# Luminosity

• Is the amount of electromagnetic energy a body radiates per unit of time. {J/s (W)}
• Is intrinsic to a body and is a measurable property which is independent of distance.

Imagine a point source of light of luminosity L that radiates equally in all directions. A hollow sphere centred on the point would have its entire interior surface illuminated. As the radius increases, the surface area will also increase, and the constant luminosity has more surface area to illuminate, leading to a decrease in observed brightness.

A is the area of the illuminated surface, a sphere of radius r.

F is the Flux of the illuminated surface, the energy radiated per second per square metre of surface from a point source such as a star. This means that Luminosity is the total flux in watts

So:

The sun has a luminosity L = 3.8 x 1026W. If Earth – sun distance is 150m km and the Sun can be considered a point source, we can show that the radiant energy flux at the surface of the Earth is about 1.3kW m-2

Hertsprung and Russell showed that that the luminosity of a star L (assuming the star is a black body – a perfect emitter and absorber – which is a good approximation) is also related to temperature T and radius R of the star by the equation:

where  ‘sigma’ is the Stefan-Boltzmann Constant  5.67 × 10−8 W·m-2·K-4

L is often quoted in terms of solar luminosities, or how many times as much energy the object radiates than the Sun, so LSun= 1

Questions:

1. Find the actual luminosity of the Sun, given a surface temperature of 6,000K and a radius of 7 x 108m
2. Compare with Sirius – a very bright star – temperature 12,000K and radius 2.22 x 109m. How much brighter is Sirius than the Sun – in solar luminosities?
3. What would be the surface temperature of a star having the same luminosity as the Sun but twice the radius. What would it look like?
• So a bigger star can be at a lower temperature and yet have the same luminosity, i.e. it looks just as bright
• A hotter star is more luminous than a cooler one of the same radius.
• A bigger star is more luminous than a smaller one of the same temperature.

A cool (red) giant star is more luminous than the Sun because, even though it is cooler, it is much larger than the Sun.

Finally, the idea of a Standard Candle is an important one in astrophysics. Certain classes of objects such as supernovae and Cepheid variable stars have properties whereby their luminosities can be determined separately from other measurements. For example, the period of a Cepheid variable star depends on it’s mean absolute magnitude; the more luminous the star, the longer the period.

If we know the luminosity and can measure the energy flux or brightness, then by comparing it with a standard candle then we can figure out how far away it is by comparing it with a standard candle of the same luminosity.