But for simplicity, we ll define significant as the time it takes a star to lose all memory of its original trajectory, i.e.,
|
|
- Jonah McKinney
- 6 years ago
- Views:
Transcription
1 Stella elaxation Time [Chandasekha 1960, Pinciples of Stella Dynamics, Chap II] [Ostike & Davidson 1968, Ap.J., 151, 679] Do stas eve collide? Ae inteactions between stas (as opposed to the geneal system potential) impotant? We can answe this question by calculating the time it takes fo a sta s obit to be significantly petubed by individual encountes with othe stas. To calculate this elaxation time, let s fist define the wod significant. One way of doing this is though total enegy: when does the kinetic enegy exchanged duing stella encountes equal the sta s oiginal kinetic enegy, i.e., T E = ( E) = E (30.01) But fo simplicity, we ll define significant as the time it takes a sta to lose all memoy of its oiginal tajectoy, i.e., T D = sin φ = 1 (30.0) We then assume that a) all deflections ae two-body encountes, b) each encounte is statistically independent, and c) close encountes ae insignificant compaed to long-ange encountes, so that duing each encounte, E E. Unde these assumptions, all the deflections ae small (sin φ 1), and we can use the Bon appoximation, whee (v init v final v). θ s b M v v ϕ 1
2 Fo a single encounte, the deflection angle, φ is elated to the initial impact paamete, b, by F = m dv dt = v = dv = 1 m F dt (30.03) Fom the geomety of the encounte F = F sin θ = F ( ) b = ( GMm ) ( ) b (30.04) Also, fom the Bon appoximation v dt = vdt = ds = dt = ds v (30.05) So v = 1 m F dt = m 0 ( GMm ) ( b ) ( ) 1 ds (30.06) v o, since = (s + b ) 1/ v = GM v Letting x = s/b 0 v = GM vb = GM vb b GM ds = (s + b 3/ ) v 0 dx GM = (1 + x 3/ ) vb 0 ds/b (1 + (s/b) ) 3/ x (1 + x ) 1/ (30.07) 0 (30.08)
3 Since fo small deflections, tan φ φ = v /v φ = GM v b (30.09) Now, let s sum this ove all possible collisions. The numbe of collisions that take place in time dt depends on the impact paamete, the distance a sta tavels in dt, and the density of stas in the stella system, N, i.e., So, to deflect the sta by 90, sin φ φ = 1 = TD N coll = (πb db) (vdt) N (30.10) 0 bmax b min (πb db)(vdt) N φ = TD bmax 0 b min (πb db)(vdt)n ( GM v b ) = 8πG M N v 3 T D bmax b min db b (30.11) As fo the limits on the log quantity, we can use the obvious fact that no deflection angle can be geate than π. Thus φ = GM v b min = π = b min = GM πv (30.1) Similaly, it is clea that the maximum impact angle must be less than the mean distance between stas, so N = 1 (4/3)π b 3 max = b max = ( ) 1/3 3 (30.13) 4πN 3
4 So giving ( 8πG M ) { N bmax πv } v 3 T D ln = 1 (30.14) GM T D = v 3 { 8πG M N ln bmax v } π GM (30.15) Obviously, the above deivation involves a numbe of appoximations. A moe igoous deivation by Chandasekha gives and T D = T E = v 3 { 8πG M NH(χ) ln bmax v } GM v 3 { 3πG M NG(χ) ln bmax v } GM (30.16) (30.17) whee H(χ) and G(χ) ae factos of the ode unity that depend on the stella distibution function. Finally, Ostike & Davidson (1968) give an impoved, ecusive expession fo the elaxation time T P = v 3 8πG M N ln { v 3 T P GM } (30.17) In the sola neighbohood, this timescale is much lage than a Hubble time. Thus, the motions of stas ae collisionless, and contolled only by the oveall Galactic potential. 4
5 The Collisionless Boltzmann Equation [Binney & Temaine 1987, Galactic Dynamics, 1987] The basis of undestanding galactic dynamics lies with the collisionless Boltzmann equation. Imagine a closed volume, V, bounded by a suface S, and containing a mass, M(t). The net amount of mass flowing though a suface is equal to the change of mass in the volume, i.e., S ρ v ds = dm dt = V ρ But the divegence theoem in mathematics says S Q ds = V dv (30.18) Q dv (30.19) so o V ρ dv + V ρ (ρv) dv = 0 (30.0) + (ρv) = 0 (30.1) 5
