Lecture Notes on Mathematical Astronomy I

Lecture notes on Mathematical Astronomy I

Maurice H.P.M. van Putten

June 14, 2015 ii

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 typical 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 integration 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 potential 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

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 subsequent 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 {i, j, k}, 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 |a| = 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 = |a||b| cos θ, (1.6) where, conversely, a · b cos θ = (a, b) = (1.7) ∠ |a||b| denotes the cosine of the angle between the two. In particular, we have √ |a| = 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,

|a × b| = |a||b| 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

|r|2 = (b − a) · (b − a) = |a|2 + |b|2 − 2a · b = a2 + b2 − 2ab cos θ (1.11) Vector ﬁelds 5 where we denote a = |a| and b = |b|. 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 = √ = Pl(cos θ) (1.12) |r| 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 allowing 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 component 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. When problems have symmetry, it is often advantageous to chose a corresponding coordinate system.

Example. Consider a particle moving in a circular orbit with radius R. In the XY-plane, the position is described by a vector with length R and poloidal angle ϕ, i.e.,

r(ϕ) = x(ϕ)i + y(ϕ)j = (R cos ϕ)i + (R sin ϕ)j = Rir (1.24) expressed in terms of a new basis vector along the radial direction, given by

ir = cos ϕi + sin ϕj. (1.25) Vector ﬁelds 7

Diﬀerentiating (1.24) with respect to ϕ,

dr = Ri , (1.26) dϕ ϕ we encounter a second basis vector orthogonal to the radial direction, given by

iϕ = − sin ϕi + cos ϕk. (1.27)

The pair (ir, iϕ) forms a set of orthonormal (orthogonal and of unit length) basis vectors, that rotate along with the position vector. In a mechanical setting, consider φ = φ(t) to be time-dependent. In this event, we have the velocity vector

dϕ(t) dr v(t) ≡ r˙(t) = = Rω(t)i (1.28) dt dϕ ϕ in terms of the instantaneous angular velocity

dϕ(t) ω(t) = . (1.29) dt

If the angular velocity is constant, then it equals the mean angular velocity

2π ω = Ω ≡ (1.30) P for an orbital period P . A commonly used alternative notation for vectors as expressed above is in terms of column vectors,

  a1 a =  a2  . (1.31) a3

As will become clear later, this notation is particularly convenient in linearly algebra, that extends the above to include linear transformations of vectors. 8 1.2 Estimating velocity

The physical world, measuring velocities from position vectors is the art of finding suitable approximations to the time-rate of change as defined by the derivative in the previous section. There are essentially two cases. First, we may have a model, such as for the apple falling from a tree in a constant background gravitational acceleration from the Earth. In this event, we may perform a model fit to data, such as measurements on the time at which the gust of wind took the apple from the tree, and proceed with the theoretical parabolic trajectory of the apple as it falls henceforth. In this event, velocities can be estimated by analytic differentiation of this trajectory based on a given acceleration g = −gk. However, there is often the need for a model independent analysis with no assumptions. For instance, we may wish to measure all the relevant physical parameters, including the gravitational acceleration. With no model used, we must resort to using approximations of the continuous derivative of position vectors. This is the art of numerical approximations, here in terms of finite differences. The derivative of a function f(t) is defined at t as the limit df(t) f(t0) − f(t) f 0(t) ≡ = lim . (1.32) dt t0→t t0 − t It means that f(t + h ) − f(t) f 0(t) = lim n (1.33) n→∞ hn ∞ exists for any sequence of small deviations {hn}n=0, hn → 0 as n → ∞. Let f(t + h) − f(t) f(t) − f(t − h) D+f(t) = lim ,D−f(t) = lim (1.34) h→0+ h h→0− h denote the right and left sided derivatives. If both exist and D+f(t) = D−f(t), then f(t) is differentiable at t in the sense of (1.32). In the real world, functions are typically differentiable except for a finite number of points, such as at discontinuities, or functions may be differentiable a finite number of times.1 1There exists a nowhere differentiable continuous function, constructed by Karl Weier- 0 strass (1872). For each real t, there exist sequences {tn} and {tn} converging to t such 0 that his function satisfies the inequality lim inf(f(tn) − f(t))/(tn − t) > lim sup f(tn) − 0 f(t))/(tn − t). Vector fields 9

Similar to the above, we define the one-sided and two-sided finite differences f(t + h) − f(t) f(t + h) − f(t − h) δ f(t) = , ∆ f(t) = . (1.35) h h h 2h How good are the approximations (1.35)? The derivative f 0(t) equals the slope of the tangent line to f(t) at t. So, the expressions (1.35) can be readily illustrated by looking at the graph of f(t), i.e., the curve (t, f(t)) in the XY-plane. Look at the lines passing through the points

A = (t, f(t)),B = (t + h, f(t + h)) (1.36) and compare it with the slope of the tangent at (t, f(t)); or consider the line passing through the points

A = (t − h, f(t − h)),B = (t + h, f(t + h)) (1.37) and do the same. You may notice that if f(t) is smooth and h is small, that ∆hf(t) does a better job than δhf(t). To estimate the degree of approximation in (1.35) to (1.32), consider the case when f(t) is smooth in the sense of having a Taylor series expansion about t. To ﬁrst, second or third order, it may have an the expansion2 1 1 f(t + h) = f(t) + hf 0(t) + h2f 00(t) + h3f (3)(t) + O(h4). (1.38) 2 6 Substitution into the expressions (1.35),

[f(t)+hf 0(t)+O(h2)]−f(t) δhf(t) = h 0 1 2 00 3 0 1 2 00 3 (1.39) [f(t)+hf (t)+ 2 h f (t)+O(h )]−[f(t)−hf (t)+ 2 h f (t)+O(h )] ∆hf(t) = 2h shows after some cancellations

0 0 2 δhf(t) = f (t) + O(h), ∆hf(t) = f (t) + O(h ). (1.40)

0 Thus, the approximation δhf(t) converges to f (t) linearly in h whereas 0 ∆hf(t) converges to f (t) quadratically in h (the errors go to zero linearly,

2A function f(t) is analytic at t if f(t) has a convergent Taylor series at t to all orders. The properties of analytic functions are explored in complex function theory. 10 respectively, quadratically in h). Hence, if the function f(t) has a second derivative at t, then it tends to be advantageous to use ∆hf(t), in that a relatively modest step size h may be providing a good approximation to f 0(t). In some instances, the approximations δhf(t) and ∆hf(t) are exact for linear, respectively, quadratic functions:

2 δhf(t) = b, f(t) = a + bt, ∆hf(t) = b, f(t) = a + bt + ct . (1.41)

Following two snapshots of the apple falling from the tree by a gust of wind, the constant horizontal velocity is adequately measured by δh and ∆h, whereas the velocity downwards (at the instance of time given by the mean of the times of the two snapshots) is adequately measured by ∆h but 2 not by δh. To see the latter, note that for f(t) = a + bt + ct , we have [a + b(t + h) + c(t + h)2] − [a − bt − ct2] δ f(t) = = b + 2ct + ch. (1.42) h h

The error in estimating the velocity at the midpoint t is ch. Instead, δh is exact for determining the velocity at the instant t + h/2 between the two snapshots, since b + 2ct + ch = b + 2c(t + h/2) - which is ∆h/2f(t + h/2). Evidently, some care in choosing h in (1.35) is needed. In case of circular orbital motion, we may estimate the velocity by ﬁnite diﬀerencing of the position vector, dr(t) r(t + h) − r(t − h) v(t) ≡ ' ∆ r = . (1.43) dt h 2h Given an orbital period P , we must insist h << P . Since the motion is periodic,

r(t) = r(t + P ), (1.44) choosing h = P/2 in (1.43) would give the estimate 0.

