Light curves of transients (like supernovae and kilonovae) are a powerful tool for understanding the underlying astrophysics of eruptions, explosions and collisions of stars. With a few simplifying assumptions, we can create analytical models of these events which are in good agreement with observations.

First Law of Thermodynamics

To model transient light curves, we just need to use the first law of thermodynamics:

\[\Delta \mathsf{Energy} = \mathsf{Heat\,Added} - \mathsf{Work\,Done}\]

or put another way:

\[\frac{\Delta \mathsf{Energy}}{\Delta \mathsf{Time}} = \frac{\Delta \mathsf{Heat}}{\Delta \mathsf{Time}} - \mathsf{Pressure}\frac{\Delta \mathsf{Volume}}{\Delta \mathsf{Time}} - \mathsf{Luminosity}\]

The luminosity on the right-hand side of the equation can be approximated as the change in energy over a characteristic timescale of the system called the diffusion time.

For transients, the left-hand side of the equation is powered by an internal heating source: radioactive decay, recombination, shock-heating, etc. Then, we assume that all of this energy goes into the expansion of material or electromagnetic radiation.

If you squint your eyes, this equation looks very similar to a low-pass filter. The input energy (the left-hand side) will be stetched out over a diffusion time.

When we solve the equation for luminosity as a function of time, we will calculate the bolometric light curve of the transient. Let’s go into more detail about each of the pieces of this equation.

Input Luminosities

First, let’s look at a specific example of input luminosity. Supernovae are largely powered by the radioactive decay of nickel and cobalt. The input luminosity from this decay is an exponential:

\[L_\mathsf{inp}=M_\mathsf{Ni} (\epsilon_\mathsf{Ni} e^{-t/\tau_\mathsf{Ni}}+\epsilon_\mathsf{Co} e^{-t/\tau_\mathsf{Co}})\]

In this equation, \(M_\mathsf{Ni}\) is the nickel mass; \(\epsilon_\mathsf{Ni}\) and \(\epsilon_\mathsf{Co}\) are the energy generation rates of nickel and cobalt; and \(\tau_\mathsf{Ni}\) and \(\tau_\mathsf{Co}\) are the decay rates. The only free parameter that we can fit in this equation is the nickel mass, which determines the overall normalization of the light curve.

Like I said, there are other common heating sources which power astronomical transients. All of them generally take the form of power laws or exponentials. You can find a few more (interactive!) examples at the end of this post.

Diffusion Process

The input luminosity gets smoothed out over the diffusion timescale:

\[\tau_{diff}=\sqrt{\frac{2\kappa M}{\beta c v}}\]

In this equation, \(\kappa\) is the opacity of the ejecta material, \(M\) is the ejecta mass, \(v\) is the ejecta velocity and \(\beta\) is a geometric term. The diffusion timescale plays a huge role in the overall width of the light curve.

What About Colors?

Everything we’ve talked about so far deals with the bolometric luminosity. In reality, we only observe a fraction of the light through a filter.

There are two ways we counteract this issue. First, we can assume some bolometric correction in order to construct the bolometric light curves from filtered data. Our team uses an alternative approach of assuming a spectral energy distribution (SED) and directly fitting the filted light curves using MOSFiT. We typically assume a simple blackbody SED.

Interactive Examples

The best way to learn more about these light curves is to play with them yourself!

Radioactive Decay Model

We discussed this model previously in the post. The free parameters of this model are the ejecta mass, ejecta velocity, opacity, nickel fraction and gamma-ray opacity.

Radioactive decay also powers kilonovae in a similar fashion. However, in the case of a kilonova, many species of elements decay simultaneous, resulting in a power law-like input luminosity function.

Magnetar Spin-down

Some bright SNe may be powered by the spin-down a millisecond pulsar with a strong magnetic field (aka a magnetar). The free parameters of this model are the ejecta mass, the ejecta velocity, the opacity, the gamma-ray opacity, the magnetic field of the magnetar and the initial spin of the magnetar. See for yourself how the magnetic field and spin can be optimized to create an extremely bright transient.

Shock Interaction with a Circumstellar Medium

The cooling of shocks resulting from the interaction of the supernova ejecta and surrounding circumstellar material (CSM). There are a ton of free parameters in this model, but some main ones are: the ejecta mass, the ejecta velocity, the shape of the supernova ejecta, the density of the CSM, and the radius of the CSM.

Resources and Thanks!

If you want to learn more details about this model, feel free to read my recent paper which goes more into the mathematics. If you found this blog post useful, you can also cite that paper and pretend you read it 0:). An older paper by E. Chatzopoulos is also really useful and did the first census (that I know of!) of many of these anlaytical models. This very early paper by Arnett is where most of the math comes from and is a very useful resource. Quick thanks to D. Kasen for writing these useful notes and to J. Guillochon and B. Metzger for teaching me all ‘bout light curve models.