6 Now expand this concept to stas flowing in a 6-dimensional phase space (x, y, z, v x, v y, v z ). Assume that the flow of stas is smooth, and is contolled by a potential pe unit mass Φ. Now let f = the stella density in the 6-D space ω = the 6-D position coodinate x = the 3-D space coodinate v = the 3-D velocity coodinate With these definitions, the continuity equation fo a fluid of stas is + (f ω) = 0 (30.) Now let s expand this out to + 6 ( α=1 ω α + f ω ) α ω α ω α = 0 (30.3) and explicitly wite out the space and velocity pats of the second pat of the equation. + 6 α=1 ω α ω α + f ẋ i + f v i = 0 (30.4) Since x and v ae independent dimensions, ẋ i = = 0 (30.5) Also, fo consevative foces with no collisions v = Φ x (30.6) 6
7 so + 6 α=1 ω α ω α + f { Φ } = 0 (30.7) Also, since the potential a paticle feels depends only on its location, and not on its velocity ( ) Φ = 0 (30.8) which leaves only 6 + ω α = 0 (30.9) ω α α=1 Finally, to make the equation moe tanspaent, we can again explicitly wite out the position and velocity pats of the sum + ẋ i + v i = 0 (30.30) o, if we substitute ẋ = v and v = Φ + which, in vecto notation is x, { v i Φ + ( v f) } ( Φ ) v = 0 (30.31) = 0 (30.3) This is sometimes called the collisionless Boltzmann equation, the Vlasov equation, o the equation of continuity. Note that this equation, when expessed in tems of a Lagangian deivative (i.e., fom the point of view of an obseve moving though space with the fluid) is ( ) ( ) v v dv = dt + dx = df x dt = 0 (30.33) 7
8 Jeans Equations As it stands, the collisionless Boltzmann equation is athe useless, as it is not only a function of 7 vaiables (many of which ae not easily obsevable), but thei deivatives. Even if f( ω) wee measuable, uncetainties due to Poisson noise would play havoc with the deivatives. Fotunately, moe tactable equations can be found by finding the moments (aveages) of the equation. Such moments poduce the Jeans Equations. Fist, note that the spatial aveage of any quantity associated with the 6-D stella fluid is simply found by integating ove velocity, i.e., Q = Qfd v/ 3 fd 3 v (30.34) Moeove, since the density of stas in the fluid, ν, is simply ν( x) = f d 3 v (30.35) the spatial mean of any quantity is Q = 1 Q fd 3 v (30.36) ν So, fo the fist Jeans equation, stat with the Boltzmann equation { + v i Φ } = 0 (30.31) and integate ove the 3 velocity dimensions d3 v + v i d 3 v Φ d 3 v = 0 (30.37) 8
9 Since time, position, and velocity ae independent quantities, we can extact those dependencies fom the integals fd 3 v + v i fd 3 v Φ d 3 v = 0 (30.38) The fist integal in the equation is simply the density, and the second is just the density times mean velocity, so (30.38) becomes ν + (ν v i ) Φ d 3 v = 0 (30.39) But if you expand that last integal d 3 v = = = dv i dv j dv k dv i dv k dv i dv k f + = 0 (30.40) (because obviously, the phase-space density of objects at infinite velocity is zeo). This leaves us with the fist Jean s equation: the 3-D equation of continuity ν + (ν v i ) = 0 (30.41) 9
10 Fo the second Jeans equation, we again stat with the Boltzmann equation + { v i Φ } = 0 (30.31) This time, howeve, we multiply though by v j befoe integating v j d3 v + v i v j d 3 v v j Φ d 3 v = 0 (30.4) Once again, the time and space deivatives can be taken outside the integal, leaving mean quantities v j fd 3 v + (ν v j ) + v i v j fd 3 v (ν v i v j ) Φ Φ v j d 3 v = 0 v j d 3 v = 0 (30.43) And, once again, we can expand and evaluate the individual integals of the last tem v j d 3 v = If j i, the esult is the same as befoe v j dv i dv j dv k = v j dv i dv j dv k (30.44) v j dv j dv k df = 0 (30.45) 10
11 But if j = i, v i dv i dv j dv k = dv j dv k v i df (30.46) We can evaluate this last tem by integating by pats v i df = v i f fdv i (30.47) The fist tem is zeo, and the second is just the space density, ν, so the second Jean s equation becomes (ν v j ) + (ν v i v j ) + ν Φ x j = 0 (30.48) The final Jeans equation comes fom noting that the anisotopic pessue tem (i.e., the stess tenso), σ ij is σ i,j = (v i v i )(v j v j ) = v i v j v i v j (30.49) So, if we substitute this into the nd Jeans second, (ν v j ) + { ( ν σ,j + v i v j )} + ν Φ = 0 (30.50) i x j If we expand this out ν v j + v j ν + { (νσij ) + ν j (ν v i ) + ν v i v } j = ν Φ x j (30.51) 11
