The HR Diagram. L f 2 L d2 N obj V d 3 N obj L3/2. Most (>90%) stars lie
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1 The HR Diagram Most (>90%) stars lie on the main sequence. A few stars are cool and extremely bright, so, by L = 4 π R 2 σ T 4, they must be extremely large. A few stars are hot, but extremely faint, so they must be very small.
2 The HR Diagram Most (>90%) stars lie on the main sequence. (But most stars with names you have heard of are giants.) You can see giants much further away! L f 2 L d2 d N obj V d 3 N obj L3/2
3 The HR Diagram The main sequence in the UBV 2-color diagram. The kink is due to the effects of the Balmer jump.
4 The HR Diagram with Iso-Radius Lines
5 For Reference 1 R 8 = cm 1 R = cm = R 8 1 A.U. = cm = 215 R 8
6 Determining Stellar Radii Stellar radii can be estimated by combining measurements of absolute luminosity with temperature. But the uncertainties in each parameter will propagate! Since virtually all stars are unresolvable even at HST (and JWST resolution), there are only a few methods that can directly measure a star s radius: Lunar occultations (for angular diameters) Objects must be bright and within 5 of the ecliptic plane Interferometry (for angular diameters) Objects must be bright Better in the IR, where the atmosphere is better behaved Baade-Wesselink (for pulsating stars) Eclipsing binary stars
7 Lunar Occultations The Moon orbits the Earth in ~ 29.5 days, i.e., at a rate of ~ 0.5 /sec. At ~ 10 pc, the angular diameter of the Sun is ~ arcsec, which translates into an occultation timescale of a few millisec. Stars must be bright and within 5 of the ecliptic plane.
8 Interferometry There are several types of interferometry: Amplitude (Michelson) interferometry Combines the light from two slits (or telescopes) whose separation can be adjusted; object size can be derived by observing the fringes as a function of image separation. The technique requires matching the path-lengths to a fraction of a wavelength; since atmospheric turbulence destroys image coherence, this technique has not produced many results.
9 Interferometry There are several types of interferometry: Amplitude (Michelson) Intensity (Hanbury Brown) At any instant, light from the source produces an interference pattern, which constantly changes (on femtosecond timescales). The spatial scale for the interference goes as ~ λ/θ, so, at any instant, two detectors separated by less than λ/θ will have their noise correlated, while detectors separated by distances greater than λ/θ will have uncorrelated noise. Thus, θ can be estimated without phase information.
10 Interferometry There are several types of interferometry: Amplitude (Michelson) Intensity (Hanbury Brown) Speckle The seeing produced by the atmosphere is the superposition of many diffraction-limited images. (In other words, much of the blurring is due to rapid image motion.) By taking a series of very short e x p o s u r e s, a n d t h e n correlating the images, one can approach the diffraction limit of the telescope. (Adaptive optics also does this.)
11 Binary Stars Most direct measurements of stellar radii and stellar masses come from the analysis of binary stars. There are several types of binaries: Visual
12 Binary Stars Most direct measurements of stellar radii and stellar masses come from the analysis of binary stars. There are several types of binaries: Visual Spectrum
13 Binary Stars Most direct measurements of stellar radii and stellar masses come from the analysis of binary stars. There are several types of binaries: Visual Spectrum Astrometric 99 Her
14 Binary Stars Most direct measurements of stellar radii and stellar masses come from the analysis of binary stars. There are several types of binaries: Visual Spectrum Astrometric Spectroscopic
15 Binary Stars Most direct measurements of stellar radii and stellar masses come from the analysis of binary stars. There are several types of binaries: Visual Spectrum Astrometric Spectroscopic Eclipsing
16 Binary Stars Most direct measurements of stellar radii and stellar masses come from the analysis of binary stars. There are several types of binaries: Visual Spectrum Astrometric Spectroscopic Eclipsing Reflection
17 Obtaining Stellar Masses There are basically 3 techniques that can be used to measure stellar masses: Analysis of stellar absorption lines (pressure broadening, etc.) One measures (log) g = G M / R 2 =4 π σ G M T 4 / L Asteroseismology of non-radial pulsators Binary Stars
18 General Information about Binary Stars Perhaps ~ 85% of all stars in the Milky Way are part of multiple systems (binaries, triplets or more) The periods (separations) of these binaries span the entire range of possibilities, from contact binaries to separations of ~ 0.1 pc. The distribution of separations appears roughly log-normal.
