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Lecture Notes on Mathematical Astronomy I

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Lecture Notes on Mathematical Astronomy I Lecture notes on Mathematical Astronomy I Maurice H.P.M. van Putten June 14, 2015 ii c 2015 Maurice H.P.M. van Putten. All rights reserved. Contents Preface vii I Classical mechanics and accretion theory 1 1 Vector fields and finite differencing 3 1.1 Vectors and vector fields . 3 1.2 Estimating velocity . 8 1.3 Estimating acceleration . 10 1.4 Integration by finite summation . 11 2 Fitting data 15 2.1 Least squares . 16 2.2 An example . 17 2.3 Exercises . 18 3 Light propagation 19 3.1 Introduction . 19 3.2 Snell's law . 20 3.3 Principles of Fermat and Huygens . 23 3.4 Momentum . 28 3.4.1 Momentum in the Euler-Lagrange formulation . 29 3.4.2 Conservation of linear momentum . 30 3.4.3 Group velocity . 31 3.4.4 Relativistic dispersion relation . 33 3.5 Exercises . 37 4 Scaling laws and dimensions 39 4.1 The pendulum . 39 iii iv 4.2 Random walks . 43 4.3 Exercises . 46 5 Euler-Lagrange equations 47 5.1 The action principle . 47 5.2 Legendre transformation . 49 5.3 Hamiltonian formulation . 50 5.4 Exercises . 51 6 Globular clusters 53 6.1 Derivation of the main results . 55 6.2 Coefficients of relaxation . 58 7 Angular momentum vector 61 7.1 Introduction . 61 7.2 Recent experiments on Mach's principle . 62 7.3 Energy and torque . 65 7.4 Coriolis forces . 67 7.5 Spinning top . 68 7.6 Exercises . 69 8 Theory of thin accretion disks 73 8.1 Some astronomical constants . 73 8.2 Thin accretion disks . 75 8.2.1 Disk luminosity . 77 8.2.2 The α-disk model . 79 8.3 Gravitational waves from binaries and disks . 79 8.4 Exercises . 82 9 Binary evolution 83 9.1 Stars and their lifetimes . 83 9.2 Roche lobes . 85 9.3 Kepler orbits . 87 9.4 Mass transfer in binaries . 91 9.5 Supernovae in binaries . 93 9.6 Exercises . 94 10 Bondi-Hoyle-Lyttleton accretion 97 10.1 Bondi accretion . 98 v 10.2 Hoyle-Lyttleton accretion . 103 10.3 Exercises . 107 11 Fluid dynamics 109 11.1 Navier-Stokes equations . 110 11.1.1 Large and small Reynolds number limits . 114 11.1.2 Vorticity equation . 116 11.2 Viscous flow past a sphere . 117 − 5 11.3 Kolmogorov k 3 turbulence energy spectrum . 118 11.4 Jeans instability . 121 11.5 Exercises . 124 A Some units and constants 127 vi Preface Modern astronomy shows an evolving Universe rife with transient sources, mostly discovered - few predicted - in multi-wavelength observations. Within this decade, our window to the Universe is expected to include observations of extragalactic sources of high-energy neutrinos and gravitational waves. In light of these developments, how should we prepare a new generation of students? These lecture notes developed out of “flash notes" for a two-semester course on mathematical astronomy to advanced undergraduate students. The lectures are build around key concepts and techniques to modeling some typ- ical astrophysical processes, intended to familiarize the student with some of the methods and techniques of formulating problems (Part I) and developing approximate solutions (Part II). As such, the discussions on specific topics are mere preliminaries to more advanced treatments, left to more specialized courses. The present Part I is organized as follows. We start with a elements of classical mechanics (Ch. 1-5), illustrated on the problem of evaporation of globular clusters (Ch. 7). We next discuss numerical root finding and inte- gration of ordinary differential equations, such as the Hamiltonian equations of motion of many particle systems (Ch. 7). We then turn our attention to angular momentum in accretion disks (Ch. 8-9) and accretion flows in binaries (Ch. 10-11). Our final chapter introduces element of fluid dynamics and turbulence. Part II is devoted to function approximations based on concepts from linear algebra and complex function theory. It includes applications to po- tential flows and the Fourier transform, illustrated on the problem of signal processing, and an introduction to linear partial differential equations. vii viii Part I Classical mechanics and accretion theory 1 Chapter 1 Vector fields and finite differencing (Quote) In modeling the physical world with moving objects - stars, fluid flow, radiation, and so on - we commonly use vectors. They are used to denote positions and velocities at different points in space and time. They may be describing a model or may be determined from measurements. In these lectures, we briefly review some of vector calculus. In a subse- quent note, we will discuss the problem of linear regression to observational data in the language of matrix algebra. 