12 multiply the 3-D continuity equation by v j v j ν + v j (ν v i ) = 0 (30.5) and subtact the two equations, we get ν v j + ν v i v j = ν Φ x j (νσ ij ) (30.53) Hee, νσ ij is the anisotopic pessue (stess) tenso. But since σ ij is symmetic, its matix can be diagonalized. This poduces the pinciple axes of the velocity ellipsoid. This is the equivalent of Eule s equation fo fluid flows ρ v + ρ (v ) v = ρ Φ p (30.54) whee ρ is the density and p the pessue. 1
13 Jeans Equations in Cylindical Coodinates The Jeans and Boltzmann equation ae almost neve used in thei Catesian coodinate fom. Fo most applications in spial galaxies, cylindical coodinates ae used; in elliptical galaxies, the equations ae given in spheical coodinates. The deivation of these equations is staightfowad, but tedious. Fo the cylindical coodinate equations, the Boltzmann equation becomes + v 1 + v θ ( v v θ + Φ v θ θ + v z ) Φ v θ z ( v z + θ Φ ) v v z = 0 (30.54) and if we assume azimuthal symmety, the Jeans equations ae ν + 1 ν v d + ν v z z = 0 (30.55) ν v + ν v + ν v v z z ( v + ν vθ + Φ ) = 0 (30.56) ν v θ + ν v v θ + ν v θv z z + ν v θv = 0 (30.57) ν v z + ν v v z + ν v z z + v v z + ν Φ z = 0 (30.58) 13
14 This last equation is often used in its simplified fom. Fist, assume the galaxy is in a steady state, so that the time deivative is zeo. Next, conside v v z. Fom symmety, this tem should be zeo in the plane of the galaxy. Above and below the plane, this tem might be zeo, but wost case is that the pincipal axes of the stella velocity ellipsoid ae otated to align with that of the spheoidal component. To estimate the effect this would have on v v z, we must fist tansfom the spheical coodinate system (v, v θ, v ϕ ) to the cylindical system (v, v θ, v z ). This is tedious, but staightfowad. If the pinciple axis of the system is aligned with the disk (i.e., if v v θ = 0), the the esult is v v z = { v v θ } z + z = { v v θ } { v v θ } z z 1 (1 + z / ) (30.59) Since most of the stas ae nea the plane of the galaxy, v v and v θ v z, so v v z { v v z } z (30.60) Now, when we evaluate the thid tem of the Jeans equation v v z z { z ( v v z )} { v v z v v z } (30.61) Since most of the stas in a disk galaxy ae nea the galactic plane, i.e., z, this tem is small. So except fo the egion 14
15 nea the galactic cente v v z 0 (30.6) Similaly, the deivative ν v v z { νz ( v vz )} z { ν v v z + ( v vz ) ( ν ν )} 0 (30.63) (And emembe this is the wost case scenaio. The close to the plane you ae, the moe likely that v v z = 0 by symmety.) So, to fist ode, the Jeans equation elates the z velocity dispesion to the galactic potential by Φ z = 1 ν ν v z z (30.64) To see the impotance of this, we can take the deivative of this equation, Φ z = { 1 ( ν v z ν z z )} (30.65) and then note that nea the galactic plane, the gadient of the potential is almost entiely in the z diection. Hence Φ z Φ = z { 1 ν z ( ν v z )} = 4πGρ (30.66) via Poisson s equation. Thus the mass in the disk can be measued via the z-motions of stas nea the galactic plane. 15
16 Jeans Equations in Spheical Coodinates The Boltzmann equation in spheical coodinates is + v + v θ ( v θ + v ϕ 1 ) Φ θ + v ϕ sin θ v + 1 ϕ + { v ϕ (v + v θ cot θ) + 1 sin θ ( vϕ cot θ v v θ Φ ) θ v θ } Φ = 0 (30.67) ϕ v ϕ and, fo steady-state systems with v = v θ = 0, the Jeans equation is ν v + ν { v ( v θ + v ϕ)} = ν Φ (30.68) Anothe way to look at this equation is to think of it in tems of hydostatic equilibium. ecall that the definition of pessue is P = 1 3 ρ v = ρ v i (30.69) whee v i epesents one component of the motion. So let P be the adial pessue impated by the stella motions, and Q be the tangential pessue tem, i.e., P = ν v Q = ν ( v ϕ + v θ ) (30.70) In this case, the Jeans equation looks is dp d + P Q = ν dφ d (30.71) 16
17 If the stella obits ae andomly distibuted (i.e., isotopic, so that ( v = vθ = v ϕ ), then P = Q, and the equation educes to the simple hydostatic equilibium. To descibe the degee of obital anisotopy in a spheical system, one often uses the paamete β, β = 1 v θ v (30.7) Fo isotopic obits, β = 0; fo puely adial obits, β = 1, and fo puely cicula obits, β =. Note that β need not be a constant thoughout the system; ealistic models of spheical systems often have β() as thei most impotant fee paamete. 17
Homework # 3 Solution Key
PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in
More informationPhysics 506 Winter 2006 Homework Assignment #9 Solutions
Physics 506 Winte 2006 Homewok Assignment #9 Solutions Textbook poblems: Ch. 12: 12.2, 12.9, 12.13, 12.14 12.2 a) Show fom Hamilton s pinciple that Lagangians that diffe only by a total time deivative
More informationToday in Astronomy 142: the Milky Way s disk
Today in Astonomy 14: the Milky Way s disk Moe on stas as a gas: stella elaxation time, equilibium Diffeential otation of the stas in the disk The local standad of est Rotation cuves and the distibution
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationThis gives rise to the separable equation dr/r = 2 cot θ dθ which may be integrated to yield r(θ) = R sin 2 θ (3)
Physics 506 Winte 2008 Homewok Assignment #10 Solutions Textbook poblems: Ch. 12: 12.10, 12.13, 12.16, 12.19 12.10 A chaged paticle finds itself instantaneously in the equatoial plane of the eath s magnetic
More informationClassical Mechanics Homework set 7, due Nov 8th: Solutions
Classical Mechanics Homewok set 7, due Nov 8th: Solutions 1. Do deivation 8.. It has been asked what effect does a total deivative as a function of q i, t have on the Hamiltonian. Thus, lets us begin with
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
More informationAs is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.
Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.
More information11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.
Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationPhysics 181. Assignment 4
Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physical insight in the sound geneation mechanism can be gained by consideing simple analytical solutions to the wave equation. One example is to conside acoustic
More informationS7: Classical mechanics problem set 2
J. Magoian MT 9, boowing fom J. J. Binney s 6 couse S7: Classical mechanics poblem set. Show that if the Hamiltonian is indepdent of a genealized co-odinate q, then the conjugate momentum p is a constant
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More informationCOLLISIONLESS PLASMA PHYSICS TAKE-HOME EXAM
Honou School of Mathematical and Theoetical Physics Pat C Maste of Science in Mathematical and Theoetical Physics COLLISIONLESS PLASMA PHYSICS TAKE-HOME EXAM HILARY TERM 18 TUESDAY, 13TH MARCH 18, 1noon
More informationAY 7A - Fall 2010 Section Worksheet 2 - Solutions Energy and Kepler s Law
AY 7A - Fall 00 Section Woksheet - Solutions Enegy and Keple s Law. Escape Velocity (a) A planet is obiting aound a sta. What is the total obital enegy of the planet? (i.e. Total Enegy = Potential Enegy
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More information1 Spherical multipole moments
Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the
More informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More informationGalaxy Disks: rotation and epicyclic motion
Galaxy Disks: otation and epicyclic motion 1. Last time, we discussed how you measue the mass of an elliptical galaxy. You measue the width of the line and apply the adial Jeans equation, making some assumptions
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationHomework 7 Solutions
Homewok 7 olutions Phys 4 Octobe 3, 208. Let s talk about a space monkey. As the space monkey is oiginally obiting in a cicula obit and is massive, its tajectoy satisfies m mon 2 G m mon + L 2 2m mon 2
More informationStress, Cauchy s equation and the Navier-Stokes equations
Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted
More information1D2G - Numerical solution of the neutron diffusion equation
DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both
More informationPROBLEM SET #3A. A = Ω 2r 2 2 Ω 1r 2 1 r2 2 r2 1
PROBLEM SET #3A AST242 Figue 1. Two concentic co-axial cylindes each otating at a diffeent angula otation ate. A viscous fluid lies between the two cylindes. 1. Couette Flow A viscous fluid lies in the
More informationA New Approach to General Relativity
Apeion, Vol. 14, No. 3, July 7 7 A New Appoach to Geneal Relativity Ali Rıza Şahin Gaziosmanpaşa, Istanbul Tukey E-mail: aizasahin@gmail.com Hee we pesent a new point of view fo geneal elativity and/o