19 General Information about Binary Stars Perhaps ~ 85% of all stars in the Milky Way are part of multiple systems (binaries, triplets or more) The periods (separations) of these binaries span the entire range of possibilities, from contact binaries to separations of ~ 0.1 pc. The distribution of separations appears roughly log-normal. The possible mass ratios of binary stars span the entire range of masses. There is a possible preference for mass ratios of near 1, but selection effects are important!
20 The Motions of Binary Stars The motions of binary stars are controlled by Kepler s laws: The orbits are ellipses with a star at a focus The orbits sweep out equal areas in equal times (M 1 + M 2 ) P 2 = a 3
21 The Motions of Binary Stars The motions of binary stars are controlled by Kepler s laws: The orbits are ellipses with a star at a focus The orbits sweep out equal areas in equal times (M 1 + M 2 ) P 2 = a 3 The separations and motions of the two stars in the center-of-mass frame are given by M 1 = a 2 = r (t) 2 M 2 a 1 r 1 (t) = v (t) 2 v 1 (t)
22 Kepler s First Law The orbit of one body, with respect to the other, is a conic section with one of the bodies at a focus. For bound systems, the conic system is an ellipse. (An ellipse is a shape defined by 2 focii; for any point on the ellipse, the sum of the distances to the 2 focii is constant.) a = semi-major axis b = semi-minor axis b/a = ellipticity ε = eccentricity ε = $ 1 & b' % a( 2 )
23 Ellipses Distance from center to focus of ellipse: a ε True Anomaly: θ Periapsis: a (1 ε) Apapsis: a (1 + ε)
24 Cartesian coordinates: Equations of an Ellipse " $ # x a % ' & 2 " + y % $ ' # b & 2 =1 Polar coordinates (more useful): r = a ( 1 ε 2 ) 1+ε cosθ External parameterization (handy for graphics): x = a cosξ y = b sinξ
25 Kepler s Second Law The line connecting the two bodies sweeps out a constant amount of area per unit time. (In other words, the system conserves angular momentum.) The angular momentum per unit mass is da = 1 2 r v dt = 1 2m r m v dt = 1 2m L dt
26 Kepler s Second Law The line connecting the two bodies sweeps out a constant amount of area per unit time. (In other words, the system conserves angular momentum.) The angular momentum per unit mass is r v dθ = r 2 dt = 2π ab P = 2π a2 ( 1 ε 2 ) 1/ 2 P
27 Kepler s 3 rd Law (as modified by Newton) The orbital period squared times the total mass of the system is proportional to the mean separation of the two bodies cubed. a 3 = k P 2 ( M 1 + M ) 2 where k = G/4π 2. Note, however, that if distance is measured in A.U., period in years, and mass in solar masses, then from scaling to the Earth-Sun system, k = 1. In any binary, the brighter of two stars is called the primary. It is usually (but not necessarily) also the most massive.
28 Defining an Orbit Seven (or six) orbital elements define a body s orbit The shape and size of the ellipse: The semi-major axis (a) The eccentricity (ε) The direction of the plane of the ellipse: The inclination (i) Longitude of ascending node (Ω) The orientation of the ellipse s major axis: The argument of periapsis (ω) The location and speed of the object: Time of periapsis (T p ) or mean anomaly at epoch (M 0 ) The period (P) if not in the Solar System In the Solar System, a implies P (since the mass is known). You therefore only have 6 variables (and thus, at minimum, you need 6 observations to define an orbit).
29 Defining an Orbit Note: The position of an orbiting body is most easily described in polar coordinates via an angle. But usually, you want position versus time.
