Lecture 5 Binary stars. 85% of all stars in the Milky Way are part of multiple systems (binaries, triplets or more) Some are close enough that they are

Lecture 5Lecture 5

Binary stars

Binary starsBinary stars

•85% of all stars in the Milky Way are part of multiple systems (binaries, triplets or more)

•Some are close enough that they are able to transfer matter through tidal forces. These are close or contact binaries.

ExamplesExamples

Two stars are separated by 3 A.U. One star is three times more massive than the other. Plot their orbits for e=0.

Example: binary star systemExample: binary star system

Two stars orbit each other with a measurable period of 2 years. Suppose the semimajor axes are measured to be a1=0.75 A.U. and a2=1.5 A.U. What are their masses?

1

2

1

2

2

1

a

a

r

r

m

m 21

322 4

mmG

aP

Visual binaries: mass determinationVisual binaries: mass determination

A perfect mass estimate of both stars is possible if:1. Both stars are visible2. Their angular velocity is sufficiently high to allow a

reasonable fraction of the orbit to be mapped3. The distance to the system is known (e.g. via parallax)4. The orbital plane is perpendicular to the line of sight

Example: SiriusExample: Sirius

Sirius A and B is a visual binary: a period of 49.94 yr a parallax of p=0.377” The angular extent of its semimajor axis is

=A+B=5.52”. aA/aB=0.466

Assume the plane of the orbit is in the plane of the sky:

1

2

1

2

2

1

a

a

r

r

m

m 21

322 4

mmG

aP

R

a

R is the distance tothe star.

Visual binaries: inclination effectsVisual binaries: inclination effects

In general the plane of the orbit is not in the plane of the sky.

Here is the true orbit

Focii



Here is the true orbit, which defines the orbital plane



Now imagine this plane inclined against the plane of the sky with angle i:

i



Now imagine this plane inclined against the plane of the sky with angle i:

i

Tru

e m

ajor

axi

s=2a

2acosi

Instead of measuring a semimajor axis length a, you measure acosiwhere i is the inclination angle


This projection distorts the ellipse: the centre of mass is not at the observed focus and the observed eccentricity is different from the true one.

This makes it possible to determine i if the orbit is known precisely enough


In practice we don’t measure a physical distance a, but rather an angular distance that we’ll call . If is the true angular distance, and is the measured (projected distance) then:

1

2

1

2

1

2

2

1

cos

cos

i

i

m

m

So the ratio of the masses is independent of the inclination effect

33

2

2

2

32

2

32

21

cos

4

4

4

i

R

GP

GP

R

GP

ammHowever, the sum of the masses is

not:

R

a cosi

ExampleExample

How does our answer for the mass of Sirius A and B depend on inclination?

pccos

101.7

cos5

i

i

Ra

R

a cosi

i

2acosi

Sirius A and B is a visual binary: a period of 49.94 yr a parallax of p=0.377” The angular extent of its observed semimajor axis is

=5.52”. A/B=0.466

ExampleExample

How does our answer for the mass of Sirius A and B depend on inclination?

m cos

101.5

/1

466.0

12

iaa

aa

a

a

m

m

BAB

B

A

A

B

ABB aa

a

GPm

/1

4 3

2

2

SunB Mi

m3cos

40.0 SunBA M

imm

3cos

84.01.2

pccos

101.7

cos5

i

i

Ra

Thus our answers are a lower limit on the mass of these stars. The measured inclination is actually i=43.5 degrees. So cos3i=0.38 and mA=2.2 Msun, mB=1.0 Msun

BreakBreak

Spectroscopic binariesSpectroscopic binaries

Single-line spectroscopic binary: the absorption lines are redshifted or blueshifted as the star moves in its orbit

Double-line spectroscopic binary: two sets of lines are visible

Java applet: http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm

1 if

zzc

v

z

r

restrest

restobs

http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm

Spectroscopic binaries: circular orbitsSpectroscopic binaries: circular orbits

•If the orbit is in the plane of the sky (i=0) we observe no radial velocity.

•Otherwise the radial velocities are a sinusoidal function of time. The minimum and maximum velocities (about the centre of mass velocity) are given by

ivv

ivv

r

r

sin

sin

2max2

1max1

Spectroscopic binaries: circular orbitsSpectroscopic binaries: circular orbits

•We can therefore solve for both masses, depending only on the inclination angle i

iG

vvPvm

iG

vvPvm

rrr

rrr

3

2max2

max1

max1

2

3

2max2

max1

max2

1

sin2

sin2

•In general it is not possible to uncover the inclination angle. However, for large samples of a given type of star it may be appropriate to take the average inclination to determine the average mass.

