Without hair, black holes are incredibly difficult objects to observe. In order to truly measure their mass, a second object hopefully one with hair, needs to be in the vicinity of the black hole. Fortunately, most stellar systems contain two or more massive bodies. Our home star the Sun is an isolated star. The closest star to us named Proxima Centauri is 4.2 light years away, but many stars including Proxima Centauri live in groups of two, three or even four stars. In fact, systems consisting of two stars called binaries are just as common as single stars. Binary systems of two stars can consist of any combination of stars, neutron stars and black holes. They are scientifically important since they allow us to measure the masses of the components in the system and in some cases determine their sizes. Since the most accurate method for determining a black hole's mass is to observe its orbital dance with a companion star, we need to examine how stars move when they have a gravitational dance partner. When a compact object, either a black hole or a neutron star are in a dance with a companion star, it is sometimes hard to determine what the identity of the compact object is. This is because neutron stars can look strikingly similar to black holes, leading to cases of mistaken identity. The weight of the compact object or really its mass, is the key difference that distinguishes between these two types of compact objects. Neutron stars cannot be heavier than three times the mass of our Sun or else they would be dense enough to become a black hole. If we see a compact object hiding in a binary system that might be a black hole, we call it a black hole candidate. If we can measure the mass of the candidate and it is larger than three times the mass of the Sun, then we are confident that the object is not a neutron star and we call it a black hole. When two stars are in a binary system, they orbit on either circular or elliptical paths around a point in space called the center of mass. Let's start off with circular orbits, with both stars moving in a circle around the center of mass. The center of mass is a balance point that always falls on a line connecting the two stars. If the two stars have the same mass, then the system is perfectly balanced. Notice in this animation, that the two stars have the same mass and move on the same circular path around the center of mass. The center of mass is always directly in-between the two stars. We call the time that it takes for a star to travel one time around the center of mass the orbital period. Notice that both stars take the same amount of time to make one full orbit. If the two stars in a binary system have different masses, the balance point between them will no longer be equal distance between them. Instead, the higher mass star will be closer to the center of mass while the lower mass star will orbit further away. In this picture, the high-mass blue star has two times more mass than the little red star. The little red star orbits in a circle with a radius that is two times larger than the radius of the circle that the big blue star orbits on. The situation is similar to when my little sister and I played together when we were children. My sister is four years younger than me. So, when I was eight years old and she was four years old, I weighed two times more than she did. Sometimes we played on a teeter-totter. If we both sat in the seats at equal distances from the pivot point, then the teeter-totter was unbalanced and I could stay low and keep my sister up high. This led to her crying, which wasn't really very nice. The only way I could stop her crying was if I moved closer to the pivot point. Once I found the correct balance point, we could then swing up and down and have fun. The kids playing on a teeter-totter are similar to two stars in a binary system. The pivot point is like the center of mass. The kids are like the stars, except that the stars move in circles while the kids move up and down. The larger mass kid or star has to be located closer to the center of mass. While the smaller mass kid or star has to be further from the center of mass. We can quantify the relative distances that the stars have to be from the center of mass in order to balance the binary. The two stars have masses M1 and M2. The first star is at a distance a1 from the center of mass while the second star is located at a distance a2 from the center of mass. The equation relating the stars, masses and locations is M1 times a1 equals M2 times a2. In order to balance the binary, we have to balance the equation so that the larger mass has the smaller distance from the center of mass. In this picture, we can see by eye that a1 is two times larger than a2. This tells us that M2 is two times larger than M1. In a circular binary, the total distance between the stars is constant in time as the stars move in their circular orbits. If we define the total distance between the stars to be a, then you can see from the picture that a equals a1 plus a2. Sometimes a binary star system consists of a bright easy to see star and an invisible star. Even though we can't see the invisible star, we can deduce its presence when we see the bright star moving in circles around a point in the sky. The empty point in the sky is the center of mass and the invisible star is also orbiting the same point. The invisible star might be a dim star like a neutron star, or it could be something that isn't really a star like a planet or a black hole. Often, we can't see the orbital motion as circles in the sky because the stars are too far away. But what we can do is detect Doppler shifts in the light emitted by the stars. When the star is moving towards us, its light is blue shifted, and when it is moving away from us, the light is red shifted. When we see light from a star periodically Doppler shifting from blue to red to blue, we can deduce that we are observing a binary star system. This is actually the most common way that binaries are detected. The motion of the planets around the Sun is similar to a binary system. Suppose that we ignored all the planets except for Jupiter, since Jupiter is the biggest planet in our solar system, then the Sun and Jupiter are like a binary system. Since the mass of the Sun is 1,000 times larger than Jupiter, we shouldn't expect that the center of mass is exactly at the center of the Sun. Jupiter is massive enough to pull the center of gravity away from the center of the Sun, somewhere closer to the Sun's surface. Johannes Kepler studied the motion of the planets around the Sun and came up with a set of three laws of planetary motion. The first law is the law of orbits that states planets orbit the Sun in an ellipse where the Sun is located at one focus. An ellipse is a shape that is sort of like a squashed circle with a short dimension and a long dimension. It has two special points inside that are called the focus points. A circle is a special type of ellipse where both dimensions are the same size and the two focus points converge to a point at the center of the circle. Although the planets travel on elliptical orbits, these ellipses are almost circular for most of the planets. The Earth's orbit is just a tiny bit elliptical and the Earth is closest to the Sun in January and furthest from the Sun in July. In a binary star system, Kepler's first law tells us that the stars orbit on ellipses with the center of mass located at one of the focus points. Kepler's second law is the law of equal areas. Imagine a line joining the Sun and the planet. As the planet moves, the line sweeps out an area. Kepler's second law is that the line sweeps out equal areas in equal times. This is easier to think about in terms of an elliptical chocolate cake. Suppose that we want to cut slices of cake from the focus to the edges of the cake but everyone should have the same amount of cake. If we cut a wedge near the focus, then we need to cut a wide wedge since the length of the cuts are short. If we cut a wedge at the opposite side of the cake, the cuts will be long, so the wedge needs to be skinny. If we cut the cake like this, everyone gets the same amount of cake and everyone is happy. Based on the equal areas law, a planet travels faster when is near to the Sun and slower when it is further from the Sun. Here on Earth that means that in January when the Earth is closest to the Sun, the Earth travels faster and in July when the Earth is further from the Sun, it travels slower. As a result, the Earth spends more time in the outer parts of its orbit. Kepler's third law originally only applied to planets, but Newton improved it so that it could be used to describe the orbit of stars in a binary system. Kepler's third law is an equation that relates the masses of the stars to the orbital period and the total distance between the stars. The equation is M1 plus M2 equals the cube of the total distance between the stars divided by the square of the orbital period of the stars. In this equation, a represents the total distance between the stars measured in astronomical units, which is the distance between the Earth and the Sun. The time for the stars to orbit around the center of mass, the orbital period, is represented by the letter P. The period is measured in units of years. The sum of the masses of the two stars is in units of the Sun's mass. We had better do an example. Suppose that we observe two stars in a binary, we can easily time how long the orbits are by watching for a while and find that the stars take two years to make one full orbit. Measuring the distance between the two stars is harder to do but we learned that the distance is 4 astronomical units. Using Kepler's third law, we can calculate the sum of the masses in the star system to be M1 plus M2 equals four cubed divided by two squared. You can probably do this math in your head or use a calculator to find that the sum of the two masses is 16 times larger than the Sun's mass. Unfortunately, this only gives us the value of M1 plus M2. So, we'll need some more information to find out the masses of each stars as individuals. Perhaps, we can measure the properties of the light from one of the stars, and it is identical to our Sun. Then it would be logical to assume that its mass is the same as the Sun. So, M1 equals MSun then it is simple to solve for the mass of the other star. M2 equals 16 minus one equals 15 solar masses. This is a simplified example but this is how astronomers measure the masses of stars and black holes. Now that we can finally deduce the mass of a black hole in a binary pair, we can now distinguish between different masses of black holes. Just like sports are often divided into weight classes, so too are black holes lumped into major categories.