30 Relating Angles to Time An anomaly is the angle with respect to the periapsis. The true anomaly (θ) is the angle to the object from the focus. The mean anomaly (M) is the angle from the center of the ellipse to a fictitious object traveling in a circular orbit with radius, a, and period, P, i.e., ( t T ) p M = 2π P The eccentric anomaly (E) is the angle from the center of the ellipse to the point on a circle with radius a which intersects the perpendicular from the semi-major axis though the object. t(θ) is hard to work with, but t(m) is trivial. t(e) connects the two. θ
31 Relating the True and Eccentric Anomaly The true anomaly, θ, can be related to the eccentric anomaly, E, using the geometry of triangles. b sin E θ r = a( 1 ε cos E) cos E ε cosθ = 1 ε cos E ( sinθ = 1 ε 2 ) 1/ 2 sin E 1 ε cos E tan 2 ( θ /2) = 1+ε ( 1 ε tan2 E /2)
32 Relating the Mean and Eccentric Anomaly The eccentric anomaly can be computed by noting that the area of an ellipse sector is equal to that of a similar sector of a circumscribed circle, reduced by b/a. In other words Area(SPX) = b a Area(SPʹX) = b a Area(C P ʹ X) Area(ΔC P ʹ S) = b π a 2 E 1 a 2π 2 aε = ab 2 Since ( E ε sin E) Area(SPX) = πab t T p P M = E ε sin E ( )( asin E) This is Kepler s Equation = ab 2 M
33 Relating Time to Position To obtain an object s position versus time Time can trivially be converted to the mean anomaly M = 2π P ( t T ) p Mean anomaly can be converted to eccentric anomaly via Kepler s equation M = E ε sin E The true anomaly can be found from the mean anomaly via geometric relations, i.e., tan 2 ( θ /2) = 1+ε ( 1 ε tan2 E /2) Finally, the radius vector is found using r = a 1 ε 2 ( ) 1+ε cosθ
34 Binary Star Velocities Note the combination of Kepler s 1 st and 2 nd law defines the velocity of the binary at any moment. For instance, at the orbit s apapsis, r = a (1 + ε), and the radius and velocity vectors are perpendicular, so sin φ = 1 and dθ r 2 dt = rv sinϕ = rv = a(1+ε)v = 2π a2 1 ε 2 ap ap ap P So Kepler s 3 rd law gives v ap = 2π a P Similarly, at periapsis 1 ε 1+ε = G(M + M ) 1 2 a 1 ε 1+ε ( ) 1/ 2 v peri = 2π a P 1+ε 1 ε = G(M 1 + M 2 ) a 1+ε 1 ε φ φ
35 Vis Viva Equation To obtain the velocity at any point in the orbit, just use energy conservation: so 1 2 v 2 GM T r = 1 2 v 2 peri v 2 (r) = G M 1 + M 2 GM T r peri = 1 2 # ( ) % 2 r 1 $ a = GM T a # % $ = GM T 2a GM T a &# ( 1+ε & % ( GM T ' $ 1 ε ' a(1 ε) 1 # 1+ε % 1 ε $ 2 1 & ( ' # 1 ε & % ( = GM T $ 1 ε ' 2a ( ) & ( where r = a 1 ε 2 ' 1+ε cosθ
36 Visual Binaries A perfect mass estimate of both stars is possible if: Both stars are visible Their angular velocity is sufficiently high to allow a reasonable fraction of the orbit to be mapped The distance to the system is known (e.g., via parallax) The orbital plane is perpendicular to the line of sight
37 Periods of Visual Binaries Note that for a solar-type visual binary at 10 pc, an α = 1 separation corresponds to a physical separation of 10 A.U. and a period of 33 years. More generally, for a distance d, P d 3/2 α 3/2. So the periods can be very long.
38 Visual Binaries One difficulty with visual binaries is that, in general, you don t know the orbital inclination. (Is the orbit elliptical or an inclined circle?) There is the true ellipse and an apparent ellipse, but the focus of the true ellipse is not the focus of the apparent ellipse. ε =0.5 i = 45 ε =0.5 face-on For visual binaries, you measure a cos i
39 Masses from Visual Binaries True major axis=2a i 2a cos i The projection distorts the ellipse: the center of mass is not at the observed focus and the observed eccentricity is not the true eccentricity. Only with long-term, precise observations can you determine the true orbit.
40 Masses from Visual Binaries The observed angular separation of a binary at a distance d will be α = d α 0 cos i. Relative masses will be unaffected by inclination M 1 M 2 = a 2 a 1 = r 2 r 1 = dα 2 dα 1 = αʹ d cos i 2 α 1 ʹ d cos i = α 2 However, our measurement of total mass will only be a lower limit. α 1 ( ) 3 M 1 + M 2 = a3 k P = dαʹ 2 k P 2 ( = dα 0 / cos i) 3 = k P 2 d cos i 3 α0 3 k P 2 The masses derived from visual binaries will depend on distance cubed and cos 3 i.
