Why astrophysics: the universe as a physics laboratory

Astrophysics is the use of physics to understand the objects and events of the universe: planets, stars, gas clouds, galaxies and the universe itself. It is an unusual science because almost everything it studies is out of reach. No one can put a star on a lab bench, change its mass and watch what happens. Instead astronomers collect the light and other signals that arrive at the Earth, sometimes after traveling for billions of years, and use the laws of physics worked out in laboratories to work backward to temperatures, masses, compositions, ages and distances. This course is about how that is done. It builds directly on the physics you already know: Newton's gravity and Kepler's laws from mechanics, blackbody radiation and the Boltzmann distribution from thermal physics, and atomic spectra, special relativity and nuclear reactions from modern physics.

In return, the universe offers conditions that no laboratory can reach. The center of the Sun has a temperature of about 15 million kelvins and a density more than 150 times that of water. A neutron star packs more mass than the Sun into a sphere about 24 km across, so a teaspoon of its matter would weigh about a billion metric tons. The gas between galaxies is emptier than the best laboratory vacuum. Black holes bend space and time so strongly that light cannot escape. Many discoveries in fundamental physics came first from the sky: helium was found in the spectrum of the Sun in 1868, decades before it was isolated on Earth; the bending of starlight by the Sun in 1919 was an early test of general relativity; the solar neutrino problem led to the discovery that neutrinos have mass; and dark matter and dark energy, still unexplained, were found by watching how galaxies move and how the universe expands. Astrophysics is not only applied physics. It is also one of the main ways we test whether our physics is complete.

The sizes of things

The first obstacle in astrophysics is scale. The numbers are so large that everyday units become useless, and astronomers use a few special units that you will meet throughout the course. The astronomical unit, AU, is the mean distance from the Earth to the Sun, now defined exactly as \(1.495\,978\,707 \times 10^{11}\) m. The light year is the distance light travels in a Julian year of 365.25 days, about \(9.461 \times 10^{15}\) m. The parsec, pc, is the distance at which one astronomical unit spans an angle of one arcsecond; it is about 3.26 light years or \(3.086 \times 10^{16}\) m. Its name comes from parallax of one second, and lesson 4 shows why it is natural. Distances inside the Milky Way are measured in parsecs and kiloparsecs, distances to other galaxies in megaparsecs, Mpc.

Masses are given in units of the Sun's mass, \(M_\odot = 1.989 \times 10^{30}\) kg, luminosities in units of the Sun's power output, \(L_\odot = 3.828 \times 10^{26}\) W, and radii in units of the Sun's radius, \(R_\odot = 6.957 \times 10^{8}\) m. The symbol \(\odot\), a circle with a dot, is the old astronomical sign for the Sun. Thinking in these units makes the universe easier to grasp: a typical star has a mass between 0.1 and 50 solar masses, the Milky Way has a mass of roughly a trillion solar masses, and the nearest star beyond the Sun, Proxima Centauri, is 1.30 parsecs away.

# Scales of the universe: distances and light travel times
c = 2.99792458e8            # m/s
AU = 1.495978707e11         # m
ly = c * 365.25 * 86400     # m
pc = AU / (3.141592653589793 / (180 * 3600))   # one AU subtends one arcsecond
print(f"1 light year = {ly:.4e} m, 1 parsec = {pc:.4e} m = {pc / ly:.3f} light years")
objects = [
    ("Moon", 3.844e8),
    ("Sun", AU),
    ("Neptune", 30.1 * AU),
    ("Proxima Centauri", 1.302 * pc),
    ("Betelgeuse", 168 * pc),
    ("Milky Way core", 8.2e3 * pc),
    ("Andromeda galaxy", 0.77e6 * pc),
    ("Virgo cluster", 16.5e6 * pc),
    ("Edge of the observable universe", 14.3e9 * pc),
]
for name, d in objects:
    t = d / c
    if t < 3600: span = f"{t:8.1f} s"
    elif t < 3.156e7: span = f"{t / 3600:8.1f} h"
    else: span = f"{t / 3.156e7:8.3g} yr"
    print(f"{name:32s} {d:10.3e} m   d/c = {span}")

