- Title: Analytical Expressions for Light-curves of Supernovae Type Ia
- Authors: Shlomo Dado & Arnon Dar
- Both Authors’ Institution: Physics Department, Technion, Haifa 32000, Israel
Big Picture (literally)
I have a violent story to tell you today! Star gets old, gets fat, and explodes. We see the explosion, know how fat the star was, and hence the energy that should be released, and can infer the distance to the star from the energy we actually measure. This in turn, repeated enough times, tells us about how quickly the Universe has been expanding in the past. Knowing the expansion history constrains the properties of Dark Energy (DE), an exotic substance that comprises 73% of the Universe (stuff we’re made of, by comparison, is a measly 4%!) That is, literally, the biggest-picture view of the importance of exploding stars. (See also this post on supernovae, and this post explaining how DE leads to accelerated expansion.)
Where does today’s paper fit in? The explosions I mention above are called Type Ia supernovae (SNIa), further detail to follow below, and we measure their light curves: the amount of light we receive from them over time. This amount as a function of time has a characteristic shape, shown below. Today’s paper develops a simple analytic model to predict this shape from basic physics. Certainly, SNIa have been known for a long time—at least since the ’80’s. So their shape is not news to anyone. The value of this paper is to provide a very simple physical model with only a few inputs and only a couple different mechanisms, and to get out of it a shape that matches observations pretty well, as the picture shows. So this is the paper for anyone who has ever seen a light curve and said, “Gee, why does it have that funny shape?”
Type Ia supernovae are violent explosions that occur when the gravity in an old star (specifically, a type known as a “white dwarf“) overwhelms the pressure even densely packed, degenerate matter can provide and the star explodes.
Why is it important that the star is old? The very oldest stars are no longer doing nuclear fusion, so they do not have heat providing a source of pressure to support them. Thus, as mass is added to them, for instance from a neighboring star’s shedding, they simply contract and become more dense. Eventually, they become so dense that particles are actually close enough to each other that quantum mechanical effects become important.
Specifically, the “Pauli exclusion principle” says that two fermions cannot be in the same overall quantum state (fermions are just particles that have half-integer values of their intrinsic spin—imagine a sphere rotating with different speeds). Electrons are fermions, and the constraint that they cannot be in the same QM state means they must have some spatial separation between each other. This in turn means that, if they are compressed enough, they’ll push back, providing what is called “electron degeneracy pressure“. However, electron degeneracy pressure is not enough when the star reaches a mass of 1.4 times the mass of the Sun, and gravity wins (this mass upper bound is called the Chandrasekhar mass—he thought of it in 1931, way before any stars like this had been seen. But on the other hand, the Chinese observed a supernova in 1054 AD!) Everything falls inward, complicated physics ensues, and the star blows itself apart.
- A super-simple supernova synthesis
Try saying those sibilants sixteen times swiftly. Then continue reading. What do we need to predict the luminosity (amount of energy emitted per unit time)? Well, we need a total amount of energy the supernovae produces, and an amount of time it takes to do that. To get the total energy, the authors assume that the SNe is powered at first by nuclear fusion of carbon and oxygen into higher mass elements. This gives a total energy. Assuming all of this energy goes into driving a huge, expanding fireball (of uniform density at any given moment), they calculate the expansion speed of the fireball, which turns out to be constant. This is important because it means that the size of the fireball is just a constant speed times time, so the size (radius, to be precise) increases linearly with time. Why do we care?
Well, this is key for two reasons. First, the fireball loses energy as it expands, so the expansion rate tells us how much energy it loses per unit time from expansion. Second, the fireball also loses energy via photons inside it random-walking their way out (like a drunkard, they stagger in random directions inside the fireball, but they will eventually reach its edge). The time it takes them to do this is bigger if the distance they must cover is bigger—so the photon diffusion time will scale with 1/time since the explosion occurred.
These two ideas let the authors set up a simple equation relating the change in energy over time to the energy of the SNe, which can be solved to give the energy as a function of time—i.e. the luminosity. For those who care for the technical, the equation is
This may look scary, but let’s look at it step-by-step. It is just accounting. Just like the money you deposit in your savings account is just what you make minus what you spend, here, the total change in energy per unit time is just the energy produced minus per unit time minus the energy lost per unit time. E represents energy, t time.
The term on the left-hand side is just the total change in energy per unit time. On the right-hand side, the first term is just the energy generation rate: all of the energy is assumed to be produced by the radioactive decay of the elements produced earlier by nuclear fusion.
The second term, with brackets, on the right encodes 2 effects. The describes energy lost because the fireball is expanding—it grows linearly with time, so it loses energy like . The 1/ is the more complicated effect described above: photons random walk out of the fireball, taking energy with them as they do so.
The authors solve this equation for how the energy behaves with time, and that gives them the light-curve, shown in the picture. From this, everything we actually observe about SNIa can be predicted, providing a great way for us to understand why the light-curve has the shape it does!