41 Spectroscopic Binaries Orbits in the plane of the sky (i = 0 ) show no radial velocity. In general, v obs = v true sin i. i =0 i =30 i =60
42 Separations of Spectroscopic Binaries For a solar-type star at ~ 10 pc, a ~ 1 year period corresponds to a separation of 0.1, and a 1 km/s velocity corresponds to ~ 5 A.U. Most spectroscopic binaries are therefore unresolved.
43 Spectroscopic Binaries Spectroscopic binaries have the opposite problem as visual binaries: we can observe stellar motions in the radial direction, but not in the plane of the sky. If z is the (Cartesian) line-of-sight coordinate, then at any time, the z-position of a star will be z = rsin( ω +θ)sini where r = a 1 ε2 1+ε cosθ The star s radial velocity will therefore be ( ) + r! After some manipulation, this becomes ( ) ( ) v z =!z = sini {!rsin ω +θ θ cos ω +θ } v z = γ + K { cos( ω +θ) +ε cosω} with K = 2π P asini 1 ε 2 K 1 and K 2 are the semi-amplitudes of the radial velocity curves of the primary and the secondary, and γ (the gamma velocity) is the systemic velocity of the system.
44 Spectroscopic Binaries Note that the maximum and minimum measured velocities will be so v max = γ + K( ε cosω +1) and v min = γ + K( ε cosω 1) K = v max v min 2 and ε cosω = v max + v min 2K with K = 2π P asini 1 ε 2 ε=0.5, ω=0 ε=0.75, ω=90 From the shape and amplitude of the velocity curve, one can measure ε, ω, P, and T p. But one can t separate a from sin i.
45 Single and Double Line Binaries If one star is much brighter than the other, the spectrum from the secondary star may not be visible. You will have a single line spectroscopic binary. If both sets of lines are visible, it is called a double line spectroscopic binary. Clearly, from the definition of the center of mass, a 1 a 2 = r 1 r 2 = v 1 v 2 = K 1 K 2 = M 2 M 1 = q and since a = a 1 + a 2, we obtain equations like " M a = a 1 + M 2 % 1 $ ' # & M 2
46 The Mass Function For single line binaries, it is impossible to obtain the ratio of the stellar masses. But one can write down a function which depends on the minimum mass that the unseen companion must have. Start with Kepler s 3 rd law: M 1 + M 2 = a3 k P 2 and substitute in the primary s semi-major axis using " M a = a 1 + M 2 % 1 $ ' # M 2 & Kepler s 3 rd law then becomes M 1 + M 2 = M + M 3! $ a # & 1 M 2 " M 2 % k P 2 M 1 + M 2 ( ) 2 = a 1 Finally, since the semi-major axis isn t observable, substitute the K 1 velocity for a 1 using K 1 = 2π a 1 sini P 1 ε 2 3 k P 2
47 Kepler s 3 rd law is then or f M 2 The Mass Function 3 M 2 ( M 1 + M 2 ) = a k P = P 3 K ε 2 2 2π 3 sin 3 i ( ) = M 2 ( M 2 + M 1 ) = P K 1 2 ( ) 3/2 ( ) 3 sin 3 i ( ) 3 / ε 2 ( ) 3 k 2π In other words, from a single-line spectroscopy binary, it is impossible to determine individual masses, mass ratios, or even total mass. Note also that the mass function depends on sin 3 i. 1 k P 2 f ( M 2 ) = M 2 3 ( M 2 + M 1 ) 2 sin3 i This (with or without the sin 3 i term) is called the mass function
48 Radial Velocity Curves for Spectroscopic Binaries Depending on the eccentricity, the radial velocity curve of a binary can have many shapes; the closer to sinusoidal, the closer ε = 0.
49 Note on Radial Velocity Curves One rarely has single time series containing the entire time period of a binary. Instead, one usually has a series of observations taken at various random phases. One must combine the data to discover the period. (This is best done in Fourier space.) But note: time series are all susceptible to aliasing (i.e., a 12 day period can look like an 18 day period). P=1.65 days P= days P=1.68 days
50 Eclipsing Binaries If a binary system is eclipsing, then we have a good estimate of its inclination. If r is the separation of the two stars, then, for at least a partial eclipse cos i < R + R 1 2 r R 1 +R 2 r For a total eclipse to happen cos i < R 1 R 2 r If r R 1 +R 2 then i ~ 90. Also, the larger the separation, the rarer the phenomenon. Many eclipsing binaries have short periods.
