## Luminosity from Stellar Spectra

Scientists use spectroscopic parallax to estimate the luminosity of a star from its spectrum (the different wavelengths shown as a band of colours when a spectrograph splits the light from a star into its electromagnetic waves).

Be careful not to confuse spectroscopic parallax with the parallax we have been discussing earlier. Scientists use spectroscopic parallax to measure the distance to stars, by assuming that spectra from distant stars of a given type are the same as those from nearby stars of the same type.

They use the Hertzsprung-Russell diagram, which gives a place to each star according to the point it has reached in its lifecycle. This method enables scientists to estimate the luminosity of a star that is far away by comparing its spectrum to those of nearer stars.

Figure 3.5: Recreated stellar spectra by class (from top to bottom): O, B, A, F, G, K, M

### Distances derived from Apparent Brightness and Luminosity

Once the luminosity of a star has been estimated, its distance can be determined by using its apparent brightness. To do this, we use the inverse square law, which states that a star's apparent brightness decreases with the square of its distance. For instance, if you take two stars of the same luminosity, they will differ in brightness by four times if one star is twice as far away as the other. To determine distance, we use the following equation:

Since the Sun is our nearest star, it is usually taken as the reference star.

By comparing another star's luminosity and apparent brightness to that of the Sun, using this formula, it is possible to determine its distance:

### Limits on Spectroscopic Parallax

Spectroscopic parallax is only accurate enough to measure stellar distances of up to about 10 kpc. This is because a star has to be sufficiently bright to be able to measure the spectrum, which can be obscured by matter between the star and the observer. Even once the spectrum is measured and the star is classified according to its spectral type there can still be uncertainty in determining its luminosity, and this uncertainty increases as the stellar distance increases. This is because one spectral type can correspond to different types of stars and these will have different luminosities.

### Example

1 - Spica

• Apparent magnitude, m = 0.98
• Spectral type is B1
• From H-R diagram this indicates an absolute magnitude, M, in the range: -3.2 to -5.0
• D = 10 (m-M+5)/5

M= -3.2, D = 10 (0.98 - (-3.2) +5)/5 = 68.54 pc

M= -5.0, D = 10 (0.98 - (-5.0) +5)/5 = 157.05 pc

The Hipparcos measurements give d = 80.38 pc

2 - Tau Ceti

• Apparent magnitude, m = 3.49
• Spectral type is G2
• From H-R diagram this indicates an absolute magnitude, M, in the range: +5.0 to +6.5
• D = 10 (m-M+5)/5

M= +5.0, D = 10 (3.49 -5.0 +5)/5 = 5.00 pc

M= +6.5, D = 10 (3.49 -6.5 +5)/5 = 2.50 pc

The Hipparcos measurements give d = 3.64 pc

 < Previous article: Apparent & Absolute Magnitude Next article: Cepheid Variables >

Last Update: 14 May 2013