A standard candle is an astronomical object of which the intrinsic brightness and distance from the Earth are known. By comparing the apparent brightness of a star with the apparent brightness of the standard candle, it is possible to work out the distance of the star from Earth. To do this, you need to know the intrinsic brightness of the star.

Two examples of standard candles are:

- Cepheid Variables – intrinsic brightness related to time period of pulsation.
- Type Ia supernovae – all have the same intrinsic brightness because stars explode after reaching the same critical mass.

## How does comparing the brightness tell us the distance?

As you move further away from a light source, the apparent intensity of the light drops following the inverse square law. This is because the light travels out in

*every*direction to cover the surface of the imaginary sphere. Here is a bullet point guide to the relation between distance and apparent brightness:- At distance
**r**, apparent brightness is**B**. - At distance
**2r**, apparent brightness is**B/4**. - At distance
**3r**, apparent brightness is**B/9**. - At distance
**4r**, apparent brightness is**B/16**. - At distance
**5r**, apparent brightness is**B/25**. - At distance
**6r**, apparent brightness is**B/36**.

For any star, the apparent brightness α intrinsic brightness / distance from Earth ²

B α I / d²

We can consider two starts A + B of *equal *intrinsic brightness to relate both to the above equation:

B(A) α I / d(A)² and B(B) α I / d(B)²

Therefore, B(A)d(A)² = B(B)d(B)². This then produces the equation:

B(A) / B(B) = d(B)² / d(A)²

## Using Parallax and Standard Candles to Measure Astronomical Distances

### Parallax

Imagine observing an object against a fixed background. If you move the position of the object relative to the fixed background changes. The further away the object, the smaller the apparent change in position. This is the basis of stellar parallax.

### The Parsec

A unit of distance in common use amongst astronomers is the parsec. As the Earth moves in its orbit around the sun, the position of nearby stars against the background of very distant stars seem to change.

The parallax angle of a star is the angle labelled Ø on the diagram. These angles are very small because nearby stars are very distant compared with the Earth’s orbital diameter. Small angles can be expressed in minutes and seconds; one minute of arc is 1/60 of a degree; one second of arc is 1/60 of a minute (1/3600 of a degree).

If the radius of the Earth’s orbit is R then the distance to a star with parallax angle Ø is:

d = R / tanØ

A star at a distance of 1 parsec (a pc) from us has a stellar parallax angle of 1 second.

## If R = 150×10^6 km, find the value of 1 parsec in meters.

To work this out, we can put the above information into the above formula:

- d = 150×10^9 (m) / tan 1/3600 (one second)
- d = 3.09×10^13 km

The length of one parsec is 3.09×10^13 km.

## Summary

- We can measure the distance of a distance star if it is a standard candle. We can use the equation B α I / d² and B(A) / B(B) = d(B)² / d(A)² where B is the apparent brightness, I is the intrinsic brightness and d is the distance from Earth.
- The apparent intensity of the light from the distance star drops following the inversely square law.
- From using distance stars as a background, we can work out the distance of nearby stars through the equation d = R / tanØ where R is the distance of Earth’s orbit from the sun, Ø is the parallax angle and d is the distance of the nearby star to the sun.
- A star of distance one parsec (3.09×10^16m) away from Earth will have a parallax angle of 1 second or 1/3600 of a degree.