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# Stellar quantities

There are two significant measures to get to grips with in astrophysics: distance and brightness.

#### Distance

The astronomical unit (AU) is defined as the average distance from the Earth to the Sun. This is 149 597 870 700 m, or the slightly more convenient 150 million km (to 3 sf).

A light year (ly) is the distance travelled by light in one year, equivalent to 9.46 × 1015 m.

To calculate we need two quantities:

• The speed of light, 3 x 108 ms-1.
• The number of seconds in one year (365 x 24 x 60 x 60), 3.15 x 107 s.

Distance = speed x time

The parsec (pc) is the largest unit of distance and is defined by the technique of parallax.

Parallax is a technique to determine the distance to stars. You might have noticed on car or train journeys that nearby objects appear to move much faster past the window in comparison to the distant landscape.

As the Earth orbits the Sun (the process that defines the year), nearby stars move relative to the backdrop of distant stars. Using the mathematical technique of similar triangles, we can observe that the angle subtended by the Earth's movement through 1 AU is equal to that subtended by the near star relative to the distant background.

$$\tan p \approx p=$$1 AU / distance to near star

Rearranging: distance to near star = 1 AU / $$p$$

One parsec is the distance at which one astronomical unit subtends an angle of one arcsecond. One parsec is equal to:

• 3.09 x 1016 m
• 6.02 x 105 AU
• 3.26 ly

One arcsecond is $$1\over 3600$$ degrees, a small angle!

Angle $$p$$ must be in radians for the small angle approximation to apply: $${1\over 3600}$$ degrees = $${1\over 3600} \times {pi\over 180}$$radians

#### Brightness

The luminosity of a star is equivalent to its power, the energy released in one second:

$$L= {\Delta E \over t}$$

This luminosity is also related to the star's temperature:

$$L=\sigma AT^4$$

• $$L$$ is the luminosity (W)
• $$\sigma$$ is the Stefan-Boltzmann constant (= 5.67 x 10-8 Wm-2K-4)
• $$A$$ is the surface area of the star (m2)
• $$T$$ is the absolute temperature of the star (K)

The apparent brightness of the star at a distance from the centre of the star is equivalent to the intensity, the power (or luminosity) per unit area moved through by the radiation:

$$b={L\over {4\pi d^2}}$$

• $$b$$ is the apparent brightness (Wm2)
• $$L$$ is the luminosity (W)
• $$d$$ is the distance that the star's radiation has travelled from its centre

It is important to distinguish between the areas involved:

• ​​​​​​​$$A$$ is the surface area of the star
• $$4\pi d^2$$ is the area of the sphere that the star's radiation has produced as it expands through space (like the skin of a balloon)