The Properties of Stars
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1 10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product of their msses nd inversely proportionl to the distnce between them. Gm F r The force on prticle outside n object with sphericl symmetry is the sme s if ll of its mss were concentrted t its center. Newton s lw of grvity, combined with his lws of motion enble us to determine the mss of celestil body by observing its effect on second celestil body. For exmple, we cn find Jupiter s mss by mesuring the orbitl rdius nd period for ech of its Glilen stellites nd using Newton s form of Kepler s third lw of plnetry motion. 4 π Jupiter GP ( 4π )( ) 11 6 ( )( ) 9 Cllisto m 6 PCllisto s Jupiter Jupiter kg Jupiter Erth sses Jupiter 18 Erth sses We ll use similr method to find the msses of strs in binry str systems. 1
2 10/11/010 Using Newton s Form of Kepler s Third Lw to esure the sses of Strs 4π In using Jupiter GP to determine the mss of Jupiter, we ssumed tht the mss of fjupiter is much greter thn the mss of ny of fits moons. However, in binry str systems, the two bodies hve comprble msses nd the relevnt form of Kepler s third lw is 4π 1 + where 1 nd re the msses of the two strs. GP P is the orbitl period of the strs nd is the verge distnce between them. ecuse the msses of strs re very lrge, but reltively smll multiple of the mss of the Sun, it is convenient to use solr mss units. In tht cse, Kepler s third lw is nd re multiples of the Sun s mss if is in U s nd P is in yers. P Center of ss r v r v The blck dot is the center of mss, nd the colored disks re two strs t distnces r 1 nd r from the center of mss. 1 nd re the msses of the two strs nd v 1 nd v their orbitl speeds.
3 10/11/010 Types of inry Str System Visul inries oth strs re visible, so their orbits cn be plotted. Spectroscopic inries The strs pper s single str but, becuse of the Doppler effect, the spectrl lines cn be seen to shift s the strs move in their orbits. Eclipsing inries The strs pper s single str, but we see the orbits edge-on, so the strs periodiclly eclipse ech other. In order to use 1 + P to find 1 nd, we hve to (1) mesure nd P nd () determine wht frction of the totl mss belongs to ech str. In order to ccomplish (), we need the concept of center of mss defined below. When two bodies move through spce nd re cted on only by their mutul grvittionl forces, there is point between them tht moves in stright line. Tht point is clled the center of mss of the two bodies. The center of mss of pir of bodies stisfies the eqution 1 r 1 r where r 1 is the distnce from 1 nd r is the distnce from. Visul inries
4 10/11/010 The figures below show the observed positions of the strs in two different visul binry systems). Poor visul binry observtions result from () the period being so long tht few observtions hve been mde or (b) the strs being so close together nd/or so fr from Erth tht their ngulr seprtions cnnot be mesured ccurtely. The solid lines in the figures represent the ellipse tht best fits the dt (colored dots nd plus signs). Visul binries hve periods between 1 yer nd thousnds of yers. Plenty of observtions of strs seprted by severl rcseconds. Few observtions of strs seprted by frction of rcsecond. Exmple 1 Consider visul binry str system in which str is 5 times frther from the center of mss thn str 1, the period is 00 yers nd the semi-mjor xis is 100 U. Clculte () the totl mss nd (b) the mss of ech str. The equtions to be used re 1 r nd r P We re given P 00 yers nd 100 U So, We re lso told tht 5, so solr msses solr msses solr msses 6 4
5 10/11/010 Spectroscopic inries In spectroscopic binry systems, the two strs re too close together to be resolved by telescope. ecuse their seprtions re less thn bout 1 U, their periods re s short s few hours or s long s few months. In the nimtion, the blck dot represents the center of mss. ctully, the center of mss moves, but we re looking t the system from the viewpoint of someone t rest reltive to the center of mss. Finding the sses of Spectroscopic inries Finding the mss of the strs in binry str system requires observtions tht give () the sum of the msses nd (b) the rtio of the msses. This cn esily be done if the system is well-observed visul binry. In tht