ASTR 200 : Lecture 13 Doppler Effect and binary motion
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1 ASTR 200 : Lecture 13 Doppler Effect and binary motion 1
2 Announcements Reminder: Midterm is in two weeks (Oct 18), in class, 50 minutes long. HW 1 and HW 2 solutions now posted on the course web site. Appeal procedure: Obtain form (in Henn 312), complete, and turn in HW and form to prof's office. HW 4 due tomorrow. HW 5 assigned Friday. Due Thursday Oct 12, with possible late submission Fri Oct 13 (FINAL deadline), because solutions posted Oct 13, 5 PM. 2 HW5 is the last homework before the exam. Won't be returned before exam, but solutions there.
3 How can one measure the distance to, or size of, a point of light??? What can we directly observe about a star? Its position on the sky Maybe parallax, if close enough Proper motion, if star is close and moving fast The radiation we receive Incident flux Spectrum Allows measurement of radial motion Allows surface T to be estimated Provides compositional information 3
4 Stellar motions The star's true motion (or star's velocity VS) can be decomposed Into two perpendicular components: The radial velocity VR (>0 if away) the tangential velocity VT, when observed at distance d generates a `proper motion' (angular rate) μ = VT/d 4 (typically given in ''/year)
5 Stellar motions The star's true motion (or star's velocity VS) can be decomposed Into two perpendicular components: The radial velocity VR (>0 if away) the tangential velocity VT, when observed at distance d generates a `proper motion' (angular rate) μ = VT/d (typically given in ''/year) For the vast majority of stars, the distance d is so large the proper motion is undetectable (ASTR 205 discusses this more) 5
6 Radial motion: from the Doppler effect 6
7 The Doppler Effect The (non relativistic) doppler effect relates the observed wavelength λo of a wave (in this case a spectral line) to that emitted λe in the lab, and the radial speed vr of the emitter relative to the observer λ λ v o e r Δλ = = λe λe c SIGNS note: vr> 0 is for motion APART, giving λo > λe UNITS note: Need v and c in the same units; if one uses consistent units for wavelengths, all will be OK. 7
8 So, can measure speeds of CARS! 8
9 The Doppler Effect 1. Light emitted from an object moving towards you will have its wavelength shortened. BLUESHIFT 2. Light emitted from an object moving away from you will have its wavelength lengthened. REDSHIFT 3. Light emitted from an object moving perpendicular to your line of sight will not change its wavelength (unless v~c, in which case use relativistic). 9
10 The Doppler shift for light 10
11 So, can measure speeds of STARS! 11
12 Measuring Radial Velocity We can measure the Doppler shift of emission or absorption lines in the spectrum of an astronomical object. Can calculate the velocity of the object in the direction either towards or away from Earth. (radial velocity) 12
13 The relativistic Doppler effect Once the emitter/observer distances are changing by speeds that become a non negligible fraction of the speed of light, the previous formulation is only a first order approximation Derivation requires special relativity, due to time dilation. We will just use the result, for a light emitter moving a speed v away from an observer, that: 1+ v / c λ o =λ e 1 v / c Notice that as v approaches c, the observed wavelength o goes to infinity. The `redshift' z is defined as : 13 λ o λ e z λe
14 Stellar Temperatures, from spectra Can match full stellar spectrum to a 'best fit' blackbody. eg: Sun's spectrum (red) Inverse wavelength (cm 1) Sometimes, this fitting is complicated by the many spectral lines present. so a 'best' match is a bit of an art... stars are not perfect blackbodies in any case. 14
15 Binary motion 15
16 Binary and multiple stars are common Why? As interstellar clouds collapse, they tend to have too much `spin' (angular momentum) to contract down to a single star+disk. Instead, the collapsing crowd fragments into multiple sub-clouds, each of which forms a star Sometimes those pairs remain bound together. [Planet formation aside : Often a pair is quite close...but sometimes they are far enough apart that each star has a (truncated) disk out of which planets might form.] 16
17 Sirius A and Sirius B A visual binary Very different luminosities orbital period ~50 years A B 17 Offset from B to A, over time
18 Binary star orbits Waaaaiiiit a minute...this as drawn as though one star is not moving. What if the mass of one of the objects is not negligible? How can one object be fixed and the other move? Well, it's never true It's a good approximation if one is very much more massive than the other In fact, both masses orbit the center of mass 18
19 The concept of center of mass The two stars both orbit the center of mass The center of mass goes to the center of the more massive object if one is dominant General formula > Often put xcm at origin, in 19 which case m1 x 1 = m2 x 2
20 If the orbits are circular, then they are 'nested', with radii inversely proportional to mass The radii (semimajor axes) of the orbits satisfy m1a1 = m2a2 The circular speed will also satisfy m1v1 = m2v2 The stars have the same orbital period P, so they always stay `directly across from each other' 20
21 Also true for elliptical orbits more massive center of mass less massive Again, the two stars are always on opposite sides of the c.o.m. (eg. both a peri at same time) The total distance r is the sum, as is a = aa+ab 21
22 rd Newton's form of Kepler's 3 Law We can write this explicitly now 2 4π 3 P= ( a A +ab ) G(m A +mb ) 2 If mb ---> 0 then aa--->0 too, and recover what we're used to. (because: a A = m b a B ) ma Allows stellar masses to be calculated 22
23 In reality the system's center of mass also moves along uniformly through space 23 Sirius A and Sirius B : (a) across the sky, (b) B relative to A on sky, (c) same as (b) without compass direction, and (d) individual orbits
24 Sirius B is very dim, so the fact that the system was binary was discovered by the 'wobble' of Sirius A's proper motion across the sky only Sirius A could be seen Extension of textbook Figure Sirius A and Sirius B : Text derives ma = 2.2 mb and ma = 2.1 solar masses mb = 0.97 solar masses
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