Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.
In order to gain credit for this exercise, you must create a PDF document which provides the answers to all the questions. Submit the PDF to the instructor via the "Assignments" tab in myCourses.
Today, we'll look at the physics of an eclipsing binary star system, and figure out how to derive fundamental properties of a system -- the mass and size of the stars, for example -- from the quantities we can easily observe.
This exercise builds upon the information you've already seen in our previous class meeting. We will once again use the star KIC 8736245 as our subject of study, and we will rely heavily on the the paper by Fetherolf et al. (AJ 158, 198, 2019) to provide background, explanations, and the right answers.
You may already have calculated some or all of the answers to the first section. If so, congratulations on taking such good notes.
The first and easiest (usually) aspect of a binary system to measure is the period of the orbit. We will use data collected by the the Kepler space telescope, which made very precise light curves. We can access some of this data at the following archive site:
Enter the name "KIC 8736245" into the "and enter target" box, then click on "Search".
When the search results appear, in the "Mission" panel, check the "Kepler" box (not the "KeplerFFI" box). A single item should apear in the "List View" panel. Click on the icon that looks like a small light curve -- it says "Timeseries Viewer" if you hover over it.
You should see a new window like this:
Your task is to measure the period of this eclipsing binary system. Note that the dips come in pairs: a deep one, then a shallow one, then a deep one, and a shallow one. A full period runs from deep dip to deep dip.
Next, we will zoom in on just one primary dip. Each group should choose one of the deep dips. Use the sliders and controls in the panels on the left to zoom in one your dip, and measure two time intervals:
You can see that the light curve has a flat bottom; the light stays (nearly) constant during the middle of the dip. Why? Take a look at this close-up view of an eclipsing system. As star "A" moves to the left, and star "B" crosses in front of it to the right, we can identify four moments when the limbs of the stars first (or last) touch each other.
Your measurement of the duration at the TOP of the eclipse corresponds to (t4 - t1), and the duration at the BOTTOM of the eclipse is (t3 - t2). We will use these times to compute the SIZE of each star in a moment.
But first, we need another piece of information: the velocity of each star in its orbit. We can derive these speeds using spectra of each star measured at several locations in the orbit. The Doppler shift of the lines yield the radial velocities of each star -- that is, the motion of each one toward or away from us.
In this paper, Fetherolf et al. provide a nice graph showing the radial velocities at eight moments during the orbit. Note that the average speed of each star is not zero, but around +12 km/sec. That means that the entire binary system as a whole is moving away from the Sun at a systemic velocity of 12 km/sec.
Figure 3a taken from
Fetherolf et al., AJ 158, 198 (2019)
Right. We're almost ready to compute the size of each star. Look closely at the motion of the two stars as the eclipse happens. Since the two stars are moving in opposite directions as they pass each other, their RELATIVE speed is simply (vA + vB). Let's fix the star B in place, and watch as the star A moves across it from right to left at this combined speed. During the time period from first to fourth contact, the star A must move a distance of RB + RA + RA + RB.
That means that we can find one equation for the radii of the stars.
In a similar way, we can derive a second relationship involving the radii and the time between second and third contact.
Hmmm. Two equations for two unknown radii. This is a job for linear algebra.
The solutions turn out to be:
Okay! Using your measurements of the time intervals (t4 - t1) and (t3 - t2), and the velocities measured by Fetherolf et al., you can compute the size of each star.
Our Sun has a radius of Rsun = 6.96 x 105 km.
In order to compute the mass of each star, we will apply Kepler's Third Law. Kepler found a very simple relationship between the period and the size of an orbit:
(1 Astronomical Unit = AU is the average distance between the Earth and Sun. For our purposes, 1 AU = 1.5 x 108 km.)
The period -- you've already measured that. But what is the semi-major axis?
In a binary star system, the semi-major axis is the separation between the two stars. If the two stars have different masses, then each will orbit the center of mass with a different orbital radius. The semi-major axis, or separation, a, is just the sum of these two radii.
So, if we could figure out the orbital radius of each star, we could add them together to find a, and then compute the total mass using Kepler's Third Law. And we CAN figure out the orbital radii; here's how. We know how fast each star is moving, based on the radial velocity graph you examined earlier. We also how long it takes each star to complete one orbit: that's just the period. So, we can calculate the circumference of each star's orbit:
It's probably easiest if you use the period in seconds, and the velocity in km/sec.
Now, the average distance between the Earth and Sun is known as 1 Astronomical Unit, or 1 AU for short. To a rough approximation,
1 AU = 1.5 x 108 km
Great! Now we can compute the mass of the two stars added together using Kepler's Third Law. We can use this simple formula as long as we remember to express the period in years and the separation in AU; it will give the mass in units of the solar mass.
But we want to know the mass of each star individually, too. Fortunately, we already possess a clue that provides the answer. The radial velocity measurements showed that one star moved more quickly than the other.
Figure 3a taken from
Fetherolf et al., AJ 158, 198 (2019)
Why is that? Because the gravitational force is the same on both stars, the less-massive object will be given a larger acceleration, and thus a larger velocity. It's a simple relationship:
At this point, you know the size of each star, and you know the separation between them. That means that you could create a picture showing the two stars to scale; in other words, a picture in which the two stars have their proper relative size and separation.
It doesn't have to be fancy. Treat each star as a simple sphere (or circle, in 2-D), make each one the right size, and place them at the proper separation. Label the more massive star "B" and the less-massive star "A".
When astronomers take images of this binary star system, we don't see two little dots of light, moving around each other. Instead, we see just a single point of light. Why is that?
The answer is simple: we lack the angular resolution to distinguish the two stars. You can easily read the letters printed on the page of a textbook if it sits two feet away from you; but you can't separate them if the book is fifty feet away. The same thing happens with star: if a binary star is very close to the Earth, we can perhaps see each of the two stars individually, as in the case of epsilon Lyrae. But if the binary system is too far from the Earth, then the images of the two stars merge and blur together.
Most people measure angles in degrees or radians, but astronomers like to use a unit called the "arc-second."
1 arc-second = 1/3600 degree = 2.77 x 10-4 degree
= 4.84 x 10-6 radian
Typical ground-based telescopes can resolve two stars about one arcsecond apart. The best space-based telescopes can do better; the Hubble Space Telescope can separate stars as close as about 0.05 arcseconds.
How far apart are the two stars in KIC 8736245? In order to compute the angle between them, we need to know two things:
You already know the separation a. But how can we determine the distance d? Fortunatly, the European Space Agency launched a space telescope mission, Gaia, dedicated to making measurements of stellar distances. Over the past nine years, Gaia has measured the distances to over a billion stars in our Milky Way. Very impressive.
You can look up Gaia measurements using SIMBAD. After finding the entry for KIC 8736245, scroll down to the "External Archives" section and look for an entry with a name starting with "Gaia EDR3". Click on the entry. It will include a value for a quantity called parallax, labelled "Plx". You can determine the distance to a star in the units called "parsecs" in the following manner:
1000
distance in parsecs = ---------------------
Plx entry in Gaia
Parsecs are very large units -- they are well-suited to expressing the distances between stars. In fact, the typical distance between stars in our Galaxy is about one parsec; that's why astronomers use them so much. One parsec is 3.08 x 1016 m.
It's a good thing that clever scientists have found ways to investigate the properties of double stars, even if we can't distinguish them optically!
How do your results for size and mass of the two stars compare to the values derived by Fetherolf et al., AJ 158, 198 (2019) ? Just find the right table ...
Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.