ASTR 200 : Lecture 13 Doppler Effect and binary motion 1
Announcements Reminder: Midterm is Oct 22, in class, 50 minutes long. More information provided next week. HW 1 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. HW2 solutions to be posted soon. HW 3 due tomorrow. HW 4 assigned Friday. 2
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 surface T to be estimated (next slide) Provides compositional information Allows measurement of radial motion 3
Stellar Temperatures, from spectra Can match full stellar spectrum to a 'best fit' blackbody. - eg: Sun's spectrum (red) 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. Inverse wavelength (cm -1 ) 4
Stellar motions The star's true motion (or star's velocity V S ) can be decomposed Into two perpendicular components: - The radial velocity V R (>0 if away) - the tangential velocity V T, when observed at distance d generates a `proper motion' (angular rate) μ = V T /d (typically given in ''/year) 5
Stellar motions The star's true motion (or star's velocity V S ) can be decomposed Into two perpendicular components: - The radial velocity V R (>0 if away) - the tangential velocity V T, when observed at distance d generates a `proper motion' (angular rate) μ = V T /d (typically given in ''/year) 6 For the vast majority of stars, the distance d is so large the proper motion is undetectable (ASTR 205 discusses this more)
7 Radial motion: from the Doppler effect
The Doppler Effect The (non-relativistic) doppler effect relates the observed wavelength λ o of a wave (in this case a spectral line) to that emitted λ e in the lab, and the radial speed v r of the emitter relative to the observer Δ λ λ e = λ o λ e λ e = v r c SIGNS note: v r > 0 is for motion APART, giving λ o > λ e UNITS note: Need v and c in the same units; if one uses consistent units for wavelengths, all will be OK. 8
9 So, can measure speeds of CARS!
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). 10
The Doppler shift for light λ 11
12 So, can measure speeds of STARS!
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) 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: λ o =λ e 1+v/c 1 v /c Notice that as v approaches c, the observed wavelength λ o goes to infinity. The `redshift' z is defined as : z λ o λ e λ e with 0 < z < infinity 14
15 Binary motion
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
Sirius A and Sirius B A visual binary A - Very different luminosities - orbital period ~50 years B 17 Offset from B to A, over time
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
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 x cm at origin, in which case m 1 x 1 = m 2 x 2 19
If the orbits are circular, then they are 'nested', with radii inversely proportional to mass The radii (semimajor axes) of the orbits satisfy m 1 a 1 = m 2 a 2 The circular speed will also satisfy m 1 v 1 = m 2 v 2 The stars have the same orbital period P, so they always stay `directly across from each other' 20
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 = a A +a B 21
Newton's form of Kepler's 3 rd Law We can write this explicitly now P 2 = 4 π 2 G(m A +m B ) ( a A +a B )3 If m B ---> 0 then a A --->0 too, and recover what we're used to. (because: ) Allows stellar masses to be calculated a A = m b a m B a 22
In reality the system's center of mass 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
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 13.7 24 Sirius A and Sirius B : Text derives m A = 2.2 m B and m A = 2.1 solar masses m B = 0.97 solar masses