If you are reading this post, I must assume you found it to do a background reading for the Summer school.
If you did not find this for the summer school, no probs! Just go ahead, and try to have a good read.
First off, the Summer school has been partitioned in 4 ways: (i). Introduction to General Relativity, Relativistic effects in Astronomy- almost purely qualitative,(ii). Electromagnetism, electromagnetic radiation in Astronomy, (iii). Basics of Astronomy- Optics, filters, and observations, (iv). Machine learning and Deep learning in Astronomy.
Please note I will not put a lot of math, but instead refer you to corresponding books to do the same. Keeping in mind the session is for First year undergrads, I am trying to do some Mathematica coding for the same (if I am not lazy, that is ).
Finally, don’t hesitate to post questions if any on the comments section!
Other lectures in this series:
If there is anyone interested to have a discussion on any of these topics, you could comment here, of just search for the title on Quora, wherein I have uploaded all of this stuff under Abstracted Abstract Science.
Electromagnetism, a phenomenon which you can use to look into the beyond!
Fundamentally, I have always had nuances with the way the subject has been taught at an introductory undergraduate level. We were always given an overview of the wave theory, and special relativity was never touched. However, once we start dwelling into astronomy, we find these have some major implications on our observation.
But here is the thing: Radiation is not just limited to electromagnetic radiation, but has many different kinds of production mechanisms, which play a major role in Astronomy.
I shall go ahead with the theory and assume you have read, and understood the basics of Relativity. If you haven’t gone through it yet, you can find the link in the description above.
Let’s begin then!
There are many different mechanisms of energy production in Astrophysical bodies, and it is a part of this produced energy that we receive which enables us to hypothesize what goes around in the universe. The predominant mechanisms of energy production are:
- Electromagnetic radiation
- Thermal radiation/Black body radiation
- Atomic/Molecular spectra
- Gravitational radiation.
We shall touch upon EM and Thermal radiation qualitatively, and have a short discussion on Gravitational radiation, but we shall not discuss Atomic spectra, as I am not really qualified to discuss the nuances in the subject. However, I will provide references for the same.
Electromagnetism is one phenomenon which plays role in varying orders of magnitude, due to its awesome permittivity and permeability constants. Together, these constants give rise to the Speed of light in a given medium. But how do we reach radiation from statics?
For starters, we have always seen Static charges and Electric fields, Moving currents and Magnetic fields, and we know Accelerated charged particles lose energy, but we never talk about Moving currents and electric fields. The reason lies in the complex mathematics that goes into deriving the Fields (or potentials).
We know the equations which define Electromagnetism- Maxwell’s equations. Here are they:
But we can do better. We know EM has a Scalar potential, and a Vector potential, and we can express the above equations w.r.t to the potentials. Doing so gets us to:
But, we now assert something called the Lorenz gauge, which basically constraints the two potentials, and we get the reduced form of Maxwell’s equations as:
If one sees the form of the given equations, they look like Wave-equation with a source – either current density or charge density. And here is where one gets the Electromagnetic wave into the picture.
There are many ways of solving these equations, but one way people go about is to think there is a Point or Delta source, and solve the equations accordingly. Upon doing some tedious math, one obtains a form for the potentials called Lienard-Weichert potential or Retarded Potentials. The bottom line in this potential form is this: there are two time-scales of variations involved- one, which relates to velocities of the source charges in the distribution, and one which relates to the time-taken for the waves to propagate to a given point.
So basically, all I am saying is, if we are to just change the charge/current configuration a bit, there is a time duration in which the change of configuration is taking place, and it takes some time for the change to be seen at some far-away point, and this time corresponding to the delay in change is the other time-scale. These two scales are almost the same if the distances involved in both the cases are comparable, and if there were no acceleration of these charged particles.
Okay, enough of lecture, let’s look at the form of potentials for the far-zone approximation:
We can derive the EM fields from these expressions, and we get a nasty set of equations out:
This condensed equation contains two terms: One term, which depends on , and one term, which depends on . If we are to look at the total flux of sorts of the fields through a sphere, we will find the flux to have a dependence of for the former case, and a Constant flux for the later case [In case you are wondering, the flux is just ]. Hence, at a sphere at infinity, we have a constant flux coming from the latter term, while the first term does not radiate. This is the cause of radiation due to moving charges.
Why acceleration, though? In Fig 5, you can see the E-field depends on , which is acceleration. Hence, the radiation component depends on the acceleration of the source particle!
How do these fields behave like? The field lines go like this:
A more extreme case of this kind of radiation is called the Synchrotron radiation, in which the charged particle is at highly relativistic velocities. But here, the source of radiation is acceleration of a relativistically moving charged particle due to a magnetic field. This kind of radiation was first seen in the LHC, where flashes of light were seen from charged particles. The radiation pattern looks like this:
Now, this effect has been seen the Crab Nebula, and was a cause of discomfort to the astronomers for a long time. How can an object be so bright, and yet not diminish in its brightness after such a long time? The answer as synchrotron was thought of by one engineer at LHC, and has long since been proved to be primarily synchrotron radiation.
