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PHY2083 ASTRONOMY Gravity Multiple Stellar systems: binary stars Sir William Herschel (1802): Catalogue of 500 new Nebulae ... and Clusters of Stars; with Remarks on the Construction of the Heavens. “ the union of two stars, that are formed together in one system, by the laws of attraction.’’ Binary stars Modern definition: ‘Two or more stars in orbit about a common centre of mass’ ~50% of all stars are part of a multiple system Historical Digression Plato (c. 350 B.C.) suggested the need for a framework (e.g. stars revolve around the Earth which is fixed) “Geocentric Universe”: fixed relationship between stars Ptolemy (c. 100 A.D.) refined the system introduced (most notably) by Hipparchus to explain the observed motions of the stars and planets. Copernicus (1473-1543) proposed a heliocentric model of planetary motion. Simpler and more elegant. Model could order planets e.g. Mercury and Venus are never seen more than 28 and 47deg. E or W of the Sun => inside orbit of the Earth. This model, however, could not predict positions. Enter Tycho Brahe (1546-1601) and Johannes Kepler Planets placed on rotating epicycles which in turn moved on a larger deferent. This explained e.g. the brightnesses of the planets as their distances from the Earth changed. Equants resulted in the constant angular speed of the epicycle about the deferent. Good agreement! Fix model with increasing complexity. Kepler’s Laws Tycho Brahe, foremost observer of his time. Measured celestial objects to an accuracy of better than 4’. Discovered SN of 1572, and concluded that the heavens were not unchanging -- a belief propagated by the Church doctrine. Kepler sought a geometrical model of the universe that would be consistent with the best observations available (i.e. Tycho’s). Kepler’s Laws (old version) As soon as Kepler rejected the idea of purely circular motion, he was able to construct a model that was consistent with Tycho’s data. •A planet orbits the Sun in an ellipse with the Sun at one focus of the ellipse • A line connecting a planet to the Sun sweeps out equal areas in equal time intervals (see later). Orbital speed of a planet depends on its location in the orbit • The average orbital distance of a planet from the Sun (a) is related to its period (P) in the form P2 proportional to a3 Generalise Kepler’s Laws to 2 bodies orbiting each other Johannes Kepler (1571-1630) Isaac Newton (1643-1727) How can we find binary systems of stars? How can we find binary systems of stars? 3 common ways: • Visually • Spectroscopically • Astrometrically What is the difference? Optical double Gravitationally bound VISUAL BINARIES: Resolvable, generally nearby stars (parallax likely to be available) Relative orbital motion detectable over a number of years. Can provide information about the angular separation of stars from their common centre of mass. ASTROMETRIC BINARIES: Only one component seen e.g. if one member significantly brighter than the other. Deduce existence of unseen star from the motion of the visible star. Combination of motion around the centre of mass + proper motion gives rise to “wobble” on celestial sphere (see Fig.). Newton’s first law: constant velocity maintained by mass unless there is a force acting on it. SPECTROSCOPIC BINARIES (single-lined, double-lined) • Unresolved • If orbital period not too long, and if the orbital motion has a component along the line-of-sight then spectral lines will exhibit a periodic shift. • Double-lined: 2 sets of lines observed • Single-lined: only 1 set of lines observed e.g. due to large difference in luminosity. SPECTROSCOPIC BINARIES ELLIPSES and ORBITS An ellipse is defined by the set of points that satisfies the following equation: r + r" = 2a Relation between! a,b,e: a = semi-major axis b = semi-minor axis e = eccentricity 0≤e<1 F,F’ = focal points b 2 = a 2 (1" e 2 ) ! Convenient to express orbit in polar coordinates, with r being the distance from the principal focus (see fig.) r"2 = r 2 sin 2 # + (2ae + r cos # ) 2 r"2 = r 2 + 4ae( ae + r cos # ) $r= ! a(1% e 2 ) 1+ ecos# (0 & e < 1) Kepler’s 1st Law: orbits are ellipses Both objects in a binary orbit move about the centre of mass in ellipses, with the centre of mass occuping one focus of the ellipse. Kepler’s 1st Law Sun a θ r Kepler s Second Law: equal areas in equal time Area swept out by a line b/w a planet and the focus is always the same for a given time interval. v1t1 = v2t2 v2t2 µ" ! A2 m1m2 m1 + m2 ! Area 1 = Area 2 A1 v1t1 dA 1 L = dt 2 µ (reduced mass) Rate of change of area swept ! by a line connecting a body to the focus of its elliptical orbit = constant = 0.5 x orbital angular momentum per unit mass Kepler’s 2nd Law δθ ∆A vt v Quick and dirty ‘derivation’ of Kepler’s third law (see also notes) v2 F = ma = m r GMm F= 2 r GM v2 = r 4 " 2 r 2 GM 2 v = = P2 r 4" 2 r 3 2 #P = GM ! Force necessary to keep mass in a circular orbit Using P = 2r/v Kepler’s 3rd law! Generalised form of Kepler s third law: 4" 2 2 P = a3 G(m1 + m2 ) P " a3 and N.B. for P in Earth years, a in AU, m in solar masses 4!2/G = 1 P " 1/ M tot (Tycho s data for solar system only) ! ! Kepler s laws: applicable to planets, binary systems, galaxy-galaxy orbits. ! Period + semi-major axis => mass Example: The orbital period of Io, one of the 4 Galilean moons of Jupiter, is 1.77 days, and the semi-major axis of its orbit is 4.22x108m. Estimate the mass of Jupiter, stating any assumptions that you made. Solution: Apply Kepler’s third law. Answer = 1.9x1027 kg ~ 318 Earth masses. Assumed that mass of Io is negligible c.f. mass of Jupiter. G = 6.67300 × 10-11 m3 kg-1 s-2 1.5 x10 11m = 1AU ans: 1.9 x 10^27 kg ~ 318 M_earth Mean Orbital Velocity (BB!) v∝ Earth Mars Jupiter � 1 a Orbital velocity v= � G(m1 + m2 ) � a � GM⊙ a Basic data of the 8 planets in the solar system Earth yrs Kepler’s 3rd Law Basic data of the 8 planets in the solar system P2 (Earth years) = a3 (in AU) Main points from table: 1) The orbits of all planets (except Mercury) are nearly circular (i.e. eccentricity ~ 0) 2) The further a planet is from the sun, the longer its orbital period i.e., in agreement with Kepler’s 3rd law. (The small deviations of P2/a3 for Uranus and Neptune are due to the gravitational forces between these 2 planets).