binary star systems - inside mines

PHGN324: Binary star systemsFred Sarazin ([email protected])Physics Department, Colorado School of Mines

Binary star systemsHow to get a lot more information on the stars in those systems


• More than 50% of all stars in our Milky Way are not single stars, but belong to binary star (or multi-star) systems.

• These stars orbit their common center of mass.

• If we can measure and understand their orbital motion, we can estimate the mass of the stars in that system.

Binary (or multi) star systems


Center of mass

• Center of mass = balance point of the system

• If MA = MB, then rA = rB

• More generally, !"!#

= $#$"

(or MA rA

= MB rB).

• The lighter star of the binary system has a larger orbit than its heavier companion.


Sirius A and B

• Visual binaries can be seen as two points of light. Over many years they move with respect to one another, hence one can determine their orbits.


Visual binary orbit measurementsSirius A and B (schematic)


Estimating stellar masses

• Recall Kepler’s 3rd law of motion:

• For the planetary motion, we assumed mplanet<<M☉. In this case, M and m are the mass of the two companions.

• Simplification (change of units):

where a is the average separation between the two stars.

T 2

a3=

4πG(M +m)

≈4πGM

𝑎&(𝑖𝑛 𝐴𝑈 )𝑇/(𝑖𝑛 𝑦 ) = 𝑀 +𝑚(𝑖𝑛 𝑀⊙ )

2


Exercise I

T 2

a3=

4πG(M +m)

≈4πGM

• Go from: to:𝑎&(𝑖𝑛 𝐴𝑈 )𝑇/(𝑖𝑛 𝑦 ) = 𝑀 +𝑚(𝑖𝑛 𝑀⊙ )

2


Exercise II

• A binary system has a period of 32 years and an average separation of 16 AU. What is the total mass of the star system?

• Star A is 12 AU from the center of mass. What are the masses of stars A and B?


Tilted orbit

• If the tilt angle i is 0 and if the distance of the system from Earth is d, then we can measure a1 and a2 (a = a1+a2) by measuring the angles subtended by the semi-major axes.

• Since m1a1=m2a2, then

• If the tilt angle i is not 0, then the mass ratio is:

• If one doesn’t know i, then the mass ratio can only be approximated

𝛼7 =𝑎7𝑑

𝛼/ =𝑎/𝑑and

𝑚7𝑚/

=𝛼/𝛼7

𝑚7𝑚/

=𝛼9/𝛼97

=𝛼/ cos 𝑖𝛼7 cos 𝑖

We assume here a tilt only along the semi-major axis. There could also be a tilt along the semi-minor axis.


Tilted orbit – effect on Kepler’s 3rd law

• Recall Kepler’s 3rd law of motion:

• From the previous slide, we have: 𝑎 = 𝑎7 + 𝑎/ = 𝛼7 + 𝛼/ 𝑑 = 𝛼𝑑

• If one considers the projection effect: 𝑎 = =>?@AB C

+ =>D@AB C

𝑑 = =>@AB C

𝑑 using 𝛼9 =𝛼97 + 𝛼9/. We get:

• In principle, careful measurements of the orbits over some time allow for the estimation of the angle i.

T 2

a3=

4πG(M +m)

≈4πGM

2

𝑚7 +𝑚/ =4𝜋/

𝐺𝛼𝑑 &

𝑇/ =4𝜋/

𝐺𝑑cos 𝑖

& 𝛼9&

𝑇/


Spectroscopic binaries

• Measurement of a, the average separation, is difficult because the stars are too close to each other. How is the measurement carried out?

• Spectroscopy of the binary system: the approaching star produces blue-shifted lines, while the receding star produces red-shifted lines in the spectrum.

• Doppler shift measurements allow for the determination of the radial velocities and the orbital period.

• From this, we should be able to deduce the orbital circumference (𝐶 = ∫ 𝑣 𝑡 𝑑𝑡L

M ) and then the average separation a.