Spectroscopic binaries: non-circularSpectroscopic binaries: non-circular

If the orbits are non-circular, the shape of the velocity curves becomes skewed in a way that depends on the orientation

e.g. e=0.4, i=30°,

axis rotation=45°

• A sinusoidal light curve means orbits are close to circular• From analysis of light curve it is possible to determine the eccentricity and

orbit orientation, but not the inclination. • In practice most orbits are circular because tidal interactions between the

stars tend to circularize the orbits

Java applet: http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm

http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm

Single-lined spectroscopic binariesSingle-lined spectroscopic binaries

In general, one star is much brighter than the other (remember faint stars are much more common than bright stars). This means only one set of absorption lines is visible in the spectrum.

The Doppler motion of this single set of lines still indicates the presence of a binary system.

We can still solve for a function of the two masses:

G

vPi

mm

m r

2sin

3max13

221

32

This is the mass function

Eclipsing binariesEclipsing binaries

A good estimate of the inclination i can be obtained in the case of eclipsing binaries, separated by distance d:

d

RRi 21cos

If d » R1+R2 (which is usually the case) then i~90 degrees

i

R1+

R2

d

To observer


A good estimate of the inclination i can be obtained in the case of eclipsing binaries, separated by distance d:

d

RRi 21cos

If d » R1+R2 (which is usually the case) then i~90 degrees

i

R1+

R2

d

To observer

Assume i=90 degrees when in reality i=75 degrees. What is the error in sin3i?

sin3(75)=0.9

So the error on the masses is only 10% if d > 3.9(R1+R2)


In the system just described, the eclipse just barely happens:To observer

Face on

So the amount of light blocked is not constant, and the light curve (total brightness as a function of time) looks something like this:


However, in the case of total eclipse the smaller star is completely obscured. In this case it is even more likely that the inclination is close to 90 degrees

To observer

Face on

And the light curve shows constant minima:


In the case of a total eclipse we can also measure the radii of the stars, and the ratio of their effective temperatures

If we assume i~90 degrees and circular orbits that are large relative to the stellar radius, then the radius of the smaller star is:

abs ttv

r 2

And for the larger star: acl tt

vr

2

Where v is the relative velocity between the two stars


Ratio of effective temperatures

4242llsstotal TRTRL

421 ll TRL

424222 sslsl TRTRRL

4

42

42

2

1

l

s

ls

ss

total

total

T

T

TR

TR

LL

LL

Note that (1) will be the deepest minimum if Ts>Tl.(often the case since the brightest, largest stars are the cool supergiants)

Alternatively (2) will be the deepest minimum if Tl>Ts

424 eTRL

(0)

(1)

(2)

Stellar massesStellar masses

•For select star systems, we can therefore measure the mass directly.

•Luminosity is closely correlated with stellar mass

Energy production rate is related to stellar mass

If the available energy is proportional to mass, how do stellar lifetimes depend on their main sequence location?

5.2ML

5ML

The main sequence revisitedThe main sequence revisited

•The main sequence is a mass sequence

More massive stars are closer to the top-left (hot and bright)

M=30MSun

M=MSun

M=0.2MSun

The main sequence revisitedThe main sequence revisited

•The main sequence is a mass sequence

More massive stars are closer to the top-left (hot and bright)

•Stars on the main sequence have radii 1-3 times that of the Sun

Supergiants have R>100 RSun

White dwarfs have R~0.01 RSun

M=30MSun

M=MSun

M=0.2MSun

DensitiesDensities

Since we know the stellar masses and radii, we can compute their average densities

Sun:

3

38

30

3

kg/m 1409

m1096.63/4

kg1099.1

3/4

R

M

Recall water has a density of 1000 kg/m3

Dry air at sea level: 1.3 kg/m3

DensitiesDensities


Supergiants (Betelgeuse):

353

3

kg/m 1041.11000101409

1000

10

SunSunSun

Sun

Sun

R

R

M

M

RR

MM

or 1/100,000 times less dense than air.

DensitiesDensities


White Dwarfs (Sirius B):

Sun

Sun

Sun

RR

MM

5106

01.0

6.0

or 850,000 times denser than water.

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Lecture 5 Binary stars. 85% of all stars in the Milky Way are part of multiple systems (binaries, triplets or more) Some are close enough that they are