Stellar Mass
Outline
1. Mizar & Alcor in Ursa Major
* Mizar is binary
2. Binary Star System: Orbit Center of Mass (CM)
M: massesr: distances from CM
M: massesa: average distance between stars (AU)P: orbit period (yrs)
3. 'Seeing' Binary Stars
* Visual Binary
> Two stars seen in telescope
> Mizar, Albireo, Sirius, Kruger 60
* Spectroscopic Binary
> One star 'seen' in telescope
> Two stars found by periodic Doppler shift of spectrum lines
* Eclipsing Binary
> One star 'seen' in telescope
> Two stars found by mutual eclipses
> Light curve
4. Binary Statistics
* More than 50% of stars are binary
Questions
1. On what basis are binary stars classified?
2. Consider the binary system depicted above, consisting of stars A and B orbiting (on circles) about a center of mass (+). We on Earth are observing the system
from the right in the diagram.
a) Which star is the more massive? How can you tell?
b) Which star has the greater orbit period?
c) When star A reaches point 'X' on its orbit, where will star B be located?
d) Suppose this system is viewed from Earth as a spectroscopic binary system so that the Doppler shifts of both stars are observed. The lines of which star show the larger Doppler shifts? Why?
e) As seen from Earth, where is star A located when its spectrum lines show maximum redshift? Maximum blueshift?
f) As seen from Earth, where is A located when its spectrum lines show zero Doppler shift?
3. Suppose the binary system depicted above is viewed from Earth as an eclipsing binary; the light curve for this system is depicted above.
a) The time interval between the centers of the two troughs represents what fraction of the binary system's orbit period?
4. Suppose in Problem 2, star A lies 2 AU from the center of mass; star B lies 4 AU from the center of mass.
a) What is the ratio of the masses of these stars?
b) The orbit period of this system is 12 years. What is the sum of the masses of this system?
Answers
1. Binary stars are classifiied as visual binaries, spectroscopic binaries and eclipsing binaries according to the method by which they are observed.
2. a) Star A is the more massive because it lies closer to the center of mass (which is indicated by the "plus" sign).
b) Orbit periods are equal.
c) Star B will be located at position 'Y' (see figure below).
d) The lines of Star B show the greater Doppler shift because B has the greater orbit speed.
e) To answer this question, we must first decide which direction (clockwise, counterclockwise) the stars are moving. The direction is arbitrary; I choose counterclockwise (see diagram below). So, maximum redshift for the lines of star A occur when it's at position 1; maximum blueshift occurs at position 2.
f) The lines of star A show zero Doppler shift when A is located at either position 3 or position 4 (see diagram above).
3. a) This time interval is one-half the system orbit period.
4. a) Star A is twice the mass of Star B: MA/MB = 4/2 = 2.
b) (MA + MB) = a3/P2. a = distance between stars; P = orbit period. In this case, a = 4 AU + 2 AU = 6 AU; P = 12 yrs. So,
(MA + MB) = 63/122 = 1.5. This result means that the total mass of the two stars is 1.5 times the mass of the sun.