Stellar Distances and Masses

1A thumb at arm's length

Hold up a thumb at arm's length and look at it with one eye, then the other: it jumps against the background. The nearer the thumb, the bigger the jump. In 1838 Friedrich Bessel used exactly this effect, with the Earth's orbit in place of the two eyes, to measure the distance to the star 61 Cygni, the first star distance ever found. Distances are the hardest and most important measurements in astronomy: without them a star's brightness tells us nothing about its luminosity (Lesson 2), and without luminosities we cannot test theories of how stars work. This lesson builds the first rungs of the cosmic distance ladder, then shows how binary stars give the masses of stars.

2Stellar parallax and the parsec

Figure 3.1: Stellar parallax (angles greatly exaggerated).

As the Earth orbits the Sun, a nearby star appears to shift back and forth against much more distant stars, returning to the same place after a year. The parallax angle $pp$ is half the total shift: the angle subtended at the star by one astronomical unit. Because $pp$ is tiny, $\tan p\approx p\backslash tan\; p\; \backslash approx\; p$ in radians, so

$$d=\frac{1\mathrm{A}\mathrm{U}}{p(\text{in rad})}d\; =\; \backslash frac\{1\backslash ,\backslash mathrm\{AU\}\}\{p\backslash ,(\backslash text\{in\; rad\})\}$$

Astronomers measure $pp$ in seconds of arc ($1^{\mathrm{\prime }\mathrm{\prime }}=\frac{1}{3600}1\text{'}\text{'}\; =\; \backslash frac\{1\}\{3600\}$ of a degree $=\frac{1}{206265}\mathrm{r}\mathrm{a}\mathrm{d}=\; \backslash frac\{1\}\{206\backslash ,265\}\backslash ,\backslash mathrm\{rad\}$) and define the parsec (parallax second) as the distance at which $p=1^{\mathrm{\prime }\mathrm{\prime }}p\; =\; 1\text{'}\text{'}$. Then simply

$$d(\mathrm{p}\mathrm{c})=\frac{1}{p{(}^{\mathrm{\prime }\mathrm{\prime }})}1\mathrm{p}\mathrm{c}=206265\mathrm{A}\mathrm{U}=3.09\times {10}^{16}\mathrm{m}=3.26\mathrm{l}\mathrm{y}d\backslash ,(\backslash mathrm\{pc\})\; =\; \backslash frac\{1\}\{p\backslash ,(\text{'}\text{'})\}\; \backslash qquad\; 1\backslash ,\backslash mathrm\{pc\}\; =\; 206\backslash ,265\backslash ,\backslash mathrm\{AU\}\; =\; 3.09\; \backslash times\; 10^\{16\}\backslash ,\backslash mathrm\{m\}\; =\; 3.26\backslash ,\backslash mathrm\{ly\}$$

Even the nearest star has . The blurring of starlight by the atmosphere limits measurements from the ground to roughly 100 pc. Satellites above the atmosphere do far better: the European Space Agency's Gaia mission has measured the parallaxes of more than a billion stars, reaching thousands of parsecs for the best-measured ones.

3Absolute magnitude and the distance modulus

Combining $b=\frac{L}{4\pi d^2}b\; =\; \backslash frac\{L\}\{4\backslash pi\; d^2\}$ with the magnitude scale of Lesson 2: moving a star from 10 pc to $dd$ changes its brightness by ${\left(\frac{10}{d}\right)}^2\backslash left(\backslash frac\{10\}\{d\}\backslash right)^2$, so

$$m-M=-2.5{\log }_{10}{\left(\frac{10\mathrm{p}\mathrm{c}}{d}\right)}^2=5{\log }_{10}\frac{d}{10\mathrm{p}\mathrm{c}}m\; -\; M\; =\; -2.5\backslash log\_\{10\}\backslash left(\backslash frac\{10\backslash ,\backslash mathrm\{pc\}\}\{d\}\backslash right)^2\; =\; 5\backslash log\_\{10\}\backslash frac\{d\}\{10\backslash ,\backslash mathrm\{pc\}\}$$

The quantity $m-Mm\; -\; M$ is the distance modulus. If we know any two of $mm$, $MM$ and $dd$, we can find the third. (This ignores dimming by interstellar dust, which makes distant stars look fainter still.)

4Standard candles

Beyond the reach of parallax, astronomers use standard candles: objects whose luminosity (absolute magnitude) can be found without knowing their distance. Measuring $mm$ then gives $dd$ from the distance modulus.

• Cepheid variables are giant stars that pulsate, brightening and fading regularly over days to weeks. In 1912 Henrietta Swan Leavitt, working at Harvard College Observatory, studied Cepheids in the Small Magellanic Cloud, all at practically the same distance from us, and found that the brighter ones have the longer periods: the period–luminosity relation. Once a few nearby Cepheids were calibrated, the period of any Cepheid gave its absolute magnitude. Hubble used Cepheids to measure the distance to Andromeda (Lesson 1).
• Type Ia supernovae, the explosions of white dwarfs, all reach nearly the same peak luminosity (absolute magnitude about −19.3), billions of times the Sun's, so they can be seen across billions of light-years.

Each method is calibrated with the one before it: parallax calibrates Cepheids, Cepheids calibrate supernovae in nearby galaxies, and supernovae reach the far universe. This chain is the cosmic distance ladder; an error in a lower rung affects every rung above it.

5Weighing stars: binary systems

More than half of all Sun-like stars have a companion. In a binary star both stars orbit their common centre of mass with the same period $TT$ (Figure 3.2). Applying Newton's law of gravitation to the pair gives Kepler's third law in its general form:

$$T^2=\frac{4{\pi }^2{a}^3}{G(M_1+M_2)}⟹M_1+M_2=\frac{a^3}{T^2}(a\text{ in AU},T\text{ in years},M\text{ in solar masses})T^2\; =\; \backslash frac\{4\backslash pi^2a^3\}\{G(M\_1\; +\; M\_2)\}\; \backslash qquad\backslash Longrightarrow\backslash qquad\; M\_1\; +\; M\_2\; =\; \backslash frac\{a^3\}\{T^2\}\backslash \; \backslash \; (a\; \backslash text\{\; in\; AU\},\backslash \; T\; \backslash text\{\; in\; years\},\backslash \; M\; \backslash text\{\; in\; solar\; masses\})$$

where $aa$ is the separation of the stars (the semi-major axis of their relative orbit). The second form follows because the same law gives $T=1T\; =\; 1$ year at $a=1a\; =\; 1$ AU for the Earth and the Sun. The individual masses come from the sizes of the two orbits about the centre of mass: $M_1{a}_1=M_2{a}_{2}M\_1a\_1\; =\; M\_2a\_2$.

Figure 3.2: A visual binary (circular orbits for simplicity).

Binaries show that the mass of a main-sequence star (Lesson 4) sets its luminosity, roughly $L\propto M^{3.5}L\; \backslash propto\; M^\{3.5\}$: a star of twice the Sun's mass is about $2^{3.5}\approx 112^\{3.5\}\; \backslash approx\; 11$ times as luminous.