1.3 Estimating acceleration

The above is readily generalized to acceleration, by considering f(t + h) − 2f(t) + f(t − h) ∆2 f(t) ≡= (1.45) h h2 Vector ﬁelds 11

This construction obtains by finite differencing of finite difference approximations to the change in slope of f(t). Consider the slope at the two midpoints t ± h/2 obtained to second oder accuracy by f(t) − f(t − h) f(t + h) − f(t) ∆ f(t − h/2) = , ∆ f(t + h/2) = . (1.46) h h h h The approximate rate of change of these two (approximations to the) slopes of f(t) at t ± h/2 is [∆hf(t + h/2) − ∆hf(t − h/2)]/h as expanded in (1.45). How good is (1.45)? Again, the answer may be obtained by inspection of the graph of f(t). However, this may be more difficult then before. Changes in slopes refer to curvature in the graph, and these are generally difficult to determine by geometrical means or inspection. If f(t) is smooth and permits a Taylor series expansion to second order, then by (1.38) h2 h3 ∆2 f(t) = h−2 (f(t) + hf 0(t) + f (2)(t) + f (3)(t) + O(h4)) − (···) (1.47), h 2 6 where ··· refer to a similar expression as the preceding term with h replaced by −h. After cancellations, we are left with

2 00 2 ∆hf(t) = f (t) + O(h ) (1.48) showing second order accuracy. 2 Are there functions for which ∆h is exact? Consider once more the linear and quadratic polynomials (1.41). By direct evaluation, we ﬁnd

∆hf(t) = 2c (1.49)

2 as an exact result. So ∆h will be just ﬁne for the measurement of the gravitational acceleration of the Earth from three snapshots taken at equidistant 2 times (t − h, t, t + ht), where the choice of t and h are immaterial. Will ∆h be exact also for cubic polynomials?

1.4 Integration by ﬁnite summation

The converse of differentiation by finite differencing is integration by finite summation. Consider, for instance, the sum

n X s1 = v(ti)∆t, v(ti) = δhx(ti), ti = ht, h = ∆t. (1.50) i=0 12

Expanding this sum following the deﬁnition of δh, we have

n n X x(ti + h) − x(ti) X s = ∆t = (x − x ), (1.51) 1 h i i−1 i=0 i=0 that is,

s1 = −x0 + x1 + (x2 − x1) + ··· + (xn − xn−1). (1.52)

The expansion on the right is a telescoping sequence, in which all intermediate terms cancel, leaving

s1 = x1 − x0. (1.53)

Clearly, in estimating the velocity by first-order finite differencing δh gives the exact result. How about ∆h? A similar calculation applied to

n X s2 = ∆hx(ti)∆t (1.54) i=0 shows 1 s = [(x − x ) + (x − x ) + (x − x ) + ··· + (x − x )] . (1.55) 2 2 2 0 3 1 4 2 n n−2 After cancellations, we ﬁnd 1 1 s = [−x − x + x + x ] = x − x − [(x − x ) + (x − x )](1.56). 2 2 0 1 n n−1 n 0 2 1 0 n n−1

As an estimate, s2 = xn − x0 + O(h) is therefore ﬁrst order accurate. In the limit as h = ∆t drops to zero, s2 agrees with s1, and we write the result as a Riemann integral

Z T s = v(t)dt, (1.57) 0 where T = nh is kept constant in the process of taking h → 0. Table 1.1 gives a summary of our results. Table 1.1 highlights some of the main issues. Fitting data 13

Table 1.1. Overview of vectors and finite differencing Vectors a and vector fields a = a(x, y, z, t) are elements of a linear vector space with a Euclidean structure (distances, norms, inner and outer products).

Vectors can be expressed in component form relative to basis vectors, that may be constant, when using a Cartesian coordinate system, or rotating, as when using spherical, cylindrical or polar coordinates.

Vector ﬁelds can be diﬀerentiated and integrated, by applying the rules of calculus to their components in a Cartesian coordinate system.

Differentiation can be approximated by finite differencing, e.g., by a one sided or two-sided difference operator δh or ∆h, respectively. For smooth functions, they are first and second order accurate. Applying these operators twice produces a finite differencing estimate for the second derivative. For linear or quadratic expressions, δh and ∆h, respectively, give exact results. The errors in general are first or second order in the step size h.

Integration by finite summation is a converse of finite differencing. It recovers exact results - the Riemann integral - in the limit as the step size goes to zero. 14 Chapter 2

Fitting data

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As an application of the ideas in the previous chapter, we will discuss the problem of fitting observational data in the language of matrix algebra. Fitting is a common as a first step in the understanding and interpretation of data. A best-fit refers to a compromise that minimizes a pre-defined error. Consider two sets of observations A and B in (xi, yi):

xi yi xi yi 1 k 1 k + (A) (B) (2.1) 2 2 + 2 2 3 3 3 3

Here, k shall refer to 1 or 2 and represents an error. These data can refer to a short time series, e.g., observations on the height z = yi of our proverbial apple falling from the tree at instances ti = xi, i = 1, 2, 3. Or it can be the pricing of a security on the stock market in three consecutive days. Either way, we wish to look at the data with the support of a line that most closely resembles running the three points - which will be approximate whenever (k, ) 6= (1, 0).

15 16 2.1 Least squares

The problem of our ﬁt can be approached as the minimization of the squares of the errors i = yi − fi, expressed by the sum

3 X 2 ∆ = (yi − fi) (2.2) i=1 where fi = f(xi) is our choice of fitting function, here the linear function f(x) = a + bx. (2.3) Thus, our error is a function of the unknown coefficients (a, b), i.e., ∆ = ∆(a, b). Best-fit in terms of least squares aims at finding (a, b) such that ∂∆(a, b) ∂∆(a, b) = 0, = 0. (2.4) ∂a ∂b

Since (2.2) is quadratic in fi = f(xi), it will be quadratic in (a, b), so that (2.4) deﬁnes two equations linear in the (a, b). In other words, (2.4) deﬁnes a system of two equations in the two unknowns (a, b), that we should be able to solve algebraically. To be precise, (2.4) is

n n X X (yi − (a + bxi)) = 0, (yi − (a + bxi) xi = 0, (2.5) i=1 i=1 where we silently generalized to an arbitrary number of n data points. Writ- ing it out gives

n n n n n X X X X X 2 yi = na + b xi, xiyi = a xi + b xi . (2.6) i=1 i=1 i=1 i=1 i=1 This may be written somewhat cleaner by using the averages

n n 1 X 1 X x¯ = x , y¯ = y , (2.7) n i n i i=1 i=1 so that (2.6) becomes n nx¯ a ny¯ Pn 2 = Pn . (2.8) nx¯ i=1 xi b i=1 xiyi Fitting data 17

Dividing the ﬁrst equation by n, we have

1x ¯ a y¯ Pn 2 = Pn . (2.9) nx¯ i=1 xi b i=1 xiyi

Inverting (2.10) gives

Pn 2 a −1 i=1 xi −x¯ y¯ = D Pn (2.10) b −nx¯ 1 i=1 xiyi where

n X 2 2 D = xi − nx¯ . (2.11) i=1

It follows that

" n n # " n # 1 X X 1 X a = y¯ x2 − x¯ x y , b = x y − nx¯y¯ . (2.12) D i i i D i i i=1 i=1 i=1

2.2 An example

Let us apply (2.12) to our data set A in (2.1) for which n = 3. For k = 1, we ﬁnd

a = , b = 1. (2.13) 3

This result for the least square ﬁt of a straight line to (2.1) brings about the balanced approach, in that the contribution of each data point is weighted evenly by 1/n in a. In viewing the deviation from the straight line to be at the mid-point x2, the slope is the same as that of the line through (x1, y1) and (x3, y3). 18

Table 1.1. Overview of least squares fit Real-world data have scatter due to a variety of causes. These may involve random errors, noise, and fluctuations due to finite sensitivities to secondary processes (e.g., temperature, pressure, light, vibration, etc.).

A first step in interpreting data is often performing a fit, and, if desired, a fit to a linear function. The best fit in least squares error is a well-defined procedure for establishing the coefficients of the linear function for which the residual discrepancies are minimal in this sense.

Least squares gives a model-independent unbiased ﬁt, wherein all points are weighted equally.

2.3 Exercises

1. Derive the best ﬁt of a linear polynomial (2.3) to the ﬁrst set A with k = 2. Sketch the result including the error δ = (y1 − f1, y2 − f2, y3 − f3).

