ASTRO 310: Galac/c & Extragalac/c Astronomy Prof. Jeff Kenney

Size: px

Start display at page:

Download "ASTRO 310: Galac/c & Extragalac/c Astronomy Prof. Jeff Kenney"

Evan Lamb
5 years ago
Views:

1 ASTRO 310: Galac/c & Extragalac/c Astronomy Prof. Jeff Kenney Class 9 September 26, 2018 Introduc/on to Stellar Dynamics: Poten/al Theory & Mass Distribu/ons & Mo/ons

2 Gravity & Stellar Systems ~Only force that affects stars & dark mater: Gravity Gas is more complicated since it is affected by other forces Newtonian gravity works well in most cases. Rela/vis/c effects important only in /ght binaries and near black holes in galaxy nuclei.

3 Behavior of stellar system and how to treat it mathema/cally depends on # of stars in system Binary stars: 2 stars Open clusters: ~ stars Globular clusters: ~ stars Large galaxies: ~ stars

4 Behavior of stellar system and how to treat it mathema/cally depends on # of stars in system Binary stars: 2 stars Open clusters: ~ stars Globular clusters: ~ stars Large galaxies: ~ stars No exact solu/on for n>2! need to either solve numerically, OR make simplifying assump/ons and solve analy/cally

5 triple systems have non- repea/ng (chao/c) orbits

6 The three- body problem dates back to the 1680s. Isaac Newton had already shown that his new law of gravity could always predict the orbit of two bodies held together by gravity such as a star and a planet with complete accuracy. The orbit is basically always an ellipse. However, Newton couldn't come up with a similar solu/on for the case of three bodies orbi/ng one another. For 2 centuries, scien/sts tried different tacks un/l the German mathema/cian Heinrich Bruns pointed out that the search for a general solu/on for the three- body problem was fu/le, and that only specific solu/ons one- offs that work under par/cular condi/ons were possible. Generally, the mo/on of three bodies is now known to be nonrepea/ng.

7 Key Q: Does star s mo/on depend much on individual star- star gravita/onal encounters, or does it depend more on the cumula/ve gravita/onal effect of all the more distant stars? Gravita1onal poten1al Φ(x) of star cluster or galaxy Φ Smooth part doesn t change (much) over /me x Steep parts due to individual stars move around as stars move The total poten/al is the sum of a smoothly varying shallow poten/al well due to the average effect of many distant stars, plus a steep poten/al well near each star. In some (but not all!) cases the steep poten/al well near each star can be ignored.

8 describe the orbital mo/on of the Sun in the Galaxy

9 How does the orbital speed of the Sun relate to the mass of the galaxy? A. instantaneous speed and radius gives total galaxy mass B. instantaneous speed and radius gives interior galaxy mass C. instantaneous speed and radius gives approximate interior galaxy mass D. orbit- averaged speed and radius gives approximate interior galaxy mass E. orbit- averaged speed and radius gives approximate total galaxy mass

10 How does the orbital speed of the Sun relate to the mass of the galaxy? A. instantaneous speed and radius gives total galaxy mass B. instantaneous speed and radius gives interior galaxy mass C. instantaneous speed and radius gives approximate interior galaxy mass D. orbit- averaged speed and radius gives approximate interior galaxy mass E. orbit- averaged speed and radius gives approximate total galaxy mass

11 Newton s Law of Gravity this gives accelera/on dv/dt on 1 star m, due to gravita/onal force exerted on m by M at distance r if there are N stars with masses m α (α = 1,2,3,.N) at posi/ons x α ; to get accelera/on of star α, add forces on star α from all the other stars:

12 it is oken convenient to write the accelera/on as the gradient in the gravita/onal poten/al if we have (or can approximate by) a con/nuous distribu/on of mater: in this case the accelera/on, or force per unit mass is: NOTE!! this F is really the accelera/on!!

