All right. Let's get onto the continuum radiation. Most of that is thermal radiation, thermal continuum of blackbody versus traditionally plot of what spectrum blackbodies look like, and that refers to a radiation field that is a thermal equilibrium with whatever matter is. It's like a completely hypothetical object except for the whole universe which is an excellent blackbody. And laboratory equivalent to it will be enclosed cavity that's kept on a fixed temperature in this tiny little hole that doesn't disturb equilibrium too much and radiation from it will have the spectrum. This sounds a little idealized, but in reality, actually, a lot of things look like this. And stars have spectra are not perfect bodies but pretty damn close. And certainly, inside stars which are optically thick, the blackbody radiation applies. So quantity of the description of this was invented by Max Planck, which was the other step towards quantum mechanics. And he purely empirically came up with the formula that describes the spectrum of blackbodies. Now, here is the set for temperatures differing by an order of magnitude and you can notice a couple things. This is a log-log plot, so strait line is a power line. And at the low frequency end, you'll see that the straight power lines have slope of minus two. That's called the particle spectrum. Now, there is a peak and there is a cutoff, which is exponential. And, in fact, that's what Planck incorporated into his formula. He introduced quantization of energy during this constant that bears his name. And nobody understood why this formula worked until it was explained by Einstein and others. So few things about Blackbody Spectrum. First, the peak where maximum is shifts with temperature. The hotter bodies will have the peak at higher frequencies. It's kind of intuitively clear. And that's called the Wien's Displacement Law. There is a directly near proportion between the two. For energies that are much less than this peak energy of constant times temperature, then the Jeans law applies. It's roughly quadratic proportion and directly proportional to the temperature. So, this is often used in radiostronomy, or even the origin of the radio waves may not be thermal at all, from their intensity, you can ask what's the equivalent temperature to which this would correspond. It's called Brightness Temperature. Another important part about blackbody radiation is to compute its energy density or total luminosity of an object. It turns out that energy density inside a blackbody cavity or star field with hot gas is proportional to the fourth power of temperature. And if you let the flight go, then of course, the emergent flux will be then also proportional to the fourth power of temperature. And the constant of proportionality's called Stefan-Boltzmann Constant. And it's called Stefan-Boltzmann formula. So luminosity for a given blackbody will be proportional to the fourth power of temperature, but also the bigger surface area, more light will gap, and so it has to be proportional to the square of the radius. And so luminosity is to really good approximation given by formula like this. Because of the square of the radius fourth power temperature, aside from four pi, that's spherical geometry, the Stefan-Boltzmann constant which has this value. Now, you can measure luminosity of anything, regardless of whether it's powered by thermal radiation or not. And then you can just ask, if this were a blackbody, what would be its temperature to give me this much luminosity? And so, regardless of what actual shape of the spectrum is, we define the effective temperature as what would be the temperature of the blackbody that emits the same kind of luminosity. And for sun, this is close to 6,000 degrees Kelvin. And the peak of that is in yellowish parts of the spectrum. And our eyes are the most sensitive during the daytime to those frequencies. Isn't that amazing how evolution works? So if we lived in some planet around the red dwarf star, we'd probably be having eyes that are sensitive to near infrared. Well, blackbodies do happen, actually. And the very best one known, better than any laboratory measurement is the cosmic microwave background, which is thermal relic of the Big Bang. You can think of the whole universe as a cavity which is internal equilibrium, and the temperature of that radiation is 2.73 degrees Kelvin. And there are no known deviations from it. It's really remarkable. Okay. There are other kinds of continuum mechanisms. The one that's really important in storm is called Synchrotron Emission. You probably know from ENM that if an electron holds a magnetic field that's, at some angle, relative to its specter of motion, there'll be a Lorentz force. That means that the electron will be accelerated. If it's accelerated, that means it's going to radiate, and exactly how depends on the speed. At low speeds, it will just keep going in circles. That's called a Cycle of Termination. As you increase the speed, approach the speed of light, it starts moving like a helix like this. And it shines light in a cone which gets tighter and tighter the closer you are to the speed of light. And that's called Synchrotron Emission. Essentially, it's like you have cosmic accelerator. Now, spectra synchrotron emission tends to be power law for the most part. There is cut off at both timelines. But power law is very different from blackbody spectrum. And so, here on log-log plot of spectral emissivity versus frequency, you can see a typical parallel which would be slope of minus one is relative to the blackbody spectra. So if you kinda match them at the peak of blackbody, then there is excess both at low frequencies and high frequencies. And this is how we find quasars by and large because their spectra not thermal. They are often close to the power spectrum. So stars don't have spectra like this, but strange things like active galactic nuclei do. This is what they look like. These are images. The top image is in radio waves at 5GHz with VLA. This is a famous radio galaxy, Cygnus A. You see there is a little core, which is the central engine, and there are these jets of the electrons that are being accelerated close to the speed of light. Then, plasma dissipates and makes this big Blobs of plasma that shine in synchrotron light. So that's what sources look like in radio. The bottom one is the jet of M87, giant elliptical galaxy in Virgo Cluster, and this visible light picture. Synchrotron emission is not confined to radio. It can be x-rays, or gamma rays, or visible light, as well as radio. Another favorite object is the crab nebula. You've probably seen the picture on the right. That's invisible light. It's this super nova remnant, filaments of exploded star. If you look in radio, you see something very similar, but this is not thermal emission. This is emission from electrons moving in a magnetic field that is confined to the supernova. The final source of continuum emission is also thermal of sorts, but not blackbody is so-called Thermal Bremsstrahlung, which is a German word that means breaking radiation. And for some reason, people still use the German word. It's simply plasma, there's electrons and positive ions. Electrons scattering off of positive ion will be accelerated. That means it's going to radiate. The higher the temperature, the more these things will happen, goes to the square of the density. So, it is essentially due to thermal motions inside this fully ionized plasma, but it's not the blackbody itself, but it's a continuum. And the typical places where we can see such things are say, clusters of galaxies which are filled with this hot gas and temperatures of millions or tens of millions of degrees Kelvin. So they shine in x-rays.