2. Derive the best ﬁt of a linear polynomial (2.3) to the second data set B in (2.1) for k = 1, 2. Sketch the results including the error δ = (y1 − f1, y2 − f2, y3 − f3). Chapter 3

Light propagation

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3.1 Introduction

Presently, the Universe is observed largely in light.1 The observed spectrum in electromagnetic waves extends from low-frequency radio waves, far infrared, to infrared, optical and the ultra-violet, to X-rays, gamma-rays up to TeV emissions. The propagation of light depends on the wave-length λ relative to the length scale L of geometries encountered in the course of propagation from source to detector. Quite generally, we distinguish

(a) λ << L, (b) λ ' L, (c) λ >> L. (3.1)

In case of (a), diﬀraction is negligible and light eﬀective propagates unen- cumbered along straight lines, at least in homogeneous isotropic media and especially so in vacuum. Case (b) describes a relatively strong interaction between light and geometry. For instance, in radio communication L = λ/2 describes the length of an antenna for an optimal transmission between sender and receiver. Case (c) describes relatively weak interactions, which may be encountered by long-wave length radio waves propagating past obstructions. Here, we focus in (a), which serves to illustrate the dual character of wave- and particle-like character. Following Planck and Einstein, we attribute en-

1Emerging are new windows of observations in Ultra-High Energy Cosmic Rays (UHE- CRs) by the Pierre Auger Observatory, in high-energy neutrinos by e.g. IceCube, and gravitational waves by the LIGO-Virgo and KAGRA experiments.

19 20 ergy and momentum to light according to

E = ~ω, p = ~k, (3.2) where ~ denotes the reduced Planck constant h/(2π). For light at a frequency ν and wave-length λ, 2π ω = 2πν, k = (3.3) λ denote the angular frequency and, respectively, the associated wave number. The velocity of light may vary in diﬀerent media. The index of refraction n denotes the attenuation in the velocity of light, i.e., c0 = c/n, where c denotes the velocity of light in vacuum. In non-dispersive media, where n is independent of frequency, the velocity of light satisﬁes ω/k.

3.2 Snell’s law

Light propagates through air similar to its propagation in vacuum to within about 0.01%. In the short wave length limit (a) in (3.1), light interacts with a mirror like the elastic bouncing of a ball oﬀ the mirror surface with no loss of energy or momentum. To begin, let θi denote the angle of the incident light in the (x, y) plane of a Cartesian coordinate system, where iy denotes the unit normal to the mirror and ix the unit vector tangent to the mirror surface and along the x−axis. The momentum vector can then be written as

pi = (p⊥)iiy + (p||)iix. (3.4)

The action of the mirror is described by the following interface (or jump) conditions, relating the momentum pr of the reﬂected light to that of the incident light by

(p⊥)r = −(p⊥)i, [p||] = 0. (3.5)

Here, we use the notion

[f] = (f)r − (f)i (3.6) for a change in the quantity f (which may be a vector or scalar quantity) upon reﬂection by the mirror. While the normal component of the momentum Light propagation 21

Figure 3.1: (Left.) Reflection of light in a mirror. Here, the normal component of the momentum pi of the incident photon is reversed in sign in the momentum pr of the reflected photon, while the component parallel to the mirror is conserved. The wave length λ (and the energy ~ω) of the photon is unchanged. (Right.) Refraction of light from air to water due to a change of the velocity of light by a factor 1/n. Since the angular frequency ω is conserved, the wave number of the refracted photon satisfies k0 = nk, where k = 2π/λ denotes the wave number of the incident photon. Commensurably, the wave length λ0 = λ/n is reduced as schematically illustrated.

“bounces oﬀ” the surface much like a ping pong ball on a hard surface, the tangential component is continuous. Light exerts a normal pressure with no shear stresses on an idealized mirror. Admittedly, the normal pressures exerted by incident light tend to be small under ordinary conditions. According to (3.2), we have p = E/c for a single photon, and hence a pressure

I P = cos2 θ (3.7) c i exerted by a beam with an intensity I (irradiance), [I] = erg s−1 cm−2. Daylight intensity from the Sun reaches about 1400 Watt per square meter (I = 1.4 × 106 erg s−1) at perihelion, corresponding to a normal pressure of about 5×10−5 dyn cm−2 (5 µPa). These pressures are generally imperceptibly small but can be measured in table top experiments. 2

2e.g. Demir, D., 2011, A table top demonstration of radiation pressure (Diplomarbeit, University of Vienna) 22

The equations (3.5) for a mirror can be extended for light propagation through an interface of air-to-water, taking into account and slow down of the velocity of light in water by the index of refraction n relative to the velocity of light in vacuum, i.e., c c0 = . (3.8) n For a given angular frequency ω (color) and wave number ω/c of the incident light, the wave number in water increases according to ω k0 = = nk. (3.9) c0

The associated momenta in air and water satisfy p = ~k in air and p0 = ~k0, respectively. With conservation of energy (ω = ω0) and continuity of the tangential momentum across the interface (no shear), we now have for an incident angle θi and an angle θr of the refracted light

0 = [p||] = (p||)r − (p||)i = ~ω/c (sin θi − n sin θr) . (3.10)

The result is Snell’s law

sin θi = n sin θr. (3.11)

Reflection of light in a mirror and propagation of light across two media according to Snell’s law illustrate the geometric-optics approximation, in which light propagates along piecewise linear trajectories. In generalizing the geometric-optics to stratified media described by a gradient in n, propagation is found to be along curved trajectories. The atmosphere is an example, where n−1 ' 2.7×10−4 at standard atmospheric conditions changes to essentially zero in the stratosphere and beyond. Light entering the atmosphere at a slight angle above the horizon (θi close to π/2) bends gently over towards the surface of the Earth (θr < θi) due to this slight discrepancy in n. It allows us to view the radiation from Sun for a few more minutes at sunset, even as it has already gone down under the visible horizon. Because n − 1 also depends on color with n − 1 relatively large for relatively long wave lengths, the effect is most pronounced towards red. A powerful principle to look at refraction across interfaces and the “bending of light” in stratified media is due to Fermat. Light propagation 23 3.3 Principles of Fermat and Huygens

Fermat’s principle posits that, in the geometric-optics approximation, light travels between two points A and B along a path of minimum time. Since it holds for each frequency of light, it equivalently states that light travels from A to B with the smallest difference in phase,3 defined by the product of the angular frequency and the time lapse δt = tB − tA, i.e., ∆φ = ω∆t (3.12) is minimal along the path from A to B traveled by light. Thus, the travel time or phase difference (3.12) assumes a (local) extremal value relative to that of neighboring paths. In a homogeneous isotropic medium, Fermat’s principle leads to the propagation of light along straight lines, since here it reduces to a principle of least distance between two points A and B. If C is any other point, then by the triangle inequality, 4 we have

|rA − rB| ≤ |rA − rC | + |rC − rB|, (3.13) where the ri refer to the position vectors to the points i =A, B and C. Snell’s law can be derived from Fermat’s principle by explicit consideration of a family of piecewise linear trajectories from A=(0, y)(y > 0) in air to B=(b, c)(c < 0) in water, deﬁned by the intersection at a point C=(0, x) at the air-water interface along the x−axis (Fig. 3.2). Thus, n = 1 in air (y > 0) above and n > 1 in water (y < 0) below the interface. The total time ∆t for the piecewise linear path satisﬁes p ∆t(x) = AC + nBC = x2 + y2 + np(x − b)2 + c2, (3.14) where n refers to the index of refraction of water. Fig. 3.2 illustrates cT (x) for the exaggerated value n = 2. The extremal value in φ(x) attains at d∆t(x) x x − b 0 = = + n = sin θi − n sin θr, (3.15) dx px2 + y2 p(x − b)2 + c2 which recovers (3.11).

3Phase is the appropriate invariant, when considering diﬀerent observers’ frame of reference. 4In ﬂat space with Euclidean metric, the triangle inequality holds locally and globally. In curved space with Euclidean signature, it holds locally. 24

Figure 3.2: (Left.) Schematic overview of two neighboring piecewise linear paths from A in air to B in water with corresponding diﬀerent intermediate points C1 and C2 at the interface on the x−axis. (Right.) Explicit calculation of light travel time from A = (0, 1) in medium 1 to B = (1, −1) in medium 2 via straight lines AC and CB with C = (0, x) on the interface between with index of refraction n1 = 1 and n2. By Fermat’s principle, the minimum in the curve shown identiﬁes C for the optimal piece wise linear path from A to B.

