Magnitude, Distance and Stellar Spectra

How Bright Is That Star?

When you look at the night sky, some stars blaze brightly while others are barely visible. But a star that appears bright might simply be close — a dim star nearby can outshine a luminous one far away. Astronomers need two different measures of brightness: how bright a star looks from Earth, and how bright it truly is.

Apparent Magnitude

Apparent magnitude $m$ is the brightness of a star as observed from Earth. The scale is inherited from the Greek astronomer Hipparchus, who ranked stars from 1 (brightest) to 6 (faintest visible to the naked eye). The modern scale extends this: Sirius, the brightest star, has $m = -1.46$; the full Moon is around $m = -12.6$; the Sun is $m = -26.7$. Crucially, the scale is logarithmic — a difference of 5 magnitudes corresponds to a factor of exactly 100 in brightness.

Key Fact

On the magnitude scale, lower numbers mean brighter objects. A difference of 5 magnitudes = factor of 100 in brightness. Objects brighter than magnitude 0 have negative magnitudes (e.g. Venus at its brightest reaches $m \approx -4.9$).

Absolute Magnitude and the Distance Modulus

Absolute magnitude $M$ is a measure of a star’s true luminosity: it is defined as the apparent magnitude the star would have if it were placed exactly 10 parsecs from Earth. By comparing $m$ and $M$ for the same star, we can calculate its distance using the distance modulus:

$$m - M = 5\log_{10}(d) - 5$$

where $d$ is the distance in parsecs. If $m > M$, the star is farther than 10 pc; if $m < M$, it is closer. The quantity $m - M$ is called the distance modulus.

The star Canopus has apparent magnitude $m = -0.74$ and absolute magnitude $M = -5.71$.

Step 1: Calculate the distance modulus: $m - M = -0.74 - (-5.71) = 4.97$

Step 2: Substitute into the formula: $4.97 = 5\log_{10}(d) - 5$

Step 3: Rearrange: $\log_{10}(d) = \frac{4.97 + 5}{5} = \frac{9.97}{5} = 1.994$

Step 4: $d = 10^{1.994} \approx 98.6$ parsecs

Stellar Parallax and the Parsec

Heliocentric parallax is the most direct method of measuring stellar distances. As Earth orbits the Sun, a nearby star appears to shift slightly against the background of distant stars. The parallax angle $p$ is half the total shift (measured in arcseconds, where 1 arcminute = 60 arcseconds). The distance in parsecs is simply:

$$d = \frac{1}{p}$$

where $p$ is in arcseconds. A star at 1 parsec has a parallax of exactly 1 arcsecond (by definition). In practice, ground-based telescopes can measure parallaxes reliably to about 100 pc. The ESA Hipparcos and Gaia satellites extended accurate parallax measurements to thousands of parsecs.

Key Fact

Parallax distance formula: $d = 1/p$ (parsecs, with $p$ in arcseconds). A star with $p = 0.1”$ is at 10 pc; a star with $p = 0.01”$ is at 100 pc. Smaller parallax angles = more distant stars.

Stellar Spectra and Spectral Classification

When a star’s light is spread into a spectrum, it shows a continuous rainbow of colours crossed by dark absorption lines. Each chemical element absorbs light at characteristic wavelengths, creating a unique pattern. By matching these patterns to known elements, astronomers can determine a star’s chemical composition. The most prominent lines in most stellar spectra are the hydrogen Balmer series in the visible range.

Stars are classified into spectral types based on which absorption lines dominate their spectra, which in turn reflects spectral type and hence surface temperature. The sequence O, B, A, F, G, K, M runs from hottest (O-type, blue, above 30,000 K) to coolest (M-type, red, below 3,500 K). The mnemonic “Oh Be A Fine Girl/Guy Kiss Me” is widely used. The Sun is a G-type star (~5,800 K, yellow-white). O-type stars show ionised helium; M-type stars show molecular absorption bands. Wien’s displacement law links peak wavelength to temperature: hotter stars peak in the ultraviolet and appear blue, cooler stars peak in the infrared and appear red.

Doppler Shifts in Stellar Spectra

If a star is moving towards or away from Earth along the line of sight (its radial velocity), all its spectral lines shift together in wavelength. A star approaching Earth shows lines shifted to shorter wavelengths (blue-shift); a receding star shows lines shifted to longer wavelengths (red-shift). The shift obeys $\Delta\lambda / \lambda = v/c$, where $\Delta\lambda$ is the wavelength shift, $\lambda$ is the rest wavelength, $v$ is the radial velocity, and $c$ is the speed of light. This same principle is used to detect exoplanets (Doppler wobble) and measure the recession of galaxies.