Kepler's Laws, Black Holes and Dark Matter

Closest to the Sun in January

Here is a fact that surprises most Canadians: Earth is closest to the Sun in early January, in the middle of our winter, and farthest in early July. Earth's orbit is not a perfect circle. (The seasons come from the tilt of Earth's axis, not from the changing distance.)

Remember from Grade 9 (SNC1W, Space Exploration): Johannes Kepler used Tycho Brahe's careful observations to show that planets move in ellipses. In this lesson you will see that Kepler's three laws all follow from Newton's law of universal gravitation, and then use them to "weigh" Jupiter, the black hole at the centre of our galaxy and the galaxy itself.

Kepler's three laws (1609 and 1619)

1. Law of ellipses: each planet moves in an ellipse with the Sun at one focus.
2. Law of equal areas: the line from the Sun to a planet sweeps out equal areas in equal times. So a planet moves fastest at perihelion (closest) and slowest at aphelion (farthest).
3. Law of periods: for all planets orbiting the Sun, T²/a³ is the same, where T is the period and a is the semi-major axis.

Kepler found these laws from data, with no idea why they worked. About 70 years later, Newton showed that an inverse-square force of gravity produces exactly these three laws. The second law comes from the fact that gravity points straight at the Sun, so it cannot speed up or slow down the planet's sideways turning (this is conservation of angular momentum, a topic for first-year university physics).

The third law from Newton

For a circular orbit (a = r), you already derived T² = (4π²/GM) r³. So:

T²/a³ = 4π²/GM

The right side contains only G and the central mass M. That is why every planet around the Sun has the same T²/a³, every moon of Jupiter has its own shared value, and every satellite of Earth has another. The law also works for elliptical orbits if you use the semi-major axis a.

Two handy ways to use it:

• Comparing two orbits around the same body: T₁²/a₁³ = T₂²/a₂³. For the Sun, with T in years and a in astronomical units (1 AU = 1.50 × 10¹¹ m, Earth's average distance), Earth gives T²/a³ = 1, so T² = a³ for every planet.
• Weighing the central body: M = 4π²a³/(GT²). Measure one orbit, get the mass.

A black hole at the heart of the Milky Way

At the centre of our galaxy, astronomers have tracked individual stars for over 30 years using infrared telescopes (infrared light passes through the dust that blocks visible light). One star, called S2, follows an ellipse with a period of about 16 years and a semi-major axis of roughly 1000 AU. Its focus is an object that gives off almost no light: Sagittarius A\*.

Using Kepler's third law with solar units (masses in solar masses, a in AU, T in years, M = a³/T²):

M ≈ (1000)³/(16)² ≈ 3.9 × 10⁶ solar masses.

About 4 million Suns packed inside S2's closest approach (about 120 AU, about four times the Sun-Neptune distance). The only object that can hold so much mass in so small a space is a black hole. The 2020 Nobel Prize in Physics was shared by three people: Roger Penrose, for showing that black holes follow from general relativity, and Reinhard Genzel and Andrea Ghez, whose teams tracked these stars. In 2022 the Event Horizon Telescope collaboration released the first image of the shadow of Sagittarius A\*. You will meet black holes again in the next lesson.

The case for dark matter

If most of a galaxy's mass were in its bright centre, stars far out would orbit slower, following v = √(GM/r) like the outer planets do. In the 1970s, Vera Rubin and Kent Ford measured how fast stars and gas orbit in spiral galaxies. Far from the centre, the speeds did not drop. The rotation curves stayed flat.

For v to stay constant as r grows, M = v²r/G must grow in proportion to r: there must be more and more mass farther out, where there is very little light. This unseen mass is called dark matter. Other evidence agrees: in the 1930s Fritz Zwicky found galaxies in clusters moving too fast to be held by their visible mass, and the bending of light by clusters (gravitational lensing) shows more mass than we can see. Current estimates suggest dark matter outweighs ordinary matter by about five to one.

Nobody yet knows what dark matter is. SNOLAB, an underground laboratory 2 km below the surface near Sudbury, Ontario, hosts experiments searching for dark-matter particles, shielded from cosmic rays by the rock above.