Type Ia supernovae as standard candles

A supernova is an explosive stellar event that becomes much more luminous in a short time. As its luminosity rises, its absolute magnitude becomes more negative: on the magnitude scale, smaller values mean greater brightness.

A type Ia supernova is a thermonuclear explosion of a white dwarf. Its brightness rises rapidly to a peak, typically over a few weeks, and then declines more slowly as the supernova fades. A graph showing how its brightness changes with time is called a light curve. If the graph uses magnitude, peak brightness appears at the minimum magnitude.

Calibrating the peak brightness

A standard candle is an object whose luminosity is known, allowing its distance to be found from how bright it appears. Type Ia supernovae are useful standard candles, but their peak luminosities are not exactly identical.

Their peak absolute magnitudes can be calibrated using the behaviour of their light curves. In particular, a supernova that fades more slowly after its peak is generally more luminous at peak than one that fades more rapidly. Measuring the light-curve behaviour therefore helps determine the appropriate peak absolute magnitude rather than assuming the same value for every event.

Distance from a type Ia supernova

The apparent magnitude $$m$$ describes how bright the supernova looks from Earth. Its absolute magnitude $$M$$ is the apparent magnitude it would have at a distance of 10 parsecs. Comparing the observed peak apparent magnitude with the calibrated peak absolute magnitude gives the distance modulus, $$m-M$$:

Here $$d$$ is the distance to the supernova. The equation can be rearranged to find it:

For a given calibrated peak absolute magnitude, a greater distance makes the supernova appear fainter and gives a larger distance modulus. The calibration is therefore essential: an incorrect estimate of the peak absolute magnitude would give an incorrect distance.

Neutron stars

A neutron star is the compact remnant left when the core of a massive star collapses. It is made predominantly of neutrons, the particles also found in atomic nuclei.

A neutron star has a mass comparable to that of a star, but it is only about tens of kilometres across. Because so much mass is concentrated in such a small volume, its matter is extraordinarily dense.

Black holes and the Schwarzschild radius

A black hole is a region bounded by an event horizon. Light from within the event horizon cannot escape to a distant observer.

One way to picture this boundary is through an escape-velocity model. Escape velocity is the speed needed to escape a body's gravitational pull. In this model, it equals the speed of light, $$c$$, at the event horizon and exceeds $$c$$ inside it. The model gives a useful picture of why light cannot escape, although the full description of a black hole requires general relativity.

Schwarzschild radius

For a non-rotating black hole, the radius of the event horizon is called the Schwarzschild radius, $$R_s$$:

Here $$G$$ is the gravitational constant, $$M$$ is the black hole's mass and $$c$$ is the speed of light in a vacuum. The radius is proportional to mass: doubling the mass doubles the Schwarzschild radius.

To calculate the radius, use consistent units. For example, for a black hole of mass $$2.0 \times 10^{30}\ \mathrm{kg}$$, using $$G = 6.67 \times 10^{-11}\ \mathrm{m^3,kg^{-1},s^{-2}}$$ and $$c = 3.00 \times 10^8\ \mathrm{m,s^{-1}}$$:

The event-horizon radius is therefore approximately $$3.0\ \mathrm{km}$$.

Supermassive black holes are found at the centres of galaxies.

Gamma-ray bursts from stellar collapse

A gamma-ray burst is a brief, intense emission of gamma radiation. Some bursts occur when the core of a supergiant star collapses, forming a neutron star or a black hole. The collapse releases energy, some of which is emitted as gamma radiation over a short time.

The energy inferred for a powerful burst can be comparable to the total energy the Sun emits over its entire lifetime. The contrast is in the timescale: a burst can release a very large amount of energy in a short interval.

Gamma-ray bursts: estimating emitted energy

An energy estimate depends on how the radiation is distributed. From the radiation detected at Earth, an estimate can be made by assuming that the burst emitted equally in every direction. This gives an isotropic energy estimate.

However, if the radiation is concentrated into a narrow jet directed towards Earth, it is not spread equally in all directions. Extending the detected intensity to every direction would then overstate the total energy emitted. A comparison with the Sun’s lifetime output must therefore be understood in terms of the distribution assumed for the burst.

Cosmic acceleration and dark energy

The expansion of the Universe has not necessarily proceeded at the same rate throughout its history. Observations of distant type Ia supernovae provide evidence that the expansion has accelerated: at later times, the distance between widely separated regions of the Universe has grown at an increasing rate.

Supernova distances and redshifts

A type Ia supernova can be used to estimate distance because its intrinsic brightness can be estimated from its observed behaviour. Comparing that estimate with how bright the supernova appears from Earth gives its distance: the fainter it appears for a given intrinsic brightness, the farther away it is inferred to be.

The supernova’s redshift shows how much its light has been stretched by the expansion of the Universe since the light was emitted. Its distance and redshift can therefore be compared with the predictions of a cosmological model—a description of how the Universe expands over time.

At the same redshift, distant type Ia supernovae appear fainter, and are inferred to be farther away, than expected in a comparable model without accelerating expansion. Their distances and redshifts are better matched by a model in which expansion has accelerated. This is evidence for acceleration, rather than just evidence that the Universe is expanding.

Dark energy and uncertainty

Dark energy is a proposed explanation for the observed acceleration. Its physical nature is not established, so the evidence for accelerating expansion does not by itself identify what dark energy is.

The inference depends on the cosmological models being compared and on reliable supernova observations. In particular, estimates must account for uncertainties in supernova brightness and its calibration, as well as effects such as dimming by dust. If these are not properly controlled, an apparent difference in distance could be mistaken for evidence of a different expansion history.