• Not quite however because we don’t know the inclination of the orbit with respect to our line of sight.


Example

Line spectrum for Mizar at two different times

“Big Dipper”(part of Ursa

Major)

• Because of the uncertainties on the tilt of the orbits with respect to our line of sight, only a limit of a can be obtained.


Special case: eclipsing binaries

• Eclipsing binaries occur when we look at the system edge-on (i.e. angle i ≈ 90º)

• Characteristic “double-dip” light curve

VW Cephei


Algol in the constellation of Perseus.

• From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane.

Special case: eclipsing binaries


Timing of the eclipse periods (example 1)

Example 1:

i ≈ 90º - mL>mS, but TL (or LL) <TS (or LS)

• Star S is moving in front of star L:• Partial: Dt12 = t2-t1 ≈ Dt34=t4-t3• Full: Dt23 = t3-t2

• Star S is moving behind star L:• Partial: Dt’12 = t’2-t’1 ≈ Dt’34=t’4-t’3• Full: Dt’23 = t’3-t’2 ≈ Dt23

• Primary “eclipse” has higher loss of brightness compared to secondary “one” because star S is brighter than star L (even if star S is bigger).

mLmS

Star L (for Large)Star S (for Small)



• Dt12 = t2-t1 (for example) can be used to measure the radius of the smaller (brighter) star.

where v is the relative velocity of the two stars (v = vs + vL).

• Similarly, Dt24 = t4-t2 can be used to measure the radius of the larger star.

𝑟O =𝑣2 (𝑡/ − 𝑡7)

𝑟R =𝑣2 𝑡S − 𝑡/ = 𝑟O +

𝑣2 𝑡& − 𝑡/

mLmS



mL

mS

Example 2:

i < 90º - mL>mS, but TL (or LL) <TS (or LS)

• Because we don’t see the binary system “edge-on”, the binary system is only partially eclipsing leading to a less well-defined timing structure than in the previous case.

• The parameters discussed before are harder to measure.


Brightness variation

mLmS

• Recall that the radiative flux per unit of area is given by: 𝑅 = 𝜎𝑇S .

• The light we receive from each star comes from a cross section of the star (𝜋𝑟/)

• Hence, when the two stars are apart:𝐵M ∝ 𝜋𝑟R/𝜎𝑇RS + 𝜋𝑟O/𝜎𝑇OS

• Primary minimum (only star L):𝐵Y ∝ 𝜋𝑟R/𝜎𝑇RS

• Secondary minimum (star S eclipses part of star L):

𝐵O ∝ 𝜋𝑟R/ − 𝜋𝑟O/ 𝜎𝑇RS + 𝜋𝑟O/ 𝜎𝑇OS

𝐵M − 𝐵Y𝐵M − 𝐵O

=𝑇O𝑇R

S

Ratio of effective temperatures of the two stars


Exercise

For a binary system, photometric observations show that:• At maximum light, the apparent magnitude is: m0=6.3• At the primary minimum, the apparent magnitude is: mP=9.6• At the secondary minimum, the apparent magnitude is: mS=6.6

Calculate the relative temperature of the two stars (L and S) in the system.


PREVIEW: Close binary star systems

• If the two stars are close to each other, the tidal forces can considerably deform one or both stars. In the case depicted above, the outer layer of a giant star for example can reach a point where some material can fall under the gravitational influence of the other smaller but denser star (such as a white dwarf, a neutron star or even a black hole).


Discovery of exoplanets

• Exoplanets can be discovered by applying the same eclipsing binary framework described in the previous slides.

• Looking for the (very) small change of the host star light curve due to the transit of an exoplanet across its apparent surface.

https://www.youtube.com/watch?v=ku7YjMol1k4

Transit of Venus across the Sun - light curve


“Tatooine” exoplanets

• Kepler-47b and Kepler-47c

• Kepler-47c in the “Goldilock” (habitable) zone

binary star systems - inside mines

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