By Joshua Winn, Princeton University
Are black holes for real? The world might still be divided on this but they are more than just a plot device in science fiction movies and books. Recent discoveries, especially the detection of gravitational waves, do not leave much room for skepticism. They are out there.
Zooming in on the Galaxy
To find some evidence for a black hole that we can understand, quantitatively, based on our studies so far, we start with an image of the Milky Way, our own galaxy, which appears as a band of light reaching across the sky.
We zoom in on the center of the galaxy, in the direction of the constellation Sagittarius, revealing that the smooth band of light is actually the summation of the light of hundreds of billions of distant stars. The black blotches are clouds of gas and dust in the foreground, blocking our view.
If we keep zooming in, we eventually arrive at a cluster of bright stars near the center. A group at UCLA, led by Andrea Ghez, has been watching these stars since the mid-1990s. The stars are being deflected by the gravity of a giant mass in the center of the image.
Observing the Star S0-2
Some of the stars are orbiting the center in Keplerian ellipses. There’s one star in particular, named S0-2, that made a complete orbit over the last 16 years. But here’s the thing, in optical and infrared images, there’s hardly any light coming from the focus of the ellipse. S0-2 and the neighboring stars are being attracted to a seemingly unremarkable spot in the image.
Let’s figure out how massive the attractor must be, based on observations of S0-2. The images show that the angular size, Delta-theta, of the long axis of the ellipse is about a quarter of an arcsecond. And we know from other observations that the distance, d, to the galactic center is 8 kiloparsecs.
With that information, we can calculate the semi-major axis, a. Delta-theta equals 2a over d, implying a is 1/2 Delta-theta times d. That’s 1/2 times a quarter of an arcsecond times 8000 parsecs, which, from the definition of parsecs, is 1000 AU.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Using Kepler’s Law to Calculate the Mass
We also observe that S0-2 takes 16 years to go around. That means P equals 16 years. And, whenever we know both P and a, we can apply Kepler’s third law to find the mass. First, let’s rearrange Kepler’s third law to solve for the mass: M equals 4pi-squared over G times a-cubed over P-squared.
At this stage, we could plug in the numbers, looking up the numerical value of G, converting a into meters and P into seconds, and so forth, but let’s make life easier by converting the equation into a scaling relation.
We see that M is proportional to a-cubed over P-squared, and we also know the answer is one solar mass for the Earth, so we can write M equals Msun times a over 1AU-cubed times P over 1 year to the minus 2 power. That’s more convenient for our problem because we already know a and P in those units. The mass of that mysterious attractor, in units of solar masses, is 1000-cubed over 16-squared, which is 4 million.
Evidence of Black Holes
So, lurking at that nondescript spot in the image—which has the official name Sagittarius A*—is something with a mass of 4 million Suns. And it’s all crammed into a space that must be smaller than S0-2’s distance of closest approach, which is about 100 AU. That’s not much bigger than the solar system. So, think about 4 million Suns, all within the solar system!
Astrophysicists have not been able to think of anything that could have so much mass within such a small space, without either glowing very brightly, or quickly collapsing under its own gravity. So, it’s either a black hole, or something even more exotic, beyond our current physical understanding.
Can Planets Collapse into a Black Hole?
Now, let’s imagine if the Sun is compressed down to a radius of 3 kilometers, turning it into a black hole. What would happen to the solar system? How long would we have, before the Earth fell into the black hole? Actually, the Earth wouldn’t fall in, at all.
The planets’ orbits wouldn’t change. The solar system would be a lot darker and colder, for sure, but the planets would continue on their usual orbits. It’s a corollary of Newton’s theorem about spherically symmetric mass distributions. It turns out that holds true in relativity, too. The force from the Sun is the same as the force from a black hole of equivalent mass.
So, in the popular imagination, black holes are like vacuum cleaners, sucking up everything around them. That’s not quite true, or at least, it’s no more true than it is for the Sun. The Sun doesn’t suck in all the planets.
To fall into the Sun, or a black hole, you need to have nearly zero angular momentum—hardly any transverse velocity—so that you follow a radial trajectory, all the way down to r equals zero. Black holes do swallow things—there are evidences for luminous hot vortices of material spiraling into black holes—but that only happens when there’s some way for the orbiting material to get rid of its angular momentum.
Common Questions about Black Holes
The solar system would become darker and colder but none of the planets would fall into the black hole. The planets would continue on their usual orbits, based on a corollary of Newton’s theorem about spherically symmetric mass distributions.
A black hole can swallow things when there’s some way for the orbiting material to get rid of its angular momentum.
To fall into a black hole, you need to have nearly zero angular momentum—hardly any transverse velocity—so that you follow a radial trajectory, all the way down to r equals zero. So, the planets’ orbits wouldn’t change.