An instructive alternative derivation of (3.15) is given in the Feynman lectures, 5 by expanding Fig. 3.2 as shown in Fig. 3.3. The extremum in s(x) attains when the displacement ∆x (thick continuous line) in C deﬁned by two neighboring paths with C1 and C2 nearby is such that the lengthening in AC is canceled by the shortening of BC in regards to travel times, i.e.,

∆AC + n∆CB = 0. (3.16)

By inspection of Fig. 3.2, these length changes correspond to the thick dashed line-segments,

∆AC = ∆x cos γi, ∆CB = −∆x cos γr. (3.17)

With γi = π/2 − θi and γr = π/2 − θr, Snell’s law (3.11) follows once more. When a plane wave of light hits a small pinhole of a size less than the wave length, the wave front tends to disperse isotropically in all directions. This is predicted by Fermat’s principle (3.12), since the total phase change

5Feynman, R.P., Leighton, R.B., & Sands, M., Lectures on Physics, Vol. I, 1963, Ch. 26 Light propagation 25

Figure 3.3: The net change in travel time results from ∆AC =∆x cos γi and ∆CB=−∆x cos γr.

now depends on the distance to the pinhole with no dependence on direction. When of negligible size, the pinhole acts like an elementary point source in Huygen’s principle, that attributes the propagation of a wave front to a time t + ∆t to the envelope of waves over a time interval ∆t emanating from elementary point sources on an existing wave front at an earlier time t (Fig. 3.4).

Table 1 summarizes this discussion. 26

Figure 3.4: (a) Schematic overview of diﬀraction of a plane wave through a pinhole. The pinhole disperses the incoming wave with momentum pi isotropically in all directions in accord with Fermat’s principle if the pinhole size is small relative to the wave length. The pinhole represents an elementary point source in Huygen’s principle (b), where propagation of a wave front is modeled as the envelope (solid curve) of a superposition of waves from a continuum of point sources (solid dots) on a preceding wave front (dashed curve). Light propagation 27

Table 1 Geometric-optics approximation

1. The geometric-optics approximation applies when the wave length λ of light is much smaller than the length scale of the geometry at hand, in which case light propagation is described by rays;

2. In homogeneous media, light rays propagate as straight lines with conservation of photon energy ~ω and momentum p = ~k; 3. Light propagation across and interface between two homogeneous media with possibly diﬀerent indices of refraction is described by piecewise linear paths satisfying Snell’s law;

4. In stratiﬁed media such as the atmosphere, the geometric-optics approximation gives rise to a continuous bending of light from low into dense air in the direction of increasing index of refraction;

5. For light passing through a small pinhole, Fermat’s principle reduces to Huygen’s principle. 28 3.4 Momentum

The above discussion on Snell’s law and its derivations uses the notion of photon momenta and travel times representing total phase change. These notions are intimately connected. Consider a plane electromagnetic wave described by

iφ iφ E = E0e , B = B0e (3.18) transverse to the direction of propagation with Poynting ﬂux 1 S = E × B. (3.19) 4π For a traveling plane wave of one frequency (monochromatic) in a homogeneous medium along the x−axis, the total phase satisﬁes

φ = ωt − kx, (3.20) where ω denotes the angular frequency and k the wave number. For a general dispersive medium,

ω = ω(k) (3.21) denotes the dispersion relation. The dispersion relation in vacuum is linear, i.e.,

ω = kc. (3.22)

Equivalently, the velocity of propagation of light of all colors travels at the same universal speed. There are presently no indications of exceptions to (3.22) up to the highest photon energies detected. That is, light at diﬀer- ent energies from distant gamma-ray bursts across cosmological distances appears to have the exact same travel times to the Earth. The vacuum is non-dispersive. In glass, water and, to a lesser degree air, (3.23) changes to

ω = kc/n, (3.23) where n depends to some degree on wave length, such that n is relatively larger for light with longer wave lengths. Equivalently, n decreases towards Light propagation 29

1 in the limit as the wave length becomes short and photon energies become large. Air, therefore, is a slightly dispersive medium as alluded to in the previous section, whereby the Sun appears reddened at dawn and sunset. In the geometric-optics approximation, we model light by (piecewise linear) trajectories that are integral curves of the photon momenta. Momentum as such appears in many ways in our modeling of the physical world. We mention here the following, that highlights just some of its aspects.

3.4.1 Momentum in the Euler-Lagrange formulation The second order equations of motion for a point particle m with kinetic energy Ek moving in an external potential U derives from the Lagrangian L = Ek − U, by the Euler-Lagrange equations d ∂L ∂U − = 0. (3.24) dt ∂x˙ ∂x Here, we may deﬁne the momentum ∂L p = (3.25) ∂x to obtain the equation expressing the rate of change of momentum d ∂U p = − (3.26) dt ∂x resulting from the force F = −∂U/∂x. To exemplify (3.25), consider Newton’s problem of an apple falling from the tree, for which 1 E = mv2,U = mgh. (3.27) k 2

With motion and force is along the z−axis with unit normal iz, dh 1 ∂U v = i ,E = mh˙ 2, = mgi . (3.28) dt z k 2 ∂z z In this event, our coordinate x in (3.25-3.26) is the height h, giving a momentum and a familiar equation of motion ∂L p = = mh,˙ mh¨ = −mg. (3.29) ∂h˙ 30

Upon taking the apple into space, g will approach zero and the Lagrangian reduces to L = Ek. It follows immediately from (3.26) that d p = 0 : p(t) = p , (3.30) dt 0 showing that the momentum is constant. This is a general property. When- ever L does not depend on a coordinate, the momentum associated with that coordinate is constant, i.e., it is a conserved quantity. To exemplify this further, consider (3.26) for Kepler’s problem of planetary motion in circular orbits expressed in polar coordinates (r, θ) about the Sun of mass M. In the approximation of negligible radial motion, the kinetic energy reduces to 1 E = mr2θ˙2, (3.31) k 2 giving the Lagrangian 1 GM L = mr2θ˙2 + . (3.32) 2 r Upon taking our coordinate x in (3.25-3.26) to be θ, we obtain

∂L 2 ˙ d pθ = = mr θ, pθ = 0, (3.33) ∂θ˙ dt since the potential energy U = −GM/r is independent of θ. The momentum pθ will be recognized to be angular momentum J = mj, where j denotes the speciﬁc angular momentum 2π dA j = r2θ˙ = r2 = 2 (3.34) P dt for an orbit with period P . By (3.33), conservation of pθ, j is constant and orbital motion traces out a constant rate of change of surface area.

3.4.2 Conservation of linear momentum

When two bodies interact, they exchange momentum by forces. Let Fij denote the force on object j by object i, where i, j = 1, 2 (i 6= j). According to Newton’s third law, forces balance

F12 + F21 = 0. (3.35) Light propagation 31

Forces describe a rate of change of momentum, F12 = (d/dt)pj, and hence

d (p + p ) = 0. (3.36) dt 1 2

We may integrate (3.36) with respect to time over a time period ∆t = t2 − t1 that covers the duration of the forces. If the forces are transient, i.e., they vanish at t < t1 and t > t2, then the resulting changes in momenta satisfy

∆p1 + ∆p2 = 0, (3.37) where

Z t2 Z t2 ∆p1 = F21dt, ∆p2 = F12dt. (3.38) t1 t1

In daily life, we experience (3.37-3.38) in transportation. In taking a subway, accelerations and decelerations are transients at departure and upon arrival at stations. In between, which we experience the absence of forces in the process of linear motion (no change in our momentum), as illustrated in Fig. 3.5. Very short-duration momentum exchanges occur in the familiar “New- ton’s cradle” (with small initial separation, avoid ambiguity in solutions).