cse, we cn plot the orbit nd mesure nd P. r 1 nd r cn be determined by observing the motion of the system long enough to locte the center of mss. For spectroscopic binries, it isn t so esy. In tht cse, we must extrct informtion from the combined spectr of the strs. Since the Doppler effect only gives the str s rdil velocity (the component long the observer s line of sight) nd most orbits re tilted, we re usully ble to only determine lower limit to the totl mss. We ll just consider the simplest cse: the ngle between the line of sight nd the orbit is 0º, nd the orbit is circulr. The figure shows the orbits from bove the orbitl plne. The red circle is the orbit of the red str nd the blue circle is the orbit of the blue str. Erth is to the right. v P Q R S Towrd Erth v The blue str () is moving wy from us, so its spectrum is red-shifted while tht of the red str () is blue-shifted. When the strs rrive t the points P, Q, R, nd S they re moving cross our line of sight so we see no redshift. 5
6 10/11/010 The Spectrum Usully, the spectrum will show two sets of lines tht chnge positions s the strs move long their orbits. In the following figures, wvelength increses towrd the right nd only the hydrogen lmer lines re shown. In ech cse, the lmer lines observed in the lbortory re displyed on the bottom for comprison with the binry s spectrum on the top. The first figure shows the spectrum t time when the strs re moving cross our line of sight so there is no wvelength shift, nd spectr of the two strs re superimposed. In prctice, the photogrphs of the strs spectr would be blck nd white, but I ve used blue for str nd red for str. λ Str Lb Str Lb t 0 (spectr of nd re superimposed) t ¼ of the period lter. Which str is moving towrd us? The Rdil Velocity Grphs The following grph shows the rdil velocities of the two strs s function of time. The time scle isn t the sme s tht used in the previous slide. Rdil Velocity vs. Time Strs moving cross our line of sight Rdil Velocity (km/s) Time (dys) Str moving wy from Erth. Str moving towrd Erth. Note: the rdil velocity of the center of mss hs been subtrcted before the grph ws drwn. 6
7 10/11/010 Clcultions From the rdil velocity grph, we cn red the orbitl velocities of the two strs v nd v s well s the orbitl period P. The rdii of the orbits re The semi-mjor xis is v P r v P r nd π π r + r The totl mss is + P The rtio of the msses is v v When the orbits re circulr nd the ngle between them nd the line of sight is zero (i.e., we re seeing them edge-on), the lst two equtions permit us to clculte the msses of the two strs. In tht cse, the strs will lso periodiclly eclipse ech other. Exmple spectroscopic-eclipsing binry str system hs period of.00 yers. The mximum rdil velocities of the strs reltive to the center of mss re 0 km/s (for str ) nd 10 km/s (for str ). Clculte () the rtio of their msses nd (b) the individul msses, ssuming tht the orbit is observed edge-on from Erth, v 10 1 () v 0 km/s, v 10 km/s v 0 (b) r r v P r π v P r π 7 ( 0km/s)( ) km P.00 ( s) s.0 10 km r.01u π km/u 7 ( 10km/s)( ) km π r + r ( ) U.01 U Sun.00 The totl mss is ( ) ( ) r km km/u 1.00U +.0 Sun.0 Sun 1.01 Sun.0 Sun 7
8 10/11/010 Eclipsing inries Flux PEG nimtion of the lgol System from Pper by londin, Richrds, nd linowski Time t : eclipse of hotter str begins t : eclipse of hotter str complete t C : The hotter str is bout to emerge from behind the cooler one. t D eclipse of hotter str ends. 8
9 10/11/010 v reltive orbitl velocity R H rdius of the hotter str R v( t t ) H R rdius of the cooler str RR v ( t t ) C C C Exmple The orbitl velocity of n eclipsing binry system is 85 km/s, nd the time for the hotter str to be complete is 4.0 hours, wht is the rdius of the hotter str? 1 v 85km/s RH vt ( t) 1 85km/s s km Exmple 4 In the sme system, the hotter str is eclipsed for 8 hours, wht is the rdius if the cooler str? 1 RC vt ( C t 6 ) 1 85km/s s km Properties of Strs Some Importnt Results 9
10 10/11/010 The ss-luminosity Reltion grph of bsolute visul mgnitude ( mesure of luminosity) is plotted s function of the logrithm of mss, the result is lmost stright line s shown below. bsolute Visul gnitude LL.5 L the luminosity of thesun the mss of the Sun /
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