Lab - Eclipsing Binary Stars Worksheet-1
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Northern Virginia Community College *
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Astronomy
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May 6, 2024
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docx
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Lab - Eclipsing Binary Stars
NAME: _____________________
(Part II of this lab are adapt from Contemporary Exercises In Astronomy
).
Background Reading: Read the chapters in your textbook about light curves, types of binary star systems, luminosity and center of mass.
Part I: Eclipsing Binary Stars Lab (12 points)
1.
Go to the NAAP Lab and Simulator Titled Eclipsing Binary Stars Lab
2.
Complete all questions in the Student Guide
.
Part II: Application (5 points)
Introduction: We know that the mass of a binary star system is given by Kepler's Law:
m
1
+
m
2
=
4
π
2
r
3
G T
2
From the conservation of angular momentum, we know that:
r
2
r
1
=
v
2
v
1
=
m
1
m
2
Therefore, if we are able to determine the period
and velocity
of the stars, we can then determine their mass. The period of the stars can be easily determined by the period of splitting of the spectroscopic binary's spectral lines. Also, it is possible to determine the velocity of the stars by the extent of red-
shift/blue-shift of the spectral lines.
Note: However, what if the binary stars weren't orbiting in a plane parallel to the observer, but rather on an angle? Can the velocity of the binary stars still be determined, and hence can its mass still be determined? The answer is yes, but an additional term must be included to account for this.( In an eclipsing binary system, the shape and depth of the eclipses can be uniquely solved to give the inclination and hence the masses of the individual stars.) We won't worry about this and assume that the stars are orbiting in a parallel plane.
See the example at the end of the lab. For simplicity, the orbit is assumed to be circular (so v = 2πr/P, where P is the period) and the stars are orbiting in a parallel plane as mentioned above. Also, because of G in the equation, all times should be changed to equivalent seconds.
Now complete either problem 1 or problem 2. Problem 3 is extra credit.
Problem 1: Data is gathered from the light curve and the radial velocity curves of an eclipsing, spectroscopic binary star system. a)
What information can be extracted from the light curve of the system? b)
What information can be extracted from the radial velocity curve?
c)
From what type of data is the radial velocity curve constructed, i.e., how are the radial velocities determined?
d)
Make a sketch of each type of graph, properly labeling each axis and identifying key quantities on each graph.
e)
How can this data be used to determine the following:
Page 1 of 3
i.
the total mass of the system ii.
the mass of each star
iii.
the semi-major axis iv.
the orbital velocity of each star
f)
How can the relative sizes of each star be determined from this data (or can it)?
Problem 2: A binary star system is composed of one star that is 2 times as massive as the other.
It has an orbital period of 250 yr and the two stars are 80.0 AU apart. Assume the orbit is circular and the orbit is in a plane parallel to the observer.
a)
What is the total mass of the system? Give your answer in solar masses.
b)
80.0x250kg c)
What is the mass of each star? Give your answer in solar masses
M
1
+ M
2
= a
3
/P
2
d)
What are the orbital velocities of each star in km/s?
200km/s
Problem 3 (3-point extra credit): Consider a binary system. Star A has a solar mass of 1.00. Star B has a solar mass of 12.0. Their mean separation is 60.0 A.U. a)
What is the position of the center of mass of the binary system relative to star A? b)
What is the position of the center of mass of the binary system relative to star B? c)
In years, what is the time period of rotation of the binary star system?
Part III: Summary (3 points)
Add your summary here. Be sure to include an overview of the detection method(s) including their limitations.
Example:
A binary system consists of two starts in circular orbit about a common center of mass, with an orbital period, P
orb
=
10
days
. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is v
1
=
20
km s
−
1
. Star 2 is an X-ray pulsar and its orbital radius about the center of mass is
r
2
=
3
×
10
12
cm
=
3
×
10
10
m
.
a)
Find the orbital radius, r
1
, of the optical star (Star 1) about the center of mass. v
1
=
2
π r
1
p
orb
r
1
=
P
orb
v
1
2
π
=
2.75
×
10
11
cm
b)
What is the total orbital separation between the two stars, r
=
r
1
+
r
2
?
r
=
r
1
+
r
2
=
2.75
×
10
11
+
3
×
10
12
=
3.3
×
10
12
cm
c)
Compute the total mass of the system, M
1
+
M
2
=
M
tot
.
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