3.4.3 Group velocity From our discussions of light, we see that the notion of particles is much broader than objects in classical mechanics. Light has “particle-like” behavior and “wave-like” behavior depending on the experiment. Consider a superposition Φ = cos φ+ + cos φ− of two waves at slightly diﬀerent phases φ± = φ ± δφ/2, satisfying

1 1 1 Φ = cos(φ + δφ) + cos(φ − δφ) = A cos φ, A = 2 cos δφ . (3.39) 2 2 2

An observer receiving this signal may measure the modulation of signal strength described by variations in the amplitude A. These measurements may, for instance, be realized by correlating Φ with the carrier cos φ or by measuring its intensity by squaring Φ. 32

Figure 3.5: (Left.) Momentum as a function of time of a passenger in a train, here shown for two trips between three consecutive stations. The distances traveled scale with the areas a and b between the graph of p = p(t) and the horizontal axis p = 0. In between, passengers travel with constant momentum and experience no forces. Momentum changes in response to forces, where positive forces integrated over time lead to acceleration and negative forces lead to braking. The changes in momentum are given by the areas A and B between the graph of f = f(t) and the horizontal axis f = 0. In traveling between two consecutive stations, the two areas A (and B) shown are equal but opposite in magnitude. (Right.) In Newton’s cradle, an impulse by one ball onto a series of balls leads to conservative transfer of linear momentum through them by consecutive kicks between them. The last ball hereby deﬂects with the same as the incoming momentum. Here, the forces are large and of extremely short duration, whose area p = R f(t)dt remains ﬁnite. Light propagation 33

The modulations A derive from slightly diﬀerent frequencies in φ± = ω±t − k±x, here for a plane wave along the x−direction, whereby dω δφ = t − x δk (3.40) dk to linear order in a variation δk = k+ −k−. Modulations A hereby propagate with the group velocity dω v = . (3.41) g dk In case of the linear dispersion relation (3.22), we recover dω/dk = c. In general, however, (3.21) is nonlinear and the group velocity can be vastly diﬀerent from the phase velocity ω/k. More general results follow, up considering (3.41) equivalently as dE v = (3.42) g dp following (3.2). For instance, for a particle of mass m, we have the kinetic energy 1 p2 E = mv2 = . (3.43) k 2 2m With p = mv, (3.42) implies the identity

v = vg. (3.44)

The velocity of a particle in classical mechanics is the group velocity of the dispersion relation given by its kinetic energy.

3.4.4 Relativistic dispersion relation The dispersion relation and group velocity of a classical point particle (3.43- 3.44) and that of light in (3.21-3.22) have a common origin in a relativistic dispersion relation. It derives from Einstein’s theory of special relativity of particle motions in Minkowski space. In Cartesian coordinates (t, x, y, z), the line-element of Minkowski space is

ds2 = −dt2 + dx2 + dy2 + dz2, (3.45) 34 describing distances across inﬁnitesimally displacements (dt, dx, dy, dz) in ﬂat space. In particular, light propagates along the light cones

ds = 0. (3.46)

b Consider a particle of rest mass m0 with a world-line x (τ) as a function of the eigentime τ, measured buy an observer comoving with the particle. As before, the motion of the particle is described by momentum, here extended to the four-momentum dxb dt dx dy dz pb = m ub, ub = = , , , (3.47) 0 dτ dτ dτ dτ dτ in terms of the tangent velocity four-vector ub along its world-line, satisfying

c t 2 x 2 y 2 z 2 −1 = u uc = −(u ) + (u ) + (u ) + (u ) . (3.48) Consequently, the four-momentum pb has the invariant

2 2 c 2 p = m0u uc = −m0. (3.49)

2 2 2 For motion along the x−direction, (3.49) reduces to −pt + px = −m0. Using the hyperbolic representation

ut = cosh λ, ux = sinh λ, (3.50) the observed three-velocity satisﬁes dx ux v = = = tanh λ. (3.51) dt ut Thus, world-lines of massive particles exist inside the light cones (3.46), satisfying causality dx/dt ≤ 1 for all λ. 6 Let us denote E = −pt and p = px, so that q 2 2 E = m0 + p . (3.52) We now have a dispersion relation E = E(p) with group velocity dE p v = = = tanh λ, (3.53) g dp E

6 The quantity pt is conserved for particles in free fall (geodesic motion) whenever the background space-time is independent of t (when (∂t)b is a Killing vector). Light propagation 35 representing the observed velocity (3.51). This justiﬁes our interpretation of (3.52) as energy with v = vg and associated momentum

px = m0ux = mv. (3.54)

Here, we introduced√ the eﬀective mass m = m0Γ in terms of the Lorentz factor Γ = 1/ 1 − v2. 2 2 For non-relativistic three-velocities (p << m0), (3.52) satisﬁes

2 px 1 2 E = m0 + = m0 + m0v = E0 + Ek, (3.55) 2m0 2

2 where E0 = m0, i.e., E0 = m0c in cgs units with the velocity of light c, denotes the rest mass energy with additional kinetic energy Ek due to motion. 2 The same follows from expanding the eﬀective mass m = m0Γ = m0(1+v /2) in the same approximation v << 1. The ultra-relativistic limit in which v approaches the velocity of light corresponds to neglecting the rest mass m0. In the (singular) limit of m0 = 0, (3.52) reduces to

E = p. (3.56)

By (3.2), we thus recover (3.22). It shows that photons are particles of zero rest mass. The above may be generalized further in the Hamiltonian formulation, where H = H(x, p) denotes the total energy given by the sum of E in (3.52) and a potential energy U = U(x) as a function of position x. The Hamilto- nian equations of motion are

dp ∂H dp ∂H = − , = . (3.57) dt ∂x dt ∂p

Here, the ﬁrst equation describes the rate of change of momentum due to a force as in (3.26), whereas the second equation generalizes (3.53). Table 2.1 summarizes this discussion. 36

3.5 3 E=(1+p2)1/2 2.5 E=p

2 E 1.5

0.5

0 −3 −2 −1 0 1 2 3

1 E=(1+p2)1/2 0.5

0 dE/dp −0.5

−1 −3 −2 −1 0 1 2 3 p

Figure 3.6: Shown are the dispersion relations of a massive particle of mass m0 = 1 (thick line) and that of massless photons (dashed lines)(top) and the group velocity vg = dE/dp of the former (bottom). The group velocity is the observed particle three-velocity v. In the non-relativistic limit, p = mv ' m0v.

Table 1.1 Momentum in dispersion relations

1. Momentum results from force integrated over time. Total momentum is conserved as a consequence of Newton’s law that forces come in pairs of like magnitude and opposite sign; a

2. Impulse momenta result from brief moments of large forces, that introduce sudden changes in momenta in magnitude and/or direction, such as in collisions;

3. Energy and momentum are related by dispersion relations, whose derivative is the group velocity. The group velocity is the observed three-velocity of particle motion.

aMore precisely, momentum is conserved as a consequence of symmetries of the Lagrangian. Light propagation 37 3.5 Exercises

1. Show that light rays passing through a slab of glass propagate with a parallel displacement.

2. Consider a an ellipse with semi-major axis a and semi-minor axis b, a ≥ b, described by

x2 (y − a)2 + = 1. (3.58) b2 a2

1. Derive (3.58) from the deﬁnition of an ellipse as a closed curve of points with constant sum of distances l1 and l2 to two focal points, say, at (−p, 0) and (p, 0) along the x−axis. √ 2. Derive that p = a2 − b2. √ 3. Show that the minor semi-axis satisﬁes b = a 1 − e2 in terms of the ellipticity e.

4. Determine the equation of an ellipse corresponding to z = 0 in (3.58).

5. Show that (3.58) reduces to a parabola for small x and y about the origin.

6. Show that light rays coming in parallel to the y−axis reﬂecting on the parabola all pass through a common point on the y−axis. How does this point relate to p?

3. As Kepler discovered, particle trajectories in a Newtonian gravitational potential trace out a constant rate of change of surface area per unit time for arbitrary ellipsoidal orbits. The same holds true for hyperbolic orbits.