More informationNewton s Laws, Kepler s Laws, and Planetary Orbits
Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationGENERAL RELATIVITY: THE GEODESICS OF THE SCHWARZSCHILD METRIC
GENERAL RELATIVITY: THE GEODESICS OF THE SCHWARZSCHILD METRIC GILBERT WEINSTEIN 1. Intoduction Recall that the exteio Schwazschild metic g defined on the 4-manifold M = R R 3 \B 2m ) = {t,, θ, φ): > 2m}
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More information, and the curve BC is symmetrical. Find also the horizontal force in x-direction on one side of the body. h C
Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Dynamics (Stömningsläa), 2013-05-31, kl 9.00-15.00 jälpmedel: Students may use any book including the textbook Lectues on Fluid Dynamics.
More informationThe Schwartzchild Geometry
UNIVERSITY OF ROCHESTER The Schwatzchild Geomety Byon Osteweil Decembe 21, 2018 1 INTRODUCTION In ou study of geneal elativity, we ae inteested in the geomety of cuved spacetime in cetain special cases
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi
ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,
More information3. Electromagnetic Waves II
Lectue 3 - Electomagnetic Waves II 9 3. Electomagnetic Waves II Last time, we discussed the following. 1. The popagation of an EM wave though a macoscopic media: We discussed how the wave inteacts with
More informationFinal Review of AerE 243 Class
Final Review of AeE 4 Class Content of Aeodynamics I I Chapte : Review of Multivaiable Calculus Chapte : Review of Vectos Chapte : Review of Fluid Mechanics Chapte 4: Consevation Equations Chapte 5: Simplifications
More informationd 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c
Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.
More informationLecture 7: Angular Momentum, Hydrogen Atom
Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z
More informationMechanics Physics 151
Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationRadial Inflow Experiment:GFD III
Radial Inflow Expeiment:GFD III John Mashall Febuay 6, 003 Abstact We otate a cylinde about its vetical axis: the cylinde has a cicula dain hole in the cente of its bottom. Wate entes at a constant ate
More informationPHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0
More informationIs there a magnification paradox in gravitational lensing?
Is thee a magnification paadox in gavitational ing? Olaf Wucknitz wucknitz@asto.uni-bonn.de Astophysics semina/colloquium, Potsdam, 6 Novembe 7 Is thee a magnification paadox in gavitational ing? gavitational
More informationPressure Calculation of a Constant Density Star in the Dynamic Theory of Gravity
Pessue Calculation of a Constant Density Sta in the Dynamic Theoy of Gavity Ioannis Iaklis Haanas Depatment of Physics and Astonomy Yok Univesity A Petie Science Building Yok Univesity Toonto Ontaio CANADA
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationRotational Motion. Lecture 6. Chapter 4. Physics I. Course website:
Lectue 6 Chapte 4 Physics I Rotational Motion Couse website: http://faculty.uml.edu/andiy_danylov/teaching/physicsi Today we ae going to discuss: Chapte 4: Unifom Cicula Motion: Section 4.4 Nonunifom Cicula
More informationPhysics 111. Ch 12: Gravity. Newton s Universal Gravity. R - hat. the equation. = Gm 1 m 2. F g 2 1. ˆr 2 1. Gravity G =
ics Announcements day, embe 9, 004 Ch 1: Gavity Univesal Law Potential Enegy Keple s Laws Ch 15: Fluids density hydostatic equilibium Pascal s Pinciple This week s lab will be anothe physics wokshop -
More informationPhysics 2212 GH Quiz #2 Solutions Spring 2016
Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 9
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Novembe 17, 2006 Poblem Set 9 Due: Decembe 8, at 4:00PM. Please deposit the poblem set in the appopiate 8.033 bin, labeled with name
More informationQuestion Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if
Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7
More informationF(r) = r f (r) 4.8. Central forces The most interesting problems in classical mechanics are about central forces.
