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Binary How to get a lot more information on the in those systems

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: systems Binary (or multi) star systems

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

• These stars their common .

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

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary 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 has a larger orbit than its heavier companion.

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems A and B

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

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems orbit measurements Sirius A and B (schematic)

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Estimating stellar masses

• Recall Kepler’s 3rd law of motion: T 2 4π 2 4π = ≈ a3 G(M + m) GM

• For the planetary motion, we assumed mplanet<

• Simplification (change of units): � (�� �� ) = � + � (�� � ) � (�� � ) ⊙ where a is the average separation between the two stars.

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Exercise I

T 2 4π 2 4π � (�� �� ) • Go from: = ≈to: = � + � (�� � ) a3 G(M + m) GM � (�� � ) ⊙

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems 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 A is 12 AU from the center of mass. What are the masses of stars A and B?

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Tilted orbit

• If the tilt angle i is 0 and if the distance of the system from is d,

then we can measure a1 and a2 (a = a1+a2) by measuring the angles subtended by the semi-major axes.

� � � = and � = � � � � • Since m1a1=m2a2, then = � � • If the tilt angle i is not 0, then the mass ratio is: � � � cos � = = � � � cos � • If one doesn’t know i, then the mass We assume here a tilt only along the semi-major axis. ratio can only be approximated There could also be a tilt along the semi-minor axis.

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Tilted orbit – effect on Kepler’s 3rd law

• Recall Kepler’s 3rd law of motion: T 2 4π 2 4π = ≈ a3 G(M + m) GM

• From the previous slide, we have: � = � + � = � + � � = ��

• If one considers the projection effect: � = + � = � using � = � + �. We get: 4� �� 4� � � � + � = = � � � cos � �

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

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

• From this, we should be able to deduce the orbital circumference (� = ∫ � � ��) 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.

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

“Big Dipper” (part of Ursa Major)

Line spectrum for at two different times • 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.

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

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Special case: eclipsing binaries in the of . • 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.

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Timing of the eclipse periods (example 1)

Example 1: Star L (for Large) Star S (for Small) i ≈ 90º - mL>mS, but TL (or LL)

mL mS • 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

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

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Timing of the eclipse periods (example 1)

• Dt12 = t2-t1 (for example) can be used to measure the radius of the smaller (brighter) star. � � = (� − � ) 2

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. � � � = � − � = � + � − � 2 2

mL mS

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Timing of the eclipse periods (example 2)

Example 2: mS i < 90º - mL>mS, but TL (or LL)

• The parameters discussed before are harder to measure.

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Brightness variation

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

mL mS • The light we receive from each star comes from a cross section of the star (��)

• Hence, when the two stars are apart: � ∝ �� �� + �� ��

• Primary minimum (only star L): � ∝ ���� � − � � = • Secondary minimum (star S eclipses part � − � � of star L): � ∝ �� − �� �� + �� �� Ratio of effective temperatures of the two stars

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

For a binary system, photometric observations show that:

• At maximum light, the apparent is: m0=6.3 • At the primary minimum, the 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.

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems 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 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 , a or even a ).

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

• 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 across its apparent surface.

Transit of Venus across the - light curve

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

Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems “Tatooine” exoplanets

• Kepler-47b and Kepler-47c

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

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