1 light year = 9.4607e+15 m, 1 parsec = 3.0857e+16 m = 3.262 light years
Moon                              3.844e+08 m   d/c =      1.3 s
Sun                               1.496e+11 m   d/c =    499.0 s
Neptune                           4.503e+12 m   d/c =      4.2 h
Proxima Centauri                  4.018e+16 m   d/c =     4.25 yr
Betelgeuse                        5.184e+18 m   d/c =      548 yr
Milky Way core                    2.530e+20 m   d/c = 2.67e+04 yr
Andromeda galaxy                  2.376e+22 m   d/c = 2.51e+06 yr
Virgo cluster                     5.091e+23 m   d/c = 5.38e+07 yr
Edge of the observable universe   4.413e+26 m   d/c = 4.66e+10 yr

The light travel times are a reminder that looking out in space is looking back in time. We see the Sun as it was eight minutes ago, the Andromeda galaxy as it was about 2.5 million years ago, and the most distant galaxies as they were when the universe was a small fraction of its present age. The last line is the radius of the observable universe, about 46 billion light years, which is larger than the 13.8 billion year age of the universe multiplied by the speed of light because space has expanded while the light was on its way, a point lesson 12 returns to. The last column is the distance divided by \(c\). For objects inside our galaxy and its neighbors that is the light travel time; for distant galaxies it is not, because the light set out when the galaxy was much closer, so the true travel time is shorter.

What we can measure

Nearly everything we know about objects outside the solar system comes from a short list of measurable quantities. The first is direction: where an object is on the sky, and how that position changes with time. The second is brightness: the energy per unit time per unit area that arrives at the telescope, called the flux \(F\) and measured in watts per square meter. The third is the spectrum: how that flux is distributed over wavelength. The fourth is time variation: whether the brightness, position or spectrum changes, periodically or suddenly. In the last few decades three new messengers have joined light: neutrinos from the Sun and from Supernova 1987A, gravitational waves from merging black holes and neutron stars since 2015, and cosmic-ray particles. Light still carries almost all of the information.

From these measurements physics turns observables into physical properties. The flux and the distance together give the luminosity, the total power an object emits. If the object radiates equally in all directions, its power spreads over a sphere of radius \(d\), so

This inverse-square law is the most used equation in the course. It shows at once why distances are so important: without \(d\) a measured flux tells us nothing about how powerful a source is. A faint star might be a weak star nearby or a powerful one far away. The spectrum gives temperature through Wien's law and Planck's law, composition through spectral lines, and speed along the line of sight through the Doppler shift. Combining temperature with luminosity gives radius through the Stefan–Boltzmann law. Periodic motion in binary stars and planetary systems gives mass through Kepler's third law. Each of these steps is a lesson in this course.

The Sun as a star

The Sun is the one star we can study in detail, and its properties calibrate everything else. Above the atmosphere the Earth receives 1361 W of sunlight on each square meter facing the Sun, the solar constant, measured from satellites since 1978. Multiplying by the area of a sphere with radius 1 AU gives the solar luminosity. The mass of the Sun follows from the Earth's orbit: for a circular orbit the Sun's gravity supplies the centripetal force, \(G M_\odot m / r^2 = m v^2 / r\), and with \(v = 2\pi r / P\) this gives \(G M_\odot = 4\pi^2 r^3 / P^2\), Kepler's third law in Newton's form. The angular diameter of the Sun, about 0.533 degrees, gives its radius; the radius and the luminosity together give its surface temperature from \(L = 4\pi R^2 \sigma T^4\).

import math
G, sigma = 6.67430e-11, 5.670374419e-8
AU, year = 1.495978707e11, 3.15576e7
S = 1361.0                                 # solar constant, W/m^2
L_sun = 4 * math.pi * AU**2 * S
M_sun = 4 * math.pi**2 * AU**3 / (G * year**2)
theta = math.radians(0.5332)               # angular diameter of the Sun
R_sun = AU * math.tan(theta / 2)
T_eff = (L_sun / (4 * math.pi * R_sun**2 * sigma)) ** 0.25
rho = M_sun / (4 / 3 * math.pi * R_sun**3)
print(f"luminosity  {L_sun:.3e} W")
print(f"mass        {M_sun:.3e} kg")
print(f"radius      {R_sun:.3e} m")
print(f"surface temperature {T_eff:.0f} K")
print(f"mean density {rho:.0f} kg/m^3 ({rho / 1000:.2f} times water)")
# how long could gravity alone keep the Sun shining? (lesson 7 does this properly)
t_KH = G * M_sun**2 / (R_sun * L_sun)
print(f"gravitational energy / luminosity = {t_KH / year:.2e} years")

luminosity  3.828e+26 W
mass        1.988e+30 kg
radius      6.961e+08 m
surface temperature 5770 K
mean density 1407 kg/m^3 (1.41 times water)
gravitational energy / luminosity = 3.14e+07 years

Four measurements made from the Earth give the Sun's luminosity, mass, radius and surface temperature. The last line is a first hint of a deep problem. If the Sun's power came from slow gravitational contraction, as Kelvin and Helmholtz proposed in the nineteenth century, it could have shone at its present rate for only a few tens of millions of years. Geologists and Darwin needed hundreds of millions, and radioactive dating later showed the Earth to be 4.54 billion years old. The answer, nuclear fusion, was suggested by Arthur Eddington in 1920 and worked out by Hans Bethe in 1939; lesson 7 develops it. This pattern, where an astronomical observation conflicts with known physics and forces new physics, recurs throughout the subject.

Order-of-magnitude reasoning

Astrophysics rewards rough estimates more than most of physics, because the quantities involved span so many powers of ten that getting the power right is often the main thing. A useful estimate keeps only the physics that matters and drops numerical factors near one. For example, the gravitational binding energy of a uniform sphere is \(\tfrac35 GM^2/R\), but for an estimate \(GM^2/R\) is enough. The pressure needed to hold up a star is about its weight per unit area. The time for a cloud to collapse under its own gravity is about \(1/\sqrt{G\rho}\), the free-fall time, where \(\rho\) is its density. These estimates often reach the right answer to within a factor of a few, and they show which quantities control the result.

A brief history of the subject

People have measured the sky for thousands of years. Babylonian astronomers recorded planetary positions on clay tablets and predicted eclipses. Around 130 BCE Hipparchus of Nicaea compiled a catalog of about 850 stars, ranked them in six classes of brightness, the origin of the magnitude system of lesson 4, and discovered the slow wobble of the Earth's axis called precession. In 1610 Galileo turned a small telescope to the sky and saw the moons of Jupiter, the phases of Venus and the stars of the Milky Way. Kepler's laws of 1609 and 1619 and Newton's Principia of 1687 turned planetary motion into physics.

Astrophysics in the modern sense began in the nineteenth century, when spectroscopy made it possible to analyze starlight. Joseph von Fraunhofer cataloged dark lines in the solar spectrum in 1814; Gustav Kirchhoff and Robert Bunsen showed in 1859 and 1860 that each element has its own lines, so the composition of the Sun could be read from its light. Photography let astronomers record faint objects and measure brightness objectively. The twentieth century added the quantum theory of atoms, which explained spectral lines, nuclear physics, which explained stellar energy, and general relativity, which made cosmology possible. Since the 1940s astronomy has expanded across the whole electromagnetic spectrum, from radio waves to gamma rays, using telescopes above the atmosphere for the wavelengths it blocks. Today the field is driven by large surveys, space telescopes such as the James Webb Space Telescope, launched in December 2021, and huge data sets analyzed by computer.

Canada and the United States have long traditions in the field. In Canada the Dominion Astrophysical Observatory near Victoria opened in 1918 with what was briefly the largest telescope in the world; the David Dunlap Observatory near Toronto followed in 1935; the Mont-Mégantic Observatory in Quebec sits at the heart of the first International Dark Sky Reserve, designated in 2007; and the CHIME radio telescope in British Columbia has detected thousands of fast radio bursts. In the United States, Mount Wilson in California, where Edwin Hubble showed in the 1920s that the universe is expanding, Yerkes Observatory in Wisconsin, and Kitt Peak in Arizona built the foundations of modern astronomy. Both countries are partners in major international observatories, among them the Canada-France-Hawaii Telescope on Maunakea and the James Webb Space Telescope, to which the Canadian Space Agency contributed the fine guidance sensor and the NIRISS instrument.

How this course is organized

The course moves outward. Lessons 2 and 3 cover how the sky is mapped and how telescopes and detectors work across the spectrum. Lesson 4 builds the distance ladder. Lessons 5 and 6 measure the properties of stars: temperature, composition and luminosity from their spectra, and masses from binary stars. Lessons 7 to 9 explain how stars work inside, how they evolve and how they die as white dwarfs, neutron stars and black holes. Lesson 10 moves to the Milky Way and other galaxies, and to the dark matter that their rotation reveals. Lesson 11 turns to planets around other stars, the newest and fastest-growing part of the field. Lesson 12 is a short tour of cosmology, which the later course on general relativity and cosmology treats in depth. Each lesson includes short programs that do the calculations astronomers do, using only Python's standard library.