13 Gravita/onal force and poten/al Since force can be expressed as the gradient of a poten/al, the gravita/onal force is conserva7ve:

14 Gravita/onal force and poten/al Since force can be expressed as the gradient of a poten/al, the gravita/onal force is conserva7ve: Work done by force independent of path taken between end points Energy is conserved

15 Gravita/onal force and poten/al Since force can be expressed as the gradient of a poten/al, the gravita/onal force is conserva7ve: Work done by force independent of path taken between end points [ if φ φ(t) ] Energy is conserved [ if φ φ(t) ]

16 Gravita/onal force and poten/al Since force can be expressed as the gradient of a poten/al, the gravita/onal force is conserva7ve: Work done by force independent of path taken between end points [ if φ φ(t) ] Energy is conserved [ if φ φ(t) ] Advantages of poten/al: Poten/al is a scalar field, simpler than vector field In many cases, best way to calculate force is to first calculate poten/al, then takes its gradient

17 preceding equa/on for φ(x) is integral equa/on. this can be expressed in another form which is some/me more convenient, a differen/al equa/on: Poisson s Equa1on

18 derive φ(r) in case of spherical symmetry

19 spherical coordinate system (r,ϕ,θ)

20 gravita/onal poten/al in case of spherical symmetry mass blob with density distribu/on ρ(r ) r integrate over all space (all r ) to learn gravita/onal poten/al at some posi/on r

21 Newton s theorem #2 for spherically symmetric gravita/onal field The gravita/onal force on a body that lies outside a closed spherical shell of ma;er is the same as it would be if all the shell s ma;er were concentrated into a point at its center.

Newton s theorem #2 for spherically symmetric gravita/onal field The gravita/onal force on a body that lies outside a closed spherical shell of ma;er is the same as it would be if all

22 Newton s theorem #2 for spherically symmetric gravita/onal field The gravita/onal force on a body that lies outside a closed spherical shell of ma;er is the same as it would be if all the shell s ma;er were concentrated into a point at its center I IN : by Newton s 2nd theorem, this integral I IN must be same as if all mater were at the center, i.e., r =0 So r - r - > r

Newton s theorem #1 for spherically symmetric gravita/onal field A body that is inside a uniformly dense spherical shell of matter experiences no net gravitational force from that

23 Newton s theorem #1 for spherically symmetric gravita/onal field A body that is inside a uniformly dense spherical shell of matter experiences no net gravitational force from that shell. force from A 1 balances force from A 2 Geometry of spherical shells precisely compensates for 1/r 2 dependence of gravitational force (more mass in A 1 but it is further away)

24 A body that is inside a uniformly dense spherical shell of matter experiences no net gravitational force from that shell for M out, force from S 1 cancels force from S 2 (equal and opposite) m S 1 S 2 r M in Geometry of spherical shells precisely compensates for 1/r 2 dependence of gravitational force (more mass in S 2 but it is further away) M out net force on m is all from M in

25 I OUT : Since contribu/on to force from mater outside r must be ZERO, the contribu/on to poten/al from mater outside r must be constant within radius r, and we can choose to evaluate integral at any radius smaller than r - > so choose r- >0 so r - r - > r

26 gravita/onal poten/al in case of spherical symmetry mass blob with density distribu/on ρ(r ) r integrate over all space (all r ) to learn gravita/onal poten/al at some posi/on r in the case of a spherically symmetric mass distribu/on:

27 circular speed: speed of test par/cle in circular orbit at radius r accelera/on = force per unit mass = F dφ v 2 a = = = c m dr r v c = r a = r dφ dr OK in general = GM in depends only on interior mass M(r) in the special case of spherical symmetry if not spherical symmetry, also depends on exterior mass! r spherical symmetry only r M in v c m M out

28 M in to first order, orbital speed depends on interior mass not exterior mass M out

29 What if mass distr. not spherically symmetric? (galaxies are NOT spherically symmetric!) How much does this change orbital speeds?

