Cosmology, Galaxies and Expansion of the Universe

Lessons

Senior 6 · Physics

Cosmology, Galaxies and Expansion of the Universe

5 lessons

Lesson 2 of 5

Stellar Radiation: the Sun, Brightness, Magnitude and Spectral Classes

25 min

Reading a star from its light

Look at the stars of Orion on a clear night: Betelgeuse, at the hunter's shoulder, is distinctly orange, while Rigel, at his knee, is blue-white. No one has visited a star, and no probe will reach one for thousands of years. Everything we know about them, their temperature, size, power, composition and motion, comes from their light. This lesson starts with the star we know best, the Sun, and develops the tools for reading any star: luminosity and brightness, the black-body laws, the magnitude scale and spectral classes.

The Sun: an ordinary star

The Sun is a ball of hot gas, mostly hydrogen and helium, held together by its own gravity and powered by nuclear fusion in its core (Unit 5). Its visible surface, the photosphere, radiates almost like a black body. The International Astronomical Union's nominal values (2015) are: luminosity $L_\odot = 3.828 \times 10^{26}\,\mathrm{W}$ , radius $R_\odot = 6.957 \times 10^{8}\,\mathrm{m}$ and effective surface temperature 5772 K. Its mass is $1.99 \times 10^{30}\,\mathrm{kg}$ .

Luminosity and brightness

A star's luminosity $L$ is the total power it radiates, in watts. What we measure on Earth is its brightness (radiant flux density) $b$ , the power received per unit area at right angles to the light. The power spreads over a sphere of area $4\pi d^2$ , so

$b = \frac{L}{4\pi d^2}$

A dim star may be powerful but far away. To know $L$ we need $b$ (measured) and $d$ (Lesson 3).

The black-body laws

A star's spectrum is close to that of a black body at its surface temperature (Figure 2.1). Two laws describe it.

Wien's displacement law: the wavelength at which the emission is greatest is inversely proportional to the temperature: $\lambda_{\max}T = 2.898 \times 10^{-3}\,\mathrm{m\,K}$ . Hot stars look blue-white, cool stars red.
The Stefan–Boltzmann law: the power per unit surface area is $\sigma T^4$ , with $\sigma = 5.67 \times 10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$ . For a spherical star of radius $R$ : $L = 4\pi R^2\sigma T^4$ .

Figure 2.1: Black-body spectra. If drawn to the same scale, the 10 000 K curve would be about 67 times higher than the 3500 K curve per unit area, because emission grows as $T^{4}$ .

Definition

The luminosity $L$ of a star is the total power it emits as electromagnetic radiation (unit W). Its brightness (flux density) $b$ at a distance $d$ is the power received per unit area, $b = \frac{L}{4\pi d^2}$ (unit $\mathrm{W\,m^{-2}}$ ). Example: the Sun's brightness at the Earth, the solar constant, is about $1361\,\mathrm{W\,m^{-2}}$ .

Worked example 1: the Sun's luminosity and temperature

(a) The solar constant is $1361\,\mathrm{W\,m^{-2}}$ at $d = 1.496 \times 10^{11}\,\mathrm{m}$ . Find the Sun's luminosity. (b) Using $R_\odot = 6.957 \times 10^{8}\,\mathrm{m}$ , find its surface temperature. (c) Find the peak wavelength of its spectrum.

Solution

(a) $L = 4\pi d^2 b = 4\pi(1.496 \times 10^{11})^2 \times 1361 = 3.83 \times 10^{26}\,\mathrm{W}$ .

(b) $T = \left(\frac{L}{4\pi R^2\sigma}\right)^{\frac{1}{4}} = \left(\frac{3.828 \times 10^{26}}{4\pi(6.957 \times 10^{8})^2(5.67 \times 10^{-8})}\right)^{\frac{1}{4}} = 5.77 \times 10^{3}\,\mathrm{K}$ .

(c) $\lambda_{\max} = \frac{2.898 \times 10^{-3}}{5772} = 5.02 \times 10^{-7}\,\mathrm{m} = 502\,\mathrm{nm}$ , in the blue-green; the eye sees the mixture of all visible wavelengths as white.

Worked example 2: how big is a red supergiant?

Betelgeuse has a surface temperature of about 3600 K and a luminosity of about $1 \times 10^{5}L_\odot$ (estimates vary). (a) Find its peak wavelength. (b) Find its radius in units of the Sun's radius.