4.8. Cental foces The most inteesting poblems in classical mechanics ae about cental foces. Definition of a cental foce: (i) the diection of the foce F() is paallel o antipaallel to ; in othe wods, fo
More informationHW Solutions # MIT - Prof. Please study example 12.5 "from the earth to the moon". 2GmA v esc
HW Solutions # 11-8.01 MIT - Pof. Kowalski Univesal Gavity. 1) 12.23 Escaping Fom Asteoid Please study example 12.5 "fom the eath to the moon". a) The escape velocity deived in the example (fom enegy consevation)
More informationPhysics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 18: System of Particles II. Slide 18-1
Physics 1501 Fall 2008 Mechanics, Themodynamics, Waves, Fluids Lectue 18: System of Paticles II Slide 18-1 Recap: cente of mass The cente of mass of a composite object o system of paticles is the point
More informationChem 453/544 Fall /08/03. Exam #1 Solutions
Chem 453/544 Fall 3 /8/3 Exam # Solutions. ( points) Use the genealized compessibility diagam povided on the last page to estimate ove what ange of pessues A at oom tempeatue confoms to the ideal gas law
More informationAnyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. --Niels Bohr. Lecture 17, p 1
Anyone who can contemplate quantum mechanics without getting dizzy hasn t undestood it. --Niels Boh Lectue 17, p 1 Special (Optional) Lectue Quantum Infomation One of the most moden applications of QM
More informationis the instantaneous position vector of any grid point or fluid
Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in
More informationTutorial Exercises: Central Forces
Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationATMO 551a Fall 08. Diffusion
Diffusion Diffusion is a net tanspot of olecules o enegy o oentu o fo a egion of highe concentation to one of lowe concentation by ando olecula) otion. We will look at diffusion in gases. Mean fee path
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationAH Mechanics Checklist (Unit 2) AH Mechanics Checklist (Unit 2) Circular Motion
AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Cicula Motion No. kill Done 1 Know that cicula motion efes to motion in a cicle of constant adius Know that cicula motion is conveniently descibed
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More informationA thermodynamic degree of freedom solution to the galaxy cluster problem of MOND. Abstract
A themodynamic degee of feedom solution to the galaxy cluste poblem of MOND E.P.J. de Haas (Paul) Nijmegen, The Nethelands (Dated: Octobe 23, 2015) Abstact In this pape I discus the degee of feedom paamete
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More informationB da = 0. Q E da = ε. E da = E dv
lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the
More informationDescribing Circular motion
Unifom Cicula Motion Descibing Cicula motion In ode to undestand cicula motion, we fist need to discuss how to subtact vectos. The easiest way to explain subtacting vectos is to descibe it as adding a
More informationLecture 23. Representation of the Dirac delta function in other coordinate systems
Lectue 23 Repesentation of the Diac delta function in othe coodinate systems In a geneal sense, one can wite, ( ) = (x x ) (y y ) (z z ) = (u u ) (v v ) (w w ) J Whee J epesents the Jacobian of the tansfomation.
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationRight-handed screw dislocation in an isotropic solid
Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We
More informationAstro 250: Solutions to Problem Set 1. by Eugene Chiang
Asto 250: Solutions to Poblem Set 1 by Eugene Chiang Poblem 1. Apsidal Line Pecession A satellite moves on an elliptical obit in its planet s equatoial plane. The planet s gavitational potential has the
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More information- 5 - TEST 1R. This is the repeat version of TEST 1, which was held during Session.
