Astronomy and cosmology

The luminosity 光度 $L$ of a star is the total power 功率 of radiation it gives out — the energy 能量 radiated per second in all directions. Unit: watt (W).

At distance $d$, this power has spread over a sphere of area $4\pi d^{2}$. The radiant flux intensity 辐射通量密度 $F$ (power per unit area) at distance $d$ is

Worked example. The Sun's luminosity is $L = 3.8 \times 10^{26}\ \text{W}$. Find the radiant flux intensity at the Earth, a distance $d = 1.5 \times 10^{11}\ \text{m}$ away.

Unit: $\text{W m}^{-2}$. This is the inverse-square law 平方反比定律 for flux: doubling the distance cuts the flux to a quarter. A telescope measures $F$; if $L$ is known, the distance follows:

The four units of ESO's Very Large Telescope in Chile, used to measure the flux from distant stars

The same power spreads over a larger area as distance grows, so flux falls as $1/d^{2}$

A standard candle 标准烛光 is an object whose luminosity is known from its type. Once you find one in a distant galaxy and measure the flux $F$ from it, you get its distance from $d = \sqrt{L/(4\pi F)}$.

Examples:

• Cepheid variables 造父变星 (pulsating stars) — the pulsation period is tightly linked to the luminosity, so the period gives $L$.
• Type Ia supernovae 超新星 — a white dwarf reaching a critical mass and exploding always has about the same peak luminosity.

A standard candle gives $L$ without first knowing the distance, so it reaches galaxies far beyond parallax 视差.

The Andromeda Galaxy, our nearest large galaxy, about 2.5 million light-years away — Cepheids in it are standard candles

For Cepheid variables the pulsation period sets the luminosity, making them standard candles

Wien's displacement law

A hot body gives out a continuous (blackbody 黑体) spectrum with a peak at a wavelength 波长 $\lambda_{\text{max}}$ set by its temperature 温度. Wien's displacement law 维恩位移定律:

Worked example. A star's blackbody spectrum peaks at $\lambda_{\text{max}} = 500\ \text{nm}$. Find its surface temperature. ($b = 2.90 \times 10^{-3}\ \text{m K}$.)

Hotter stars peak at shorter wavelengths: a cool red star ($\sim 3000\ \text{K}$) peaks in the infrared; the Sun ($\sim 5800\ \text{K}$) peaks near $500\ \text{nm}$; a hot blue-white star ($\sim 20{,}000\ \text{K}$) peaks in the ultraviolet. Measuring $\lambda_{\text{max}}$ gives the surface temperature.

The Pillars of Creation in the Eagle Nebula — clouds of gas and dust lit by hot, newly formed stars

A hotter black body radiates more, and its peak wavelength shifts towards the blue (Wien's law)

Stefan–Boltzmann law

A star, treated as a blackbody sphere of radius $r$ and surface temperature $T$, has luminosity

where $\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\ \text{K}^{-4}$ is the Stefan–Boltzmann constant 斯特藩-玻尔兹曼常量 (the Stefan–Boltzmann law 斯特藩-玻尔兹曼定律). Two strong dependences:

• $L \propto r^{2}$ — twice the radius, four times the luminosity (same $T$).
• $L \propto T^{4}$ — twice the temperature, sixteen times the luminosity (same $r$).

Worked example. A star has radius $r = 7.0 \times 10^{8}\ \text{m}$ and surface temperature $T = 5800\ \text{K}$. Find its luminosity. ($\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\ \text{K}^{-4}$.)

Estimating a star's radius

Combine the two laws:

1. measure $\lambda_{\text{max}}$ → get $T$ from Wien's law.
2. find $L$ (e.g. from flux $F$ and distance $d$: $L = 4\pi d^{2} F$).
3. solve the Stefan–Boltzmann law for $r$: $r = \sqrt{L/(4\pi \sigma T^{4})}$.

This is how astronomers estimate radii of stars they cannot see as a disc.

Cosmological redshift

The spectral lines 谱线 of light from distant galaxies are seen at longer wavelengths than their known laboratory values — the whole spectrum is stretched towards the red. This is redshift 红移.

The hydrogen absorption lines of a distant star are shifted to longer wavelengths — a redshift

Reading it as a Doppler shift, the galaxy is moving away. For $v \ll c$:

where $\Delta\lambda = \lambda_{\text{observed}} - \lambda_{\text{emitted}}$ and $v$ is the speed of recession 退行. Example: light emitted at $4.62 \times 10^{-7}\ \text{m}$ but seen at $4.91 \times 10^{-7}\ \text{m}$ gives $\Delta\lambda = 0.29 \times 10^{-7}\ \text{m}$ and

Why redshift means an expanding Universe

Almost every distant galaxy is redshifted (a few near ones are blueshifted 蓝移 by local motion). So galaxies are, on average, moving apart — not just from us but from each other. The Universe is expanding, with the space between galaxies stretching. More distant galaxies are redshifted more.

The Hubble Ultra Deep Field — almost every point of light is a whole galaxy, most of them redshifted and receding

Hubble's law

The link between recession speed $v$ and distance $d$ is Hubble's law 哈勃定律:

where $H_{0}$ is the Hubble constant 哈勃常数 ($\approx 2.3 \times 10^{-18}\ \text{s}^{-1}$). Always use SI units. Example: a galaxy receding at $1.9 \times 10^{7}\ \text{m s}^{-1}$ is at $d = v/H_{0} \approx 8.3 \times 10^{24}\ \text{m}$.

Hubble's law: a galaxy's recession speed is proportional to its distance, $v = H_0 d$

From Hubble's law to the Big Bang

Hubble's law means the galaxies were once together. Running the expansion backwards, all distances shrink to zero at $t = -1/H_{0}$ — the Universe was once a tiny, hugely dense, hot point. This is the Big Bang 大爆炸. The age of the Universe (for steady expansion) is about

The expansion, the redshift of galaxies, the cosmic microwave background 宇宙微波背景, and the hydrogen/helium abundances are the main evidence for the Big Bang.

Distance ladder

Astronomers combine methods, each calibrated by the one below:

1. parallax — for nearby stars.
2. standard candles (Cepheids, Type Ia supernovae) — for galaxies.
3. Hubble's law ($d = v/H_{0}$, with $v$ from redshift) — for very distant galaxies.

The distance ladder: each method is calibrated by the one below and reaches further out

• $F = L/4\pi d^2$ (inverse-square): a standard candle has known luminosity $L$, so the distance follows from the measured flux $F$.
• Wien's law ($\lambda_{max} \propto 1/T$) gives surface temperature; Stefan's law $L = 4\pi r^2 \sigma T^4$.
• Redshift $z = \Delta\lambda/\lambda \approx v/c$; Hubble's law $v = H_0 d$ leads to the age of the universe (evidence for the Big Bang).