Exercises

Multiple Star Systems and Star Clusters

Consider the following questions concerning the proper motion of stars. It may be of use to remember that the Moon (and the Sun) subtend approximately 0.5 degrees of angle on the celestial sphere.

1. Barnard's Star is the star with the largest observed proper motion (10.3 seconds of arc per year). How long will it take Barnard's Star to change its position on the celestial sphere by the equivalent of the diameter of the Moon?

2. The distance to Barnard's Star is 1.83 parsecs. Combine this with the information in the preceding exercise to determine the component of velocity perpendicular to our line of sight in km/second. The following diagram may be helpful.

Use this Star Velocity Calculator.

3. The approximate mass of Sirius A is 2.1 solar masses and that of Sirius B is about 1.0 solar masses. If the total separation between Sirius A and Sirius B at a particular time is 20 AU, what is the distance of each star from the center of mass for the system? You may find it helpful to use this Center of Mass Calculator.

4. If the total mass of the Sirius binary is 3.2 solar masses and the period for rotation is 50.1 years, what is the mean separation in AU for the two stars (the length of the semimajor axis)? The Kepler's Laws Calculator may be useful for this problem.

5. Assume for simplicity that for a visual binary one star is much more massive than the other so that it lies approximately at the center of mass and the other star orbits it. You do not know the tilt angle of the orbits with respect to the line of sight, but the orbit appears to be elliptical. If you can follow the motion of both stars over a long enough period of time, how could you decide whether the orbit is truly elliptical or is just a flattened perspective of a tilted circular orbit? Hint: At about what point does a star or planet orbit in Kepler's laws?

6. Consider a circular orbit tilted by an angle i, and in the following diagram.

Use simple trigonometry to find a formula for the ratio of the apparent semimajor to apparent semiminor axis of the elliptical projection of the orbit that you would see because of the tilt. What is this ratio if i = 30 degrees? Draw the corresponding figure.

7. Why don't any visual binaries have short periods such as a few days or weeks as some of the other types of binary stars do?

8. Go to the Sirius Binary System animation. Hide Sirius B, hide the orbits and the c.m. What information could be determined if Sirius A and its motion were all our telescopes could observe?

9. Sirius A is a main sequence star of spectral class A, but Sirius B is a white dwarf that has already finished its main sequence lifetime. The mass of Sirius B is approximately half that of Sirius A. We know that stellar lifetimes are shorter for higher mass stars and longer for lower mass stars. How do you explain the apparent discrepancy?

10. In this Binary Systems Applet, choose the binary star system "Capella". What is the ratio of the masses of the two stars? What is the distance of Capella A from the center of mass? What is the distance of Capella B from the center of mass? What is the ratio of these distances? Do these distances change with time? Why or why not?

11. In this Binary Systems Applet, choose the binary star system "57 And". What is the ratio of masses in this case? Stop the motion when the stars are at apastron (furthest distance apart). What is the distance of each star from the center of mass? What is the ratio of these distances? Now stop the motion near periastron (closest approach) and determine the distances and ratio in this case.

12. In this Binary Systems Applet, choose the binary star "epsilon ceti". What is the ratio of masses in this case? What is the consequence of this ratio in terms of distances from the center of mass?

13. From the mass-luminosity relation, approximately how much more luminous would you expect a main-sequence 20 solar mass star to be relative to the Sun?

14. The least luminous main sequence stars have masses of roughly 1/10 solar mass. Estimate the luminosity of such a star relative to the Sun using the mass-luminosity relation.

15. If we determine for a binary system that the period is 40 years and the average separation is 25 AU, what is the total mass of the binary system? You may find it useful to use the Kepler's Law Calculator in doing this problem.

16. Why do we need to know the true distance to a visual binary in order to compute the masses of the stars?

17. How could an astrometric binary become a visual binary?

The following two exercises use this Java applet.

18. Set the mass of one star to be three times greater than the other, keeping all the other parameters the same. How does the orbital period change compared to the case of equal masses? Why? What is the ratio of the distances from the center of mass for this case? Which star is closer to the center of masss and why?

19. Keeping the 3:1 mass ratio, change the eccentricity to make the ellipses much more elongated (e.g., 0.8 to 0.9). Did the orbital period change? Why or why not? Did their distances from the center of mass change? Why or why not?

20. In the Java applet that lets you explore the possibility of a stable planetary orbit in a binary star system, notice how the planetary surface temperature changes as it orbits the two stars. If life requires the presence of liquid water (temperature range 273 - 373 K, or 0 to 100 C), would this be easily possible in a typical binary star system? Set the two stars closer together (initial a smaller) and determine if this makes a stable orbit and the correct temperature range for liquid water more likely or less likely.

21. In the Java applet that lets you explore the orbits of some newly discovered planet candidates in orbit around other stars, most of the orbital periods are much shorter than Earth's orbital period around the Sun. Why is this the case? Which one on the list takes longer than Earth to orbit? Why?

22. A spectral line normally at 3000 Angstroms is Doppler shifted to 2995 Angstroms. What is the radial component of velocity for the source?

23. In a double-lined spectroscopic binary, why do the lines merge to one during certain points in the stars' orbits?

24. How do we determine the period of a spectroscopic binary from the velocity curve? For HD 80715 shown below, what is the period as determined from its velocity curve?

25. Assume a spectroscopic binary to have circular orbits with the line of sight in the plane of the orbits (no tilt). From the Doppler shift of spectral lines, star 1 has a maximum radial velocity of 75 km/s and star 2 has a maximum radial velocity of 100 km/s. If the period is three days, what is the total mass of the system?

26. Assume an eclipsing binary system in which the orbits are circular and the eclipses as viewed from Earth are total, with the plane of the orbits lying in the line of sight.

If the magnitude of the orbital velocity of the smaller star is known from Doppler methods to be 100 km/s and the time measured from point 3 to point 4 in the adjacent diagram is 2 hours, what is the radius of the smaller star? Express the answer both in kilometers and in units of solar radii.

27. If in the preceding exercise the time required from when the eclipse begins until the smaller star begins to emerge from behind the larger is 7 hours, what is the radius of the larger star expressed in kilometers and in units of solar radii?

28. In eclipsing binaries some of the light curves have flat bottoms and some have pointed bottoms on the minima. What causes the difference?

29. How do eclipsing binaries and spectroscopic binaries differ? All eclipsing binaries are also spectroscopic binaries but not all spectroscopic binaries are eclipsing. Explain.

30. What might cause a star in a binary system to fill its Roche lobe?

31. If a small mass star orbiting a larger mass star is twice as close at periastron than it is at apastron, how does its velocity change through its orbit?

32. If our Sun will shrink by a factor of 100 when it becomes a white dwarf, then how fast will it rotate on its axis? The Sun's current rotation period is approximately 25 days and its current radius is 700,000 km (ignore the mass loss processes and assume the white dwarf is one solar mass).

33. The orbital period for the Algol binary is 68.75 hours. Use Kepler's laws to deduce the average separation. The stars both are "subgiants", which means they are a little larger than main sequence stars (the B8 star has a diameter of 3 solar diameters and the K2 star a diameter of about 3.5 solar diameters). Use this information and the average separation deduced above to make a realistic plot (to scale) of the two stars and their separation. Assume the orbits to be circular for simplicity. Now try to do the same thing for the Sirius binary (Sirius A is a main sequence star a little larger than the Sun and Sirius B is a white dwarf about the size of the Earth and they are separated on average by a distance of about 20 AU). Do you see from this exercise the very different natures of these two binary systems?

SOLUTIONS