- 5 - TEST 1R This is the epeat vesion of TEST 1, which was held duing Session. This epeat test should be attempted by those students who missed Test 1, o who wish to impove thei mak in Test 1. IF YOU
More informationKEPLER S LAWS OF PLANETARY MOTION
EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee
More informationPHYS 705: Classical Mechanics. Small Oscillations
PHYS 705: Classical Mechanics Small Oscillations Fomulation of the Poblem Assumptions: V q - A consevative system with depending on position only - The tansfomation equation defining does not dep on time
More informationAP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws.
AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function
More informationPartition Functions. Chris Clark July 18, 2006
Patition Functions Chis Clak July 18, 2006 1 Intoduction Patition functions ae useful because it is easy to deive expectation values of paametes of the system fom them. Below is a list of the mao examples.
More informationMONTE CARLO SIMULATION OF FLUID FLOW
MONTE CARLO SIMULATION OF FLUID FLOW M. Ragheb 3/7/3 INTRODUCTION We conside the situation of Fee Molecula Collisionless and Reflective Flow. Collisionless flows occu in the field of aefied gas dynamics.
More informationCentral Force Motion
Cental Foce Motion Cental Foce Poblem Find the motion of two bodies inteacting via a cental foce. Examples: Gavitational foce (Keple poblem): m1m F 1, ( ) =! G ˆ Linea estoing foce: F 1, ( ) =! k ˆ Two
More informationSection 11. Timescales Radiation transport in stars
Section 11 Timescales 11.1 Radiation tanspot in stas Deep inside stas the adiation eld is vey close to black body. Fo a black-body distibution the photon numbe density at tempeatue T is given by n = 2
More informationVectors, Vector Calculus, and Coordinate Systems
! Revised Apil 11, 2017 1:48 PM! 1 Vectos, Vecto Calculus, and Coodinate Systems David Randall Physical laws and coodinate systems Fo the pesent discussion, we define a coodinate system as a tool fo descibing
More informationOn the Sun s Electric-Field
On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a
More informationSection 26 The Laws of Rotational Motion
Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to
More informationChapter 7-8 Rotational Motion
Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque,
More information16.1 Permanent magnets
Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and
More informationLecture 22. PE = GMm r TE = GMm 2a. T 2 = 4π 2 GM. Main points of today s lecture: Gravitational potential energy: Total energy of orbit:
Lectue Main points of today s lectue: Gavitational potential enegy: Total enegy of obit: PE = GMm TE = GMm a Keple s laws and the elation between the obital peiod and obital adius. T = 4π GM a3 Midtem
More informationUniversal Gravitation
Chapte 1 Univesal Gavitation Pactice Poblem Solutions Student Textbook page 580 1. Conceptualize the Poblem - The law of univesal gavitation applies to this poblem. The gavitational foce, F g, between
More informationLab #0. Tutorial Exercises on Work and Fields
Lab #0 Tutoial Execises on Wok and Fields This is not a typical lab, and no pe-lab o lab epot is equied. The following execises will emind you about the concept of wok (fom 1130 o anothe intoductoy mechanics
More informationA 1. EN2210: Continuum Mechanics. Homework 7: Fluid Mechanics Solutions
EN10: Continuum Mechanics Homewok 7: Fluid Mechanics Solutions School of Engineeing Bown Univesity 1. An ideal fluid with mass density ρ flows with velocity v 0 though a cylindical tube with cosssectional
More informationKEPLER S LAWS AND PLANETARY ORBITS
KEPE S AWS AND PANETAY OBITS 1. Selected popeties of pola coodinates and ellipses Pola coodinates: I take a some what extended view of pola coodinates in that I allow fo a z diection (cylindical coodinates
More informationPhysics C Rotational Motion Name: ANSWER KEY_ AP Review Packet
Linea and angula analogs Linea Rotation x position x displacement v velocity a T tangential acceleation Vectos in otational motion Use the ight hand ule to detemine diection of the vecto! Don t foget centipetal
More informationChapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E)
Geneal physics II (10) D. Iyad D. Iyad Chapte : lectic Fields In this chapte we will cove The lectic Field lectic Field Lines -: The lectic Field () lectic field exists in a egion of space suounding a
More informationCHAPTER 25 ELECTRIC POTENTIAL
CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When
More information= 4 3 π( m) 3 (5480 kg m 3 ) = kg.
CHAPTER 11 THE GRAVITATIONAL FIELD Newton s Law of Gavitation m 1 m A foce of attaction occus between two masses given by Newton s Law of Gavitation Inetial mass and gavitational mass Gavitational potential
More information