30 Rota/on curve of exponen/al disk A C B B+C have similar veloci/es at R>4R disk à litle addi/onal mass and Keplerian behavior (V~R - 1/2 ) beyond R>4R disk for A,B,C A peaks at R=2R disk A ~15% higher than B (at peak) A slower than B at R<1R disk à Slowed by exterior mass A faster than B at R>1R disk à sped by interior mass concentrated in disk Binney & Tremaine 1987 A: flatened exponen/al disk B: spherical distribu/on with same interior mass as exponen/al disk C: point mass with same total mass as exponen/al disk 30

31 Orbital speeds depend mostly on interior mass only. How can we measure the total mass of the galaxy?

32 Orbital speeds depend mostly on interior mass only. How can we measure the total mass of the galaxy? 2 ways: 1. measure orbital speeds at large r, beyond all the mass (so that M in = M tot ) (need test par/cles at large r not always available) 2. measure escape speed (can do this from inside mass distribu/on!)

33 escape speed: escaping from a con/nuous distribu/on of mater E tot = K + P E = 0 escape condi/on E 1 = E 2 conserva/on of energy K 1 + P 1 = K 2 + P 2 K 1 - K 2 = P 2 - P 1 K(r) K( ) = P( )- P(r) ½mv 2 esc 0 = 0 mφ v esc = 2 φ m v esc r M in Mout con/nuous distribu/on of mass M(r) described by poten/al φ(r) φ(r) and therefore v esc depends on both M in and M out

34 poten/al energy defined to be work done against gravity in assembling distribu/on of mater from infinity for single star of mass m: PE = m φ(x)

35 gravita/onal poten/al in case of spherical symmetry mass blob with density distribu/on ρ(r ) r integrate over all space (all r ) to learn gravita/onal poten/al at some posi/on r in the case of a spherically symmetric mass distribu/on: interior mass exterior mass

36 M in to first order, orbital speed depends on interior mass not exterior mass M out but escape speed depends on both!

37 circular & escape speeds v c = r dφ/dr = GM in /r v esc = 2 φ(r) if all mass is interior then v esc = 2GM in /r but in general v esc 2GM in /r exterior mass beyond r: M out = M(>r) important for escape speed but not circular speed! (strictly true in case of spherical symmetry; if not spherically symmetric, circular speed depends a litle on exterior mass)

38 Suppose we can only observe stars in the solar neighborhood. how can we es/mate the escape speed & therefore also the total mass of the galaxy?

39 Suppose we can only observe stars in the solar neighborhood. how can we es/mate the escape speed & therefore also the total mass of the galaxy? Observe speeds of high velocity stars. The fastest stars that are bound to the galaxy have speeds just less than the escape speed.

40 hypervelocity stars in Milky Way orbital speed of Sun = 220 km/s speed of hypervelocity stars ~ km/s velocity vectors of 4 hypervelocity stars near the Sun

41 ar/sts impressions of hypervelocity stars escaping Milky Way

42 How do we know if fast star is escaping or not?

43 How do we know if fast star is escaping or not? 2 ways: 1. examine distribu/on of stellar veloci/es, search for break in distribu/on at escape speed 2. examine direc/on of mo/on stars moving outward may be escaping, stars moving inward may be bound

44 Mira variable Red Giant star 1.2 M sun moving at 130 km/s through galaxy (not fast enough to escape) making gaseous tail 13 light- years long UV image from GALEX telescope UV emission is from molecular Hydrogen

45 UV image from GALEX telescope UV emission is from molecular Hydrogen

46 high- velocity neutron star (& pulsar) plowing through the ISM: Guitar Nebula moving at 1600 km/s through the ISM (greater than escape speed from Galaxy)

47 high- velocity neutron star (& pulsar) plowing through the ISM: Guitar Nebula

48 Guitar Nebula: HST image of head of Nebula, showing most recent part of trail created by fast- moving neutron star

ASTRO 310: Galac/c & Extragalac/c Astronomy Prof. Jeff Kenney. Class 15 October 24, 2018 Tidal Interac/ons for Galaxies & Star Clusters

ASTRO 310: Galac/c & Extragalac/c Astronomy Prof. Jeff Kenney Class 15 October 24, 2018 Tidal Interac/ons for Galaxies & Star Clusters /mescales of Local Group a small loose group of galaxies calculate