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Binary Stars
Lecture 10
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LIFE ON A PLANET IN A BINARY STELLAR SYSTEM
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Importance of Binary Stars
• 75 % of all stars are “binary stars”
• Studies give:
– Stellar Masses (Visual & Spectroscopic)
– Stellar Radii (Eclipsing)
Types of binaries
• Visual Binary
– Stars are separated in a telescope.
• Spectroscopic Binary
– See two sets of spectral lines Doppler shifted due to orbital motion.
• Eclipsing Binary (rare)
– Stars cross in front of one another.
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- Both stars seen - Orbital motion observed
1980
1970
1960
2000
2010
2020
2030
Apparent Orbit - projected path on the sky.
Examples: Cen A & B Sirius A & B
Visual Binary
Time Ap
par
ent M
agn
itu
de
7
7.5
8
Schematic Light Curve
6
6
5
5
1
1
2
2
3
3
4
4 1 2 3
Examples: Algol Persei Lyrae
- Stars cross in front of one another
Eclipsing Binary
Simulation
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2 4 6 8 10 1212
12.2
12.4
12.6
12.8
13
M( ), .1
M( ), 15
M( ), 25
Ap
par
ent M
agn
itu
de
Time
0o
15o
25o
Star 1: R = 2 Rsun T = 5000 K Star 2: R = 4 Rsun T = 3500 K Separation 10 Rsun
Angles are the orbital inclination. 0o is edge-on.
Eclipsing Binary Light Curve
1
2
3
4
Rad
ial V
elo
city
1
2
2
3
4
4
Doppler shift due to orbital motions.
30 km/sec
60 km/sec
-10 km/sec
Time (Days) 0 140
4
Period
Example: Aurigae
Spectroscopic Binary
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Masses of Binary Stars
• Newton’s laws allow us to determine the total mass in a binary system.
• For star of mass MA and MB, the total mass is related to the period, P, and the average distance between the stars, a.
2
3
P
aMM BA
Kepler’s Laws of Planetary Motion
• They apply to binary stars
1. Law of Ellipses
2. Law of Areas
3. Harmonic Law
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1. Law of Ellipses
• Orbits are ellipses
Eccentricity
e=0.7
e=0.4
e=0.0 (circle)
2. Law of Areas
• Equal areas in equal time.
t(2)
Area 2 t(1)
Area 1
t(1) = t(2) => Area 1 = Area 2
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2. Law of Areas (continued)
• Orbital speeds
Slowest Fastest
Apastron
Periastron
3. Harmonic Law
– P = period of revolution in years
– a = mean separation in AU
– MA and MB = mass in solar masses
(Orbital Period)2 (Mean Distance)3
BA MM
aP
32
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Example:
• If a visual binary has a period of 32 years and an average separation of 16 AU then
M M
M M
A B
A B
16
32
3
2
4 Msun
M M
M M
A B
A B
16
32
3
2
4 Msun
Newton => Kepler !
• Deriving Kepler’s third law
• Force = mass x acceleration
F ma
Acceleration is change in velocity with time.
Force in Newtons.
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Circular Motion
F ma mr
v2
Accelerationv
2
rr
m v
This is the (centripetal) force necessary to keep the mass in a circular orbit.
Newton’s Gravity
F ma mr
v2
FGMm
r
2but
GMm
r
m
r2
2v
So that
v2 GM
r
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Putting it together
v or v
P
r r
P
2 2
The velocity and the orbital period, P, are related. The circumference is 2r, so
v
22 2
2
22 34 4
r
P
GM
rP
G
r
M
Thus
Kepler’s Third Law! Or P r2 3
v or v
P
r r
P
2 2
v
22 2
2
22 34 4
r
P
GM
rP
G
r
M
Orbital Elements
- Apparent Orbit - as seen on the sky
- Real Orbit
• Size of major axis
• Eccentricity of Orbit
• Period of Revolution
• Orbit orientation angle
• Inclination of Orbit
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Notes on binary masses
• Don’t mistake the mass for magnitude!
• P is measured in years, a in AU and mass in solar masses in this formula.
M MA Ba
P
3
2
We can only get the total mass.
Variable Star Classes
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Pulsating Variable Stars
Type Period Mag. Change Spectrum
Classical Cepheids (I) 2-80 days 1 F, G Supergiants
Type II Cepheids 1-100 days 1 F, G, K giants
Long Period Variables 90-600 days 3-6 M giants
RR Lyrae 1-24 hr 1 A, F giants
Canis Majoris 3-6 hr 0.1 B
Semiregular ~ 100 days 1 M
Irregular irregular 1-6 G, K, M
Period Luminosity Relation