1. Give the sign of the total energy of closed, ellipsoidal orbits and for open, hyperbolic orbits.

2. Newton’s gravitational potential is a function of distance only. Restore the complete kinetic energy in (3.31) by including (1/2)mr˙2. Show that j in (3.34) is a constant of motion. 38

c 2 4 Show that the invariant p pc = −m0 implies that the product of the phase velocity vp = E/p and the group velocity vg = dE/dp satisﬁes vpvg = 1.

5. Light refraction in Newton’s prism emulates a rainbow: blue light is refracted relatively more than red light. Sketch the dispersion relation ω = ω(k) for glass. For reference, some particular glass used in prisms has n ' 1.54 for blue (400 nm) and n ' 1.51 for red light (700 nm), i.e., n varies by some 2% over the range of visible light. Compare your result with the dispersion relation of vacuum.

6. The dispersion relation of a plasma is of the canonical form q 2 2 2 ω = ω0 + c k . (3.59)

−3 For a fully ionized plasma with a free electron density n ([n] =cm ), ω0 is given by the plasma frequency

2 2 4πne ωp = . (3.60) me

Let ω = 2πνp. Show that νp ' 9 Hz for a plasma with n equal to 1 electron per cubic meter. Sketch the three dispersion relations for vacuum, a prism and a plasma in one dispersion diagram. What can you conclude for similar- ities and diﬀerences between the ﬁrst and the last?

7. Based on (3.59-3.60), explain the natural phenomenon of whistlers in the atmosphere following lightning, in the form of radio chirps decreasing from a few to about 1 kHz over a duration of a few seconds.

8. Explain why at sunset the sun looks reddened due to chromatic refraction in the atmosphere. [Hint. Applying Fermat’s principle to families of light rays with one kink.] Chapter 4

Scaling laws and dimensions

(Quote)

In this ﬁrst topic, we shall discuss some scaling laws. The motion of a pendulum and its viscous damping in air are an illustrative starting point, also as a ﬁrst encounter with viscosity in mediating (angular) momentum transport. In subsequent chapters, we shall apply some of these insights to accretion, which are ubiquitous in astrophysics.1

4.1 The pendulum

We can predict the essential behavior of a pendulum including, with some care, damping within a factor of a few by dimensional analysis. To begin, we note the principle variables describing its motion:

• The length l of the rope (assumed to be weightless),

• the mass m of the suspended weight, e.g., a sphere of radius R,

• the angle θ, describing the deﬂection of the rope to the vertical,

• the gravitational acceleration g (pointing down to the surface of the Earth).

1Rowan, L., Clery, D., & Coontz, R., 2005, “Everywhere You Turn,” Science, special addition “Disks in Space.”

39 40

These variables have the following dimensions

[l] = cm, [m] = g, [θ] = 1, [g] = cm s−2. (4.1)

Let us seek a combination, that is representative for the period P of the oscillation, as it swings forth and back, taking θ to oscillate between ±θ0, where the amplitude θ0 may be slowly decreasing due to damping. A natural choice is s l P = c , (4.2) 1 g where c1 is a dimensionless constant. You may recall the exact result of the p angular frequency ω = g/l, whereby c1 = 2π - a factor of order unity. As m swings through air, it experiences a viscous force similar to the force, when dropped into a bottle of viscous fluid. For the latter, at the age of 21, Leonard Euler (1707-1783) introduced the natural number e to describe the settling of the velocity to a constant velocity under the influence of the gravitational force. 2 To this end, we will use the formal experimental definition of the dynamical viscosity µ in terms of the tangential force per unit surface area A that develops between two plates, when sliding tangentially past each other with a relative velocity v at a separation L, v F = µA = Aτ. (4.3) L Here, τ denotes the surface stress. We tacitly assume that motion is sufficiently slow, such that the flow between the plates is linear. This is also referred to as a Stokes flow. In this low velocity limit of stationary flow, the inertia of the fluid between the plates is not relevant, known as lubrication theory. By (4.3), µ is dimensionful,

[µ] = pressure × s = g cm−1 s−1. (4.4)

The volume density of heat generated in (4.3) satisﬁes F v 1 q˙ = = τ 2. (4.5) AL µ

2Euler, L., 1728, “Meditatio in Experimenta explosione tormentorum nuper instituta (Meditation on experiments made recently on the ﬁring of cannon). Scaling laws and dimensional analysis 41

To describe the viscous force on m, we seek a combination of the dimensions of m, its linear size, say its radius R, and its velocity v to create a coeﬃcient k such that kµ corresponds to a force [F ] =g cm s−2. Following (4.3) and assuming a Stokes ﬂow, the viscous force is linear in v and it will scale with Rα for some positive α > 0. By inspection, the simplest choice is

Fv = c2µvR, (4.6) where c2 is some dimensionless constant. An exact analysis shows c2 = 6π. For a practical discussion, let us look up some values of µ and the associated kinematic viscosity µ ν = , [ν] = cm2 s−1, (4.7) ρ where ρ denotes the mass density of the ﬂuid.3 At standard ambient conditions (P = 1 atmosphere, T = 20 Celsius) we have for air

ρ = 1.1694 mg cm−3, µ = 1.7527 × 10−4 g cm−1 s−1, ν ∼ 0.14988 cm2 s−1(4.8).

In light of NASA’s recently started Mars exploration with Curiosity, let us also mention the corresponding quantities for CO2. At standard conditions, we have

ρ = 1.8467 mg cm−3, µ = 1.4633 × 10−4 g cm−1 s−1, ν ∼ 0.07924 cm2 s−1(4.9), since Mars’ atmosphere is 95% CO2 (at a low pressure of 0.6% atm). In applying our viscous force (4.6) to the problem of a swinging pendulum, it is important to check if the Stokes flow approximation is valid. For flow past a sphere, this approximation is quite restrictive. It is expressed by the condition of low Reynolds number, the dimensionless ratio of kinematic to viscous momentum transport, approximately satisfies vR Re = < 0.5, (4.10) ν using R to refer to the characteristic linear size of m. In other problems, such as flow in a pipe, we may calculate Re using the pipe diameter. In this event, there exists a critical Reynolds number of about 2500 marking the onset of turbulence, away from low Reynolds number Poisseuille flow.

3http://webbook.nist.gov/chemistry/ﬂuid 42

Consider, now, (4.2) scaled to a tiny pendulum,

1 l 2 P = 0.2 s, (4.11) 1 cm −2 using g ' 980 cm s . For an amplitude θ0 in the angular deﬂection, we have a velocity amplitude

1 θ l 2 v = θ lω = 3 0 cm s−1, (4.12) 0 0.1 1 cm where θ0 is in radians. By (4.10) and (4.8), it follows that 1 θ l 2 R Re = 0.37 0 , (4.13) 1 deg 1 cm 1 mm where we converted to θ0 in degrees. Only a very tiny pendulum with very small amplitude oscillations operates in the Stokes regime! How long will the pendulum swing, before it effectively comes to rest? In the Stokes regime (4.10), viscous damping is described by a corresponding time constant τ, mv m ρ R 2 = 6πµRv : τ = = 126 −3 s, (4.14) τv 6πµR 10 g cm 1 mm where we scaled to the mass density of silver. Thus, the effect of damping becomes appreciable after about 2 min, i.e., after about 1 −1 −2 P l 2 ρ R N = = 630 (4.15) τ 1 cm 10 g cm−3 1 mm periods of oscillation. A typical pendulum is much larger than our tiny pendulum with a swing of a few degrees. For these, damping is stronger than that described above as the associated Reynolds number exceeds (4.10) up to possibly several orders of magnitude. We then abandon our viscous force equation (4.6), in favor of a parameter regime governed by oscillatory flow if not fully turbulent flows. The resulting drag force obeys an entirely different scaling. For high velocities, it scales with the square of the velocity. This phenomenon of having different scaling behavior in different parameter regimes is not uncommon. In fact, we shall encounter it in our discussion of accretion disks further below. So, beware, surprises can and do crop up when dimensional analysis allows more than one scaling relation. Scaling laws and dimensional analysis 43 4.2 Random walks