Solution

(a) $\lambda_{\max} = \frac{2.898 \times 10^{-3}}{3600} = 8.05 \times 10^{-7}\,\mathrm{m} = 805\,\mathrm{nm}$ , in the near infrared: the visible part of its light is dominated by red and orange.

(b) Dividing $L = 4\pi R^2\sigma T^4$ for the two stars: $\frac{R}{R_\odot} = \sqrt{\frac{L}{L_\odot}}\left(\frac{T_\odot}{T}\right)^2 = \sqrt{1 \times 10^{5}} \times \left(\frac{5772}{3600}\right)^2 = 316 \times 2.57 = 8 \times 10^{2}$ . Placed where the Sun is, Betelgeuse would reach beyond the orbit of Mars.

The magnitude scale

About 2100 years ago the Greek astronomer Hipparchus ranked the stars from first magnitude (brightest) to sixth (faintest visible). The modern scale keeps this backwards ordering but makes it exact: a difference of 5 magnitudes is defined as a brightness ratio of exactly 100. One magnitude is therefore a factor of $100^{\frac{1}{5}} = 2.512$ :

$m_1 - m_2 = -2.5\log_{10}\frac{b_1}{b_2} \qquad\Longleftrightarrow\qquad \frac{b_1}{b_2} = 10^{-0.4(m_1 - m_2)}$

The apparent magnitude $m$ describes how bright a star looks from Earth: Sun −26.74, Sirius −1.46 (the brightest star at night), Vega 0.03, faintest stars visible to the eye in a dark sky about +6. The absolute magnitude $M$ is the apparent magnitude the star would have at a standard distance of 10 pc; it measures luminosity, and is used in Lesson 3.

Worked example 3: comparing brightnesses

(a) How many times brighter does the Sun ( $m = - 26.74$ ) appear than Sirius ( $m = - 1.46$ )? (b) A star appears 40 times fainter than Vega ( $m = 0.03$ ). What is its apparent magnitude?

Solution

(a) $\frac{b_{\text{Sun}}}{b_{\text{Sirius}}} = 10^{-0.4(-26.74 + 1.46)} = 10^{0.4 \times 25.28} = 10^{10.11} = 1.3 \times 10^{10}$ .

(b) $m - 0.03 = -2.5\log_{10}\frac{1}{40} = +4.0$ , so $m = 4.0$ (to 2 s.f.). Fainter stars have larger magnitudes.

Spectral classes

Passed through a spectrometer, starlight shows a continuous spectrum crossed by dark absorption lines, produced by atoms and ions in the cooler outer layers absorbing particular wavelengths. Which lines are strong depends mainly on the temperature, not on the composition, which is similar for most stars (about three-quarters hydrogen and a quarter helium by mass). Stars are sorted into seven main spectral classes, each divided into subclasses 0 to 9; the Sun is G2.

Class	Surface temperature / K	Colour	Strongest lines	Example
O	above 30 000	blue	ionised helium	stars of Orion's Belt (O and B)
B	10 000 to 30 000	blue-white	neutral helium, hydrogen	Rigel
A	7500 to 10 000	white	hydrogen (strongest of all)	Sirius, Vega
F	6000 to 7500	yellow-white	hydrogen, ionised calcium	Procyon
G	5200 to 6000	yellow	ionised calcium, metals	the Sun
K	3700 to 5200	orange	neutral metals	Arcturus
M	below 3700	red	molecules such as titanium oxide	Betelgeuse, Proxima Centauri

The hydrogen lines in visible light (the Balmer series) come from atoms whose electron starts in the second energy level. In cool stars few atoms are excited that far; in the hottest, most hydrogen is ionised. So the hydrogen lines are strongest in between, at about 10 000 K, in class A. The order was worked out in 1901 by Annie Jump Cannon, who classified more than 300 000 stars.

Definition

A star's spectral class is a letter (O, B, A, F, G, K, M, from hottest to coolest) given according to the pattern of absorption lines in its spectrum, which depends mainly on its surface temperature. Example: Vega's very strong hydrogen lines make it class A0, with a surface temperature of about 9600 K.

Warning

Common mistakes. (1) Confusing luminosity (power emitted, W) with brightness (power received per square metre, $\mathrm{W\,m^{-2}}$ ). (2) Forgetting that larger magnitudes mean fainter stars, and that the scale is logarithmic: 5 magnitudes is a factor of 100, not 5. (3) Using °C in Wien's or Stefan's law: temperatures must be in kelvin. (4) Thinking different spectral classes mean different compositions: mainly they mean different temperatures.

Tip

The class order O B A F G K M is remembered by the phrase "Oh Be A Fine Guy (or Girl), Kiss Me", hottest first. For any star question, link the three formulas: $b = \frac{L}{4\pi d^2}$ , $L = 4\pi R^2\sigma T^4$ , $\lambda_{\max}T = 2.898 \times 10^{-3}\,\mathrm{m\,K}$ .

Stars: Crash Course Astronomy #26, presented by Phil Plait. Watch for the spectral classes, why the hydrogen lines peak in A stars, and how colour reveals temperature.

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Practice

Distinguish between the luminosity and the brightness of a star.
Find the Sun's brightness at Mars, 1.52 AU from the Sun (solar constant at Earth $1361\,\mathrm{W\,m^{-2}}$ ).
Rigel has a surface temperature of about 12 100 K. Find its peak wavelength and state the region of the spectrum.
Two stars have the same radius, but one is twice as hot as the other. Compare their luminosities.
Vega has $m = 0.03$ and Polaris $m = 1.98$ . Which appears brighter, and by what factor?
Arrange in order of decreasing surface temperature: a K star, an A star, an M star, a B star, a G star.
A white dwarf has a surface temperature of 25 000 K and a luminosity of $0.030L_\odot$ . Find its radius in units of $R_\odot$ (take $T_\odot = 5772\,\mathrm{K}$ ).
Mugisha says: "Sirius looks brighter than Rigel, so Sirius must give out more power." Evaluate his statement.

Show answersHide answers

Luminosity: the total power a star emits (W), a property of the star. Brightness: the power received per unit area at the observer ( $\mathrm{W\,m^{-2}}$ ), which also depends on the distance: $b = \frac{L}{4\pi d^2}$ .
$b = \frac{1361}{1.52^2} = 589\,\mathrm{W\,m^{-2}}$ .
$\lambda_{\max} = \frac{2.898 \times 10^{-3}}{12\,100} = 2.40 \times 10^{-7}\,\mathrm{m} = 240\,\mathrm{nm}$ : ultraviolet (the star looks blue-white).
$L \propto T^4$ at fixed $R$ : $2^{4} = 16$ times as luminous.
Vega (smaller magnitude): $10^{0.4 \times (1.98 - 0.03)} = 10^{0.78} = 6.0$ times brighter.
B, A, G, K, M.
$\frac{R}{R_\odot} = \sqrt{0.030} \times \left(\frac{5772}{25\,000}\right)^2 = 0.173 \times 0.0533 = 9.2 \times 10^{-3}$ : about one-hundredth of the Sun's radius, roughly the size of the Earth.
Not valid. Apparent brightness depends on distance as well as luminosity: $b = \frac{L}{4\pi d^2}$ . Sirius is very close (2.6 pc), Rigel far away (about 260 pc); Rigel is in fact thousands of times more luminous. To compare power we need the distances, or the absolute magnitudes.

Key points

Luminosity $L$ (W) is the star's power; brightness $b = \frac{L}{4\pi d^2}$ is what we receive.
Black-body laws: $\lambda_{\max}T = 2.898 \times 10^{-3}\,\mathrm{m\,K}$ (colour gives temperature); $L = 4\pi R^2\sigma T^4$ (with $L$ and $T$ , gives radius).
Magnitudes: $m_1 - m_2 = -2.5\log_{10}\frac{b_1}{b_2}$ ; 5 magnitudes = factor 100; brighter means smaller $m$ . Absolute magnitude $M$ is $m$ at 10 pc.
Spectral classes O B A F G K M, hottest to coolest, set by temperature; the Sun is G2.

Sources and credits

17.2: The Brightness of Stars, 17.3: Colors of Stars and 17.4: The Spectra of Stars (and Brown Dwarfs), from Astronomy 2e by Andrew Fraknoi, David Morrison and Sidney C. Wolff, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Solar values: IAU 2015 Resolution B3 (nominal solar luminosity, radius and effective temperature).
Image: SDO 20130714 000007 4096 4500 (AIA), NASA/SDO and the AIA, EVE and HMI science teams, Wikimedia Commons, public domain.
Video: Stars: Crash Course Astronomy #26, CrashCourse, embedded from YouTube.

The diagram, the worked examples and the practice exercises are written by Ganzaa.

1Reading a star from its light

2The Sun: an ordinary star

3Luminosity and brightness

4The black-body laws

5The magnitude scale

6Spectral classes

Reading a star from its light

The Sun: an ordinary star

Luminosity and brightness

The black-body laws

The magnitude scale

Spectral classes