysis tells us the the characteristic length scale of diffusion is
which is independent of the emitting frequency and comparable to at keV energies. At higher energies, then, diffusion will be the limiting factor for effective source size while for lower energies convection will be the limiting factor. Since both results have depend on the magnetic field as , we can use effective source sizes to infer local values of the magnetic field, a parameter that is otherwise hard to get at.
The cooling time also has an impact on the shape of the spectra. To calculate the total from our source, we would integrate the specific intensity produced by these electrons over the solid angle subtended by the source:
Where D is the distance to the source. The integration over A is in principle of the area of the entire source, but as we have shown above this is limited to a region of width or before the radiation will effectively shut off. For the case of a spherical shock and constant specific intensity, then, the integral becomes
for convection-limited electrons and
for diffusion-limited electrons.
Thus, the two transport mechanism have qualitatively different effects on the shape of the spectrum. In one case there is an extra steeping by a power of while for the other the spectrum remains unchanged. We can use these differences in many situations to infer properties of diffusion and convection in the plasma. This effect is also important in other astronomical objects, such as relativistic jets and pulsar wind nebulae, but these will have different geometries which will change the degree of steepening when integrating over the area of the source.
Special Relativity Derivations
As an exercise in special relativity, we’ll derive what is by investigating two parallel plates–the bottom one having charge density and the top having We know in a motionless frame that the field between the plates is:
In a frame () where we are moving at , these become:
Jumping into one more frame () where relative to frame , we have:
We need to figure , and our first instinct might be to say , but that is wrong. We have to reference it from :
Now we do some algebra:
Thus we have:
And, of course, . The only thing we need to get now is . For this we’ll talk about a solenoid aligned with the direction with turns per unit length. The field of this solenoid in the rest frame is:
Jumping to a frame where , . Using that:
We find that:
Recall that we’ve derived all of this for boosts in the direction. To be completely general, we’ll write them for any direction: