Stellar distances

6.2 Stellar distances

To measure stellar distances the method of parallax is used. Suppose that you had to measure the width of a river. Fix on a point directly across the river. Then walk along the river bank until, looking back at it over your shoulder, the point lies at an angle of 45° to the side of the river bank. You will have walked a distance equal to the width of the river. So the method of parallax requires one to observe an

Introduction to Astronomy and Cosmology

object from two positions some distance apart (forming a baseline) and measure the change in angle. The further away the object, the greater the baseline required. In the case of stars, there is no perceptible change in angle from points across the Earth, so a considerably bigger baseline is required. Happily, one is available to us, the distance across the Earth’s orbit around the Sun (Figure 6.1). The major- ity of stars are too far away to show any change in their observed position when observed, say, in the spring and autumn when the Earth is at opposite sides of the Sun. These stars can thus be considered as reference points against which the movement in position of (at present) a relatively few nearby stars can be measured. The angular change in their position, coupled with knowledge of the Earth’s orbit, can thus be used to fi nd their distance.

The measured angles are very small, an arcsecond or less, and depend on the exact times at which the measurements were made. To determine the distance, the change in angle of a star’s position (which has been measured using a baseline of ∼2 AU) is converted into what is termed the parallax of the star which is the angu- lar shift in position of the star that would be observed with a baseline of exactly

1 AU. As the angles are (very) small, one can then immediately derive its distance from θ ⫽ A/d, where θ is the parallax in radians, A is the astronomical unit and

d is the distance. If A is in kilometres, then d would also be in kilometres.

Figure 6.1 The method of parallax.

The Properties of Stars

As an example, let us consider a star whose parallax is 1/10th of an arcsecond:

θ ⫽ 0.1/(3600 ⫻ 57.3) rad ⫽ 4.85 ⫻ 10 ⫺7 rad

As 1 AU is 1.49598 ⫻ 10 8 km, the distance, d, is thus given by:

d ⫽ 1.49598 ⫻ 10 8 /4.85 ⫻ 10 ⫺7 km ⫽ 3.084 ⫻ 10 14 km

One light year is 9.46 ⫻ 10 12 km; this is equal to 32.6 light years.

6.2.1 The parsec

As the angle and distance are inversely related, it is possible to defi ne a unit of distance such that a star located at this distance would have a parallax of 1 arcsec. This unit is called the parsec , and is the unit of distance normally used by profes- sional astronomers. The simple relationship is:

d ⫽ 1/p

where d is in parsecs and p is the parallax of the star in arcseconds. (The parsec is usually abbreviated to pc.)

So a star which subtends an angle of 1/10th of an arcsecond will be at

a distance of 10 pc. We have just calculated that this distance is 32.6 light years, so:

1 pc is equal to 3.26 light years

The nearest star has a parallax of 0.772 arcsec which corresponds to a distance

d given by:

d ⫽ 1/0.772 pc ⫽ 1.295 pc

⫽ 1.295 ⫻ 3.26 light years ⫽ 4.22 light years

Introduction to Astronomy and Cosmology

It is a star in the α Centauri multiple star system and, not surprisingly, has been given the name Proxima Centauri.