1.1 Vectors and vector fields In a three dimensional Cartesian coordinate system (x; y; z) with unit vectors fi; j; kg, a vector a can be expanded in terms of its coefficients (a1; a2; a3) as a = a1i + a2j + a3k: (1.1) Ordinarily, we work in Euclidean space, which means that the length of a vector is defined as the square root of the sum of the squares of its coefficients, q 2 2 3 jaj = a1 + a2 + a3: (1.2) Two vectors a and b span a plane, given by linear combinations c = λa + µb; (1.3) 3 4 where λ and µ are real numbers, and defined as c = (λa1 + µb1)i + (λa2 + µb2)j + (λa3 + µb3)k: (1.4) If a and b are linearly independent, then (1.4) genuinely defines a two- dimensional plane. Conversely, we say that they are linearly dependent if λa + µb = 0 (1.5) for some λ and µ. Two linearly independent vectors a and b span a parallelogram. The projection of a onto b is defined by the inner product a · b = jajjbj cos θ; (1.6) where, conversely, a · b cos θ = (a; b) = (1.7) \ jajjbj denotes the cosine of the angle between the two. In particular, we have p jaj = a · a: (1.8) In three dimensions, there exist vectors that are orthogonal to a pair a and b. They are orthogonal to the plane spanned by a and b. The outer product a × b = (a2b3 − a3b2)i + (a3b1 − a1b3)j + (a1b2 − a2b1)k (1.9) defines a vector orthogonal to both according to the right handed rule, in the direction of movement of a corkscrew turned from a to b, whose length is the area of the parallelogram spanned by a and b, ja × bj = jajjbj sin θ: (1.10) Example. A corollary to the above is a formula for the distance between the end points of two vectors a and b, given by the length of the separation vector r = b − a. According to (1.8), we have jrj2 = (b − a) · (b − a) = jaj2 + jbj2 − 2a · b = a2 + b2 − 2ab cos θ (1.11) Vector fields 5 where we denote a = jaj and b = jbj. It is interesting to consider the recip- rocal, familiar from discussions on the Newtonian gravitational potential. If b > a, then we have an expansion 1 1 1 X al+1 = p = Pl(cos θ) (1.12) jrj a2 + b2 − 2ab cos θ a b in the Legendre polynomials Pl(x), where P0(x) = 1;P1(x) = x, and 1 1 1 P (x) = (3x2 − 1);P (x) = (5x3 − 3x);P = (35x4 − 30x2 + 3); ···(1.13) 2 2 3 2 4 8 Example. Consider a binary of two stars with masses M1 and M2. Let a and b point to M1 and, respectively, M2. The center of mass of the binary is defined by M1 M2 rCM = a + b: (1.14) M1 + M2 M1 + M2 You will note that M1 and M2 act as weights, the sum of which is one. The center of mass is relatively close to M1 if M1 > M2. Since vectors can be added and subtracted (the space of vectors is linear), we may choose a coordinate system following a translation such that rCM = 0. In this center of mass frame, we have M1a + M2b = 0: (1.15) Vector fields assign vectors to all points in some region of space, by allow- ing vector coefficients to be functions of the coordinates. Thus, we consider a(x; y; z) = a1(x; y; z)i + a2(x; y; z)j + a3(x; y; z)k; (1.16) or including a parametrization such as a(x; y; z; t) = a1(x; y; z; t)i + a2(x; y; z; t)j + a3(x; y; z; t)k: (1.17) It now becomes meaningful to differentiate vector fields, e.g., with respect to one of the coordinate variables, @xa = (@xa1)i + (@xa2)j + (@xa3)k: (1.18) Similarly, we introduce integration of vector fields, by integration each com- ponent using the rules of calculus. 6 Example. Let r = x(t)i+y(t)j+z(t)k denote a time-dependent position vector. Differentiation with respect to time defines the velocity vector dr(t) v(t) = r_ ≡ =x _(t)i +y _(t)j +z _(t)k: (1.19) dt Recall the motion of an apple falling from a tree. If the apple was taken off the tree by a gust of wind, then the initial velocity may have been non-zero. In this event, we have an equation for the acceleration along the vertical direction k and conservation of momentum in the direction of the gust, say, along the x−direction i, that is a(t) ≡ ¨r(t) = −gk; [r_(t)]x ≡ r_ · i = V; r(0) = hk: (1.20) Integrating once, we have for the velocity vector v(t) = r_(t), v(t) = V i − gtk: (1.21) Integrating once more, we have the position vector 1 r(t) = hk + V ti − gt2k: (1.22) 2 With a change of variables x = V t, the latter can also be expressed as g r(x) = hk + xi − x2k; (1.23) V 2 showing explicitly the parabolic shape of the trajectory of the apple as it falls to the ground.
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