Astronomy

Magnitudes and spectra

Apparent and absolute magnitude, distance modulus, colour and spectral type, Wien’s law.

Basics

Apparent magnitude

Apparent magnitude m is how bright a source looks in the sky. Five magnitudes are a factor of 100 in flux (Pogson), so one magnitude is about 2.512. Smaller numbers are brighter; negatives occur (the Sun is about −26.7). Name the band (V, G, …) or comparisons slip. Moonlight and skyglow raise the limiting magnitude.

Absolute magnitude and distance

Absolute magnitude M is m if the source sat at 10 parsecs. The distance modulus m − M = 5 log₁₀(d/10 pc) estimates distance. At equal luminosity, a farther star looks fainter. Interstellar extinction makes objects look farther than they are. Standard candles (Cepheids, Type Ia supernovae) put ticks on this same backbone.

Spectral type and colour

Stellar spectral types OBAFGKM run from hot to cool. A stars show strong Balmer lines; M stars show molecules. A colour index (B−V) is a magnitude difference and a rough temperature scale. That temperature/colour is the horizontal axis of the HR diagram. Metallicity and gravity tweak the lines. “Redder” means “cooler” only if extinction is small.

Blackbodies and lines

A stellar photosphere is roughly a blackbody, so Wien’s displacement law ties colour to temperature. Absorption lines on the continuum record photospheric composition. Emission lines are common in thin hot gas such as nebulae. Redshift slides the whole line set and becomes a cosmological ruler. A spectrograph is a compromise among slit, grating, and detector.

Formulas

Pogson ratio

m₁ − m₂ = −2.5 log₁₀ (F₁ / F₂)

Magnitude difference vs flux ratio. Negative means 1 is brighter.

Symbols

m apparent magnitude
F flux

Distance modulus

m − M = 5 log₁₀ (d / 10 pc)

d in parsecs. If extinction A is present, add +A on the right.

Symbols

M absolute magnitude
d distance (pc)

Wien displacement

λ_max T ≈ 2.897×10⁻³ m·K

Peak wavelength of a blackbody. First ruler for stellar colour.

Symbols

λ_max wavelength of maximum
T absolute temperature

Key table

Vega	near the old zero point; actually V ≈ 0.03
1 pc	parallax of 1″. ≈ 3.26 ly ≈ 3.086×10¹⁶ m
O vs M	O: tens of thousands of K, UV. M: a few thousand K, toward IR