The whole of such radiation has been explained in some great online repositories, and are given in the references.
Now, EM is not the only kind of radiation; we also have something called the Blackbody radiation. Any ideal blackbody- a body which can emit energy and absorb energy with 100% efficiency- emits energy which follows a certain kind of spectrum, known as the blackbody spectrum [Please note this: the source of energy is thermal energy, has a certain spectrum, etc. But the information is carried to us through photons, which is nothing but oscillating electric and magnetic fields]. The spectrum looks like this:
This radiation has its source in the thermal energy of the body, and the energy emitted depends on the Temperature of the body. However, the power spectrum peaks at a particular wavelength, given by the Wien’s displacement law:
Every spectrum of an astrophysical is a combination of the Blackbody spectrum [if a star], along with Synchrotron and line emissions. There is another phenomenon called the Bremsstrahlung, which basically translated to Breaking radiation. Basically, any sudden deflection in the path of a particle causes it to release radiation [which again has special charachteristics]. This phenomenon is used in X-ray machines in medical diagnosis, wherein electrons are deflected due to Fe nuclei, causing radiation. The production mechanism is roughly like this:
The phenomenon is explained in detail in , and I shall refrain from explaining the same here.
But how do we differentiate between the above kinds of radiations? We simply look at the power spectrum, and see ‘how-much’ of the spectral contribution is from each of these processes. For example, the thermal and synchrotron radiations have distinct power spectrum:
We can use such information, and here is an example of the spectrum of Crab nebula:
We can see the synchrotron radiation fits the first peak very well, and the high-frequency stuff is basically from Cosmic Microwave background radiation, Synchrotron self-compton, etc.
Couple this with peaks and troughs due to emission and absorption lines from Nebulae, etc, one gets horrible looking curves like these:
Generally, when a body emits radiation, the E-field components may be resolved as Ex or Ey, along the two principal axes. Now, these two components may or may not have any relation to each other- or in other words, Ex and Ey could either have a phase relationship, or may randomly vary without caring about each other.
If Ex and Ey have a phase relationship, the resulting Electric field vector will oscillate over time, tracing an Ellipse, called the Polarization Ellipse. In such a case, we charachterize the Electric field with a set of parameters called the Stoke’s parameters, which are defined as:
These parameters have different values depending on the kind of polarization, which are summarized below:
From the above figures, we can see how different polarizations reflect onto the Stokes parameters.
But wait! Why are we even talking about this? Do we even measure the Stoke’s parameters?
Well, infact we do have something called the Polarization grating, which gives out Polarized light, and enables us to measure the Stoke’s parameters. These help us talk about the physics that goes in the astrophysical system. For example:
- Synchrotron radiation is polarized.
- Thermal radiation is randomly polarized.
With these Stokes parameters we can also determine how much of the light received is Polarized, and can potentially separate components of various processes in the background. For instance, if one were to consider Dipole antennae, each configuration would capture one particular polarization; so a set of different antennae could capture both the polarizations!
We have almost forgotten something: Gravitational Radiation.
This is the kind of radiation which is emitted due to rapid perturbation of spacetime continuum. For instance, if two Black-holes decide to start doing a Tango, they will start losing mass as they start coalescing into each-other. This lost mass is given out as stretches in space-time, and comes out as Gravitational waves.
The mechanism of GW, in a mathematical sense, could be called as a Tensor Perturbation to the Einstein equations. Google up the entire Jargon of a line, and try to not lose your head. However, this GW was what was detected by LIGO, as small changes in their setup.
Now, GW also have polarization. They have “+” mode and “x” mode polarizations, and they are always polarized. No question of a ‘randomly polarised’ wave:
The detection is done by having a set of Michelson interferometers, which basically measure the changes in path-length between two rays of light in the setup. Now, as the GW passes through the setup, the distortion shows up as a change in path length of the light rays, and the signal is known as a Chirp signal, and looks like:
And that’s Gravitational waves for you!
So, that’s a very very brief introduction to Radiative processes in Astrophysics. I do not claim to be an expert, but there are excellent books out there discussing this topic in great detail, covering the mathematics, which I have shamelessly skipped. However, I am appending a list of good references used, which most of you will find useful. And so, let me conclude with this Summary Image:
- Radiative processes in astrophysics, Rybicki and Lightman
Extra Good reads:
- Classical Electromagnetism: John David Jackson. Also look out for Hitler’s reaction to Jackson memes 😛
- Radiative processes in astrophysics by Gabriele Ghisellini
- Radiative processes by Frank H Shu.
- Tools of Radio Astronomy by Wilson, Rohlfs and Huttemsteir