51 If the separation becomes too small, the stars can become tidally distorted and/or interact Detached: the stars are separate and do not affect one another. Semi-detached: one star is spilling mass (i.e., accreting) onto the other Contact: two stars are present inside a common envelope (i.e., it is a common-envelope binary).
52 Partial and Total Eclipses The shape of the light curve during eclipse defines whether the eclipse is total or partial.
53 Partial and Total Eclipses In practice, other effects, such as star-spots, tidal distortions, limbdarkening, and hot spots may also effect the shape of the light curve.
54 Depth of Eclipse From symmetry the area blocked during the primary eclipse is exactly the same as that of the second eclipse. Consequently, when calculating the amount of light that is eclipsed, L = 4 π R 2 σ T 4, the radius doesn t matter only the temperature counts.
55 Total Eclipses Let F = 1 be the relative flux from a system out of eclipse, F T, the relative flux during total eclipse, and F a the flux during the annular eclipse. If we designate Star 1 to be the larger star, and Star 2 the smaller star, then During the total eclipse F 1 = F T and F 2 = F F T During the annular eclipse π R 1 F a = F 1 + F 2 π R 2 Therefore, the ratio of the stellar radii is simply 2 F 1 =1 R 2 R 1 2 F 1 κ = R $ 2 = 1 F a & R 1 % F T ' ) ( 1/ 2
56 Total Eclipses For systems, with i = 90, the eclipse begins when sinθ e = R 1 + R 2 a and becomes full when a R 1 sinθ i = R 1 R 2 a In the more general case (i 90 ), the eclipse beings when the projected separation is $ δ 2 e = sin 2 θ e + cos 2 θ e cos 2 i = R + R ' 1 2 & ) % a ( and is full when % δ 2 i = sin 2 θ i + cos 2 θ i cos 2 i = R R ( 1 2 ' * & a ) θ e θ i R 2 e
57 Solving Eclipsing Binaries Note that for total eclipses, the relative sizes of the stars and the orbit, along with the inclination of the system, can be computed without any information other than the light curve: Depth of total eclipse: F 1 = 1 F 2 The ratio of the stellar sizes: R 2 = 1 F 1/ 2 # & a % ( R 1 $ F T ' # The angle of egress: sin 2 θ e + cos 2 θ e cos 2 i = R + R & 1 2 % ( $ a ' The angle of ingress: $ sin 2 θ i + cos 2 θ i cos 2 i = R R ' 1 2 & ) % a ( If you work with the quantities, R 2 /R 1, R 2 /a, i, and F 1 /F 2, there are 4 equations and 4 unknowns. If the system is also a spectroscopic binary, you then known sin i, and can measure precise masses and radii.
58 Eclipsing Spectroscopic Binaries If a system is both an eclipsing and spectroscopic binary, then everything can be measured for the system: From the eclipse, you can measure R 2 /R 1, R 2 /a, i, and F 1 /F 2 From the velocities, you can obtain absolute size. For example, if we assume i ~ 90 and a circular orbit, then measuring individual stellar radii is simple. R 1 = v ( 2 t c t a ) R 2 = v ( 2 t t b a)
59 Kepler Data
60 Aside Variable Star Names The naming convention for variable stars stars that change their brightness due to eclipses, outbursts, etc., is as byzantine as it gets. Stars with existing names (α Lyrae or Mira) or Flamsteed numbers (55 Cancri) keep their names. The first new variable found in a constellation is given the designation R as in R Cor Bor. Then comes S, T, etc., through Z. The next (10 th ) new variable discovered in a constellation is given the name RR, as in RR Lyrae. Then comes RS, RT, etc., through RZ. The next (19 th ) new variable discovered in a constellation is given the name SS, as in SS Cygni. Then comes ST, SU, etc., through SZ. Then TT, TU, etc., through TZ, until finally, ZZ as in ZZ Ceti. The next (55 th ) new variable discovered starts at the beginning of the alphabet, with the letters, AA as in AA Cyg. After that comes AB through AZ, BB through BZ, CC through CZ until QQ through QZ. (But the letter J is always skipped!) When the 335 th new variable is discovered, the system gives up, and names become V335, V336, etc., as in V1500 Cygni.
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