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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