The microphysics of molecular viscosity is random walks. At its most basic N level, it is described by a random series of points {xi}i=1 in space. Here, space may be of any dimension, i.e., the real line, the plane or three dimensional space. In a fluid, these random position would mark collisions between molecules, or the deflections of grains in Brownian motion. We next define ∆i = xi − xi−1 to be the change in position between xi and xi−1 and the following statistical moments q 2 hxii = x0, h∆ii = 0, h∆m∆ni = 0 (m 6= n), λ = h∆i i, (4.16) where the dispersion λ denotes the mean free path length. The vanishing of the cross correlations (m 6= n) represents the absence of any memory in the collisional process. If xi denotes the position of a particle at the i−th collision, we can estimate the dispersion D of the net displacement starting from the “telescope” expansion

xi = xi − xi−1 + xi−1 − xi−2 + ··· + x0 = ∆i + ∆i−1 + ··· + ∆1 + x0.(4.17) We have

2 2 2 2 2 2 2 xi = ∆i + 2 h∆i∆i−1i + ∆i−1 + ··· ∆1 + x0 = iλ + x0, (4.18) where the dots now include both variances of the ∆j and cross corrections between ∆m and ∆n (2 ≤ j ≤ i − 2, m 6= n, 1 ≤ m, n ≤ i). In the second equation in (4.18), we used our assumption of no memory and stationarity of the mean free path length. Also, we note the identity

2 2 2 2 2 2 2 (xi − x0) = xi − 2 hxix0i + x0 = xi − 2x0 hxii + x0 = xi − x(4.19)0 by deﬁnition of x0 in (4.16). If follows that 2 2 (xi − x0) = iλ , (4.20) and hence the variance of the distance of the end point xN to the starting 4 point x0 satisﬁes √ p 2 D = h(xN − x0) i = λ N. (4.21)

4In three dimensions, D = λpN/3, since the variance in each coordinate displacement is 1/3 times the variance in displacement, following spherical averaging over all directions of the displacement. See further Chandrasekhar, S., 1943, Rev. Mod. Phys., 15, 1. 44

For a stationary process, N ∝ t. It recovers the familiar statement describing propagation by molecular diffusion, √ D(t) ∝ t. (4.22) The above is a purely statistical frame work, and hence is quite general. Any specific reference to a medium would enter in λ and the collision rate, N = t/τ, where τ denotes the mean time between two consecutive collisions. Quite commonly, λ and τ are a function of temperature and pressure of the medium. A popular illustration of the above is the time it takes for a newly created photon in the center of the Sun to migrate, by a random walk, to the surface and escape into outer space. As a self-gravitating blob of gas it shows remarkable stability, which can be seen to rely entirely on the process of nuclear fusion in the core. By the Virial Theorem, the total kinetic energy Ek of the particles in the Sun, mostly protons, equals minus one-half the total gravitational binding energy, U = −GM 2/(2R) 5 1 GM 2 E = − U = , (4.23) k 2 4R 10 where R = R ' 7 × 10 cm denotes the Solar radius. For an ideal gas, we have Ek = PV , which we here use to define a mean pressure GM 2 ρ 3M P = 4 = nkBT, n = , ρ = 3 , (4.24) 16R mp 4πR denotes the pressure and its relation to the particle density n and temperature 3 T and V = (4π/3)R denotes the volume of the Sun. It gives an associated mean temperature

P mp GMmp −10 6 kBT = = ' 8 × 10 erg : T ' 5 × 10 K, (4.25) ρ 4R −16 −1 where kB = 1.38×10 erg K denotes the Boltzmann constant. The mean free path length of the photons is determined by the Thomson cross section −25 2 σT = 6.65 × 10 cm in their elastic collisions with protons, γ + p → γ + p (4.26)

5Here, the factor of one-half can be seen to derive from integration of the binding energy dU = −(GM/R)dM between M and a small mass element dM in case of a constant radius R. Scaling laws and dimensional analysis 45 according to an optical depth of unity: 1 λ = ' 2 cm. (4.27) nσT

The total number of scatterings involved in the migration of a photon√ to the surface of the star by way of a random walk follows from (4.21), R = Nλ. The corresponding diffusion time satisfies L Nλ R2 t = = = ' 1011s ' 3kyr. (4.28) d c c λc This time scale of 103−4 years can be put into perspective with the Kelvin- Helmholtz time scale, defined as the time it takes for the Sun to radiate off all its thermal energy by the escaping photons,

2 Eth GM tKH = = ' 8Myr. (4.29) L 4R L It follows that t d ∼ 10−3. (4.30) tKH The result shows that random walks of photons are sufficiently fast to keep the Sun in thermodynamic equilibrium, between heat produced in the center and heat radiated off from its surfaces. This energy transport from the core to infinity does not involve or invoke the heat capacity of the Sun. For the above, a more refined analysis would take into account that the density in the core is about 58 times the mean density in the Sun, giving rise to a significant increase in the diffusion time for photons to escape into the stellar envelope, see6. The results above can put into broader context by further recalling the main sequence lifetime TMS of the Sun of burning hydrogen into helium. 7 Given the Sun’s mass, TMS is about the age of the Universe. Summarizing the above, we have

ts << td << tKH << TMS, (4.31) where we included the sound crossing time scale of the Sun.

1. For (4.31), derive characteristic values for each of the time scales men- tioned. Chapter 5

Euler-Lagrange equations

In this second topic, we discuss elements of classical mechanics as it follows from the action principle. From the point of view of classical mechanics, the origin in an action principle is somewhat mysterious and appears to be a largely mathematical concept. However, the action principle is a remarkably powerful concept, that facilitates the set up of initial value problems (IVPs) for the evolution of systems with arbitrarily many particles, particularly in the case of zero or small dissipation. Specifically, the Euler-Lagrange equations of motion give us suitable second order ordinary differential equations (ODEs) and the Hamiltonian equations of motion do the same in terms of first order ODEs. The origin of the action principle derives from relativity, i.e., Lorentz invariance of the Lagrangian density, and the principle of stationary phase in a path integral formulation due to Richard P. Feynman. We will only scratch the surface of this intriguing topic, later, when discussing Huygens’ and Fermat’s principle. Here, we focus on the classical equations of motion implied by the action principle.

5.1 The action principle

Classical mechanics derives from magic: the classical trajectories of particles satisfy the action principle

Z T δS = 0,S = L(t)dt, L(t) = L(x(t), x˙(t)), (5.1) 0

47 48 Lectures in Astrophysics (c)2013 van Putten where L refers to the Lagrangian. As shown in class, it leads to the Euler- Lagrange equations d ∂L ∂L − = 0, (5.2) dt ∂x˙ ∂x which deﬁnes a 2nd order ordinary diﬀerential equation (ODE). We obtain an Initial Value Problem (IVP) with initial data of the form

x(0) = x0, x˙(0) = x1, (5.3) describing the initial coordinate positions and coordinate velocities of the particle. Example. The Lagrangian L = Ek − U is the diﬀerence of the kinetic energy and the potential energy. For a pendulum, we have 1 L = ml2θ˙2 − mgl(1 − cos θ) (5.4) 2 and the Euler-Lagrange equations (9.17) become g θ¨ + ω2 sin θ = 0, ω2 = . (5.5) l It describes the nonlinear pendulum equation in view of the sine function. By 1 3 5 a Taylor series expansion about θ = 0, sin θ = θ − 6 θ + O(θ ), we have ω2 θ¨ + ω2θ = θ3 + O(θ5) (5.6) 6 as a suitable starting point for perturbation theory. Noether’s theorem follows readily from the Euler-Lagrange formulation. N N In case of N particles with positions {xi}i=1 and velocities {x˙ i}i=1, we have d ∂L ∂L − = 0 (i = 1, 2, ··· N). (5.7) dt ∂x˙ i ∂xi If ∂L = 0 (5.8) ∂xi for some i, then d ∂L ∂L = 0 → = const.. (5.9) dt ∂x˙ i ∂x˙ i

That is, if xi is ignorable - a symmetry of the problem - it comes with a conserved quantity! In (5.9), this quantity is a conserved momentum. Euler-Lagrange equations 49 5.2 Legendre transformation

The Euler-Lagrange equations of motion represent a second order ODE. A common procedure for casting it in ﬁrst order form is by a transformation of variables

∂L (x, x˙) → (q, p), p = . (5.10) ∂x˙

Here, we take a derivative ∂L/∂x˙ as a new independent variable in passing fromx ˙ to p. To illustrate this step, let us turn to functions of one variable. If f(x) is convex, i.e., f 00(x) > 0, then f 0(x) is monotonically increasing whereby it is invertible, i.e.:

s(x) = f 0(x) (5.11) deﬁnes a 1-1 map between x and s. We are therefore at liberty to consider f(x) = f(x(s)) ≡ f ∗(s). Consider further

g(s) = sx − f(x), (5.12) where x = x(s). It is instructive to write (5.12) in symmetric form1

g(s) + f(x) = sx, (5.13) where we are at liberty to consider s = s(x) or x = x(s). With (5.11) in hand, diﬀerentiation of (5.13) with respect to s gives its counter part

x(s) = g0(s) (5.14) and diﬀerentiation twice, also with respect to x, gives the slopes

ds dx = f 00(x), = g00(s), (5.15) dx ds the product of which equals 1, following our assumption that f 00(x) is nonzero.

Given L = L(q, q˙) (following q = x), the Hamiltonian formulation casts the above in ﬁrst order form following a Legendre transform ∂L H =qp ˙ − L(q, q˙), p ≡ . (5.16) ∂q˙ We have the total variation ∂L ∂L ∂L ∂L dH = pdq˙ +qdp ˙ − dq − dq˙ = p − dq˙ +qdp ˙ − dq. (5.17) ∂q ∂q˙ ∂q˙ ∂q

Here, the ﬁrst term on the right hand side vanishes by our deﬁnition of p in (5.16). For solutions satisfying the Euler-Lagrange equations, we further have ∂L d ∂L = =p. ˙ (5.18) ∂q dt ∂q˙

Consequently, (5.17) reduces to

dH =qdp ˙ − pdq.˙ (5.19) showing that

H = H(q, p). (5.20)

The Legendre transformation (5.16) hereby obtains a reformulation in the canonical variables (q, p) of position and momentum, instead of position and velocity (q, q˙) in the original Euler-Lagrange formulation. Moreover, (5.19) gives the Hamiltonian system of equations ∂H ∂H p˙ = − , q˙ = . (5.21) ∂q ∂p We thus arrive at a first order initial value problem specified by initial data p(0) = p0, q(0) = q0. An extension to N particles is straightforward: ∂H ∂H p˙ni = − , q˙ni = (n = 1, 2, ··· N, i = x, y, z) (5.22) ∂qni ∂pni Globular clusters 51 with associated initial positions and momenta for each of the particles at t = 0. This is a system of 6N equations. 4 6 Globular clusters are old, dense systems of N ∼ 10 − 10 stars. If mi denotes the mass of each star, the Hamiltonian, given by the sum of the kinetic and gravitational binding energies, satisfies

2 N pi Gmimj H = Σi=1 − Σi>j , rij = |rj − ri| . (5.23) 2mi rij The ages of the 140 odd clusters around the Milky Way approach the age of the Universe. However, they are open self-gravitating systems. These systems have the peculiar property of having negative speciﬁc heat: they heat up when energy is extracted. And some energy is emitted into the galactic halo by escaping high-velocity stars, in response to which the globular cluster contracts, conform the Virial Theorem. A remarkable feature of escaping stars is the formation of tidal streams, apparent in a number of globular clusters. In class, we will highlight some of this by a discussion on the thermodynamics of evaporation, and the associated Ambartsumian-Spitzer and Kelvin-Helmholtz time scales. We shall explore some of it in the next Chapter.

5.4 Exercises

1. The following exemplify applications of Noether’s theorem: • Extend the Lagrangian for the pendulum by including motion about the vertical axis. Show that Noether’s theorem implies conservation of angular momentum about this vertical axis. For small angular momentum, obtain solutions to the equations of motion. • Obtain the Lagrangian for the canon ball. Ignoring friction in air, show that the horizontal momentum of the ball is conserved by Noether’s theorem. Obtain the horizontal distance of the canon ball traversed when it reaches ground level as a function of maximal height. • Obtain the Lagrangian for the Earth-Moon system. Reduce it to a one-body problem by expressing it relative to the center of mass, in which the sum of the momentum of the Earth and the Moon is zero. Derive the equivalent Hamiltonian and obtain the general solution. 52 Lectures in Astrophysics (c)2013 van Putten Chapter 6

Globular clusters

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In this second topic, we discuss globular clusters. Some 150 are known today as satellites to our own Milky Way. This number should be viewed as relatively few in light of some 12,000 globular clusters around the giant elliptical galaxy M87 in the Virgo cluster. The dynamics of globular clusters is described by the Hamiltonian

2 N pi N N Gmimj H = Σi=1 − Σi=1Σj>i < 0 (6.1) 2mi |ri − ri| for gravitationally bound systems of N stars, where N ∼ 104 − 106. Looking at (6.1) as a mathematical expression for the total energy, who would have thought it comes to life with the pristine beauty1 of M80 and NGC 6752 in Fig. 6.1? Some key issues in the structure and evolution of globular clusters as open self-gravitating systems are: 1. They are gravitationally bound (H < 0), giving rise to negative specific heat: energy extraction by heat flux and/or escapers makes H more negative, resulting in contraction and heating in the dynamical equilibrium described by the Virial Theorem; 2. Weak collisionless gravitational scattering between the stars tends to preserve thermal equilibrium, producing heat flux from excess heating

1“If one cannot see gravity acting here, he has no soul.” Richard P. Feynman, Six Easy Pieces, Chapter 5.

Figure 6.1: A sample of two famous globular clusters. (Left.) M80 (NG 5.53 6093, in Scorpius, D ∼ 10 kpc, M ∼ 10 M , age ∼ 12.5 Gyr) with KM 5.33 morhology. (Right.) NGC 6752 (in Paco, D ∼4 kpc, M ∼ 10 M , age ∼ 11.8 Gyr) with Post Core Collapse morphology (PCC). (Morphology data from Chernoﬀ, D., & Djorgovski, S., 1989, ApJ, 339, 904.)

radially outwards from the core to the outer layers and an occasional ejection of stars (escapers) into the galactic halo. The ﬁrst leads to core-collapse (CC) (gravitothermal catastrophe 2), the latter leads to evaporation.

3. The time to CC is about 20 (14) times the relaxation time as deﬁned by the the half-light (virial) radius, which is shorter than the evaporation time. Indeed, they appear with both smooth King morphology (KM) and a more central Post Core Collapse morphology (PCC);

4. Lucky - or unlucky - escapers tend to produce tidal tails3 and, ultimately, end up in the galactic halo;

2Lynden-Bell, D., Wood, R., 1968, MNRAS, 138, 495; Lynden-Bell, D., Eggleton, P.P., 1980, MNRAS, 191, 483; Lynden-Bell, D., 1988, Physics and Chemistry, arXiv/cond- mat/9812172; Antonov, V.A., Vest. Leningrad Univ. 7, 135 (Translation 1995, IAU Symposium 113, 525 (1962)) 3for a sample, see, e.g., Table 1 in van Putten, M.H.P.M., 2012, New A, 17, 411 Globular clusters 55 6.1 Derivation of the main results

A. Stars within the cluster interact by weak gravitational interactions, mainly 2-body encounters producing deflections in hyperbolic trajectories of one star, say of unit mass, passing another, of mass m. It is described by a (positive) total energy H and angular momentum l, given by 1 1 Gm 1 H = r˙2 + r2φ˙2 − = v2, l = r2φ,˙ (6.2) 2 2 r 2 where v denotes the velocity at infinity. We can apply a Möbius transformation u = b/r, where u = u(φ) in terms of the polar angle φ and b denotes the distance at nearest approximation (periastron), transforming (6.2) into

Gmb Gmb sin φ u00 + u = : u = 1 + . (6.3) l2 l2 sin(∆φ/2)

We can choose our polar coordinate system (r, φ), so that for our initial (i) and ﬁnal states (f) satisfy ui = uf = 0 (ri,f = ∞) with ∆φ ∆φ r˙ = ∓v, