Binary star systems How to get a lot more information on the stars in those systems Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Binary (or multi) star 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. 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 binary system has a larger orbit than its heavier companion. Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems 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. Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Visual binary 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<<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. 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 system? • 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 Earth is d, then we can measure a1 and a2 (a = a1+a2) by measuring the angles subtended by the semi-major axes. �7 �/ � = and �/ = 7 � � �7 �/ • Since m1a1=m2a2, then = �/ �7 • If the tilt angle i is not 0, then the mass ratio is: � �9 � cos � 7 = / = / �/ �97 �7 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: � = �7 + �/ = �7 + �/ � = �� => => => • If one considers the projection effect: � = ? + D � = � using �9 = @AB C @AB C @AB C �97 + �9/. We get: 4�/ �� & 4�/ � & �9& � + � = = 7 / � �/ � 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 orbital period. • From this, we should be able to deduce the orbital L circumference (� = ∫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. Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems Example “Big Dipper” (part of Ursa Major) Line spectrum for Mizar 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 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. 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) <TS (or LS) 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 • 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). 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. � � = (� − � ) O 2 / 7 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. � � � = � − � = � + � − � R 2 S / O 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) <TS (or LS) mL • 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. 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: � = ��S . mL mS • The light we receive from each star comes from a cross section of the star (��/) • Hence, when the two stars are apart: / S / S �M ∝ ��R ��R + ��O ��O • Primary minimum (only star L): � ∝ ��/��S S Y R R �M − �Y �O = • Secondary minimum (star S eclipses part �M − �O �R of star L): � ∝ ��/ − ��/ ��S + ��/ ��S Ratio of effective temperatures of O R O R O O 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 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. 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 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). Fred Sarazin ([email protected]) Physics Department, Colorado School of Mines PHGN324: Binary star systems 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. Transit of Venus across the Sun - 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.