Earth Science Extras
by Russ Colson
X-ray image of Sagittarius A* at the core of the Milky Way galaxy. Long wavelength X-rays in red, short wavelength X-rays in blue, image from the Chandra spacecraft: NASA/CXC/MIT/F. Baganoff et al.
I suggest in the lecture that you create some of your own problems to solve including 1) calculating the speed of the Moon using an expression for the acceleration due to gravity from Newton and 2) calculate the speed of stars near the Milky Way's core black hole. Open-ended questions of this sort are not at all the same as a tidy 'class problem' where someone else has gathered all the information for you and provides you with the correct equation where all you have to do is plug the numbers into the equation and get the answer. Doing science as a researchers or a teacher often requires that you go out and gather information before you can answer a question, pursue a research project, or develop a classroom lesson. Getting the information is a lot harder than having it provided for you because you have to find the right information, the values may not be in compatible units, and the values may not exactly correspond to what you need for a particular equation. In other worrds, you have to actually think.
Therefore, if you have not already done so, I encourage you to try to do the two problems suggested in the lecture before proceeding with the exercises below (which will allow you to solve real science questions, but may provide some of the "data" for you in a nice tidy package that is really not at all like science comes at you in real life). Don't think looking up information online and solving the problem is a quick step. You have to address all kinds of related questions and make a variety of scientific judgement calls. For example, I suggest that you find the speed of a star "near" the black hole at the galaxy core, but what does 'near' mean? If you choose our own Sun, your calculation of speed is likely to be quite far off because there are so many factors affecting our Sun's speed other than rotation around the black hole (including a lot of intervening mass form other stars). Just for comparison, it took me 4-5 hours to put together the information that I used for the exercises below and then more hours to figure out how to make that information work in a question.
Notice that at at a distance of 384400 km from Earth, the Moon's orbital path will be about 2 x 3.1416 x 384400km in length = 2415262 km. (using the equation for the circumference of a circle)
At at speed of about 1018m/s = 1.018km/s, it will take 2415262km/1.018km/s to travel around the Earth, or 2372556s. Converting seconds to days = 27.5 days, the time for the Moon to travel around the Earth (the actual value for the sidereal period of the Moon is 27.3 days).
As discussed in the lecture, the mass of the black hole at the core of the Milky Way at Sagittarius A* has been estimated based on the motion of the stars surrounding it. I suggested that you create some math challenges for yourself by trying to estimate the velocity of stars near the black hole based on their distance. Below is one example of this type of calculation.
The orbit of S2 around the black hole at Sagittarius A* examined in the prevoius question is not at all circular. To calculate the mass of the black hole, the expression used for a circular problem,
v2 = GM/r, is not quite adequate. Therefore, to measure the mass, scientists had to take into account the elliptical path of the star. This can be done using Kepler's laws of planetary motion (applied to the black hole of course). Kepler lived before Isaac Newton and his three laws were in fact a major foundation of previous knowledge on which Newton based his development of the mathematical framework for the laws of gravity.
Keplers three laws are the following:
1) every planet's orbit is an ellipse with the Sun at a focus
2) a line joining the Sun and a planet sweeps out equal areas in equal times (meaning that a planet will move slower when it is farther from the Sun--something we would also predict from Newton's expression for gravitational acceleration)
3) the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit (meaning that P2 = ka*3, where P = the orbital period of a planet, a* is the length of the semi-major axis of the ellipse (half the long axis), and k is a constant.
The expression in Kepler's third law is true for any objects orbiting our Sun, or objects orbiting a different star that has the same mass as our Sun, but would not predict the movement of planets around a different star with a diffferent mass unless we changed the value of k. Newton's laws of gravitation gives us the mathematical framework to do this. If we combine Newtons expressions to Kepler's third law, we get an expression that allows for the central mass to change.
P2 = 4π2a*3/(GM), where G = gravitational constant and M is the central mass (this expression is true only where the orbiting masses are small compared to the central mass).
Notice that the constant, k, in Kepler's expression is now equal to 4π2/(GM).
Rearranging this expression to solve for mass (we are wanting to measure the mass of the black hole) gives us
M = = 4π2a*3/(GP2),
The problem below comes from Schödel, R. et al. (2002). Closest Star Seen Orbiting the Supermassive Black Hole at the Center of the Milky Way. Nature, 419, 694-696.
To measure the mass of the central black hole near Sagittarius A*, Schodel et al (2002) measured the orbit of star S2 as shown below.
Graph from Schödel, R. et al. (2002)
Notice that the semi-major axis (a*) is given as an angle--that's what can be measured using a telescope looking from Earth. To convert that number into kilometers, we need to know how far the star S2 is from Earth. Schödel et al estimated the semi-major axis to be 5.5 light days. As seen in the figure, they estimated the period to be 15.2 years. Using these measured value, see if you can reproduce their estimate for the mass of the central black hole.
last updated 3/24/2023. The image of Sagittarius A* comes from the Chandra spacecraft, courtesy of NASA/CXC/MIT/F. Baganoff et al. The graph for the orbit of S2 comes from Schodel et al (2002). Other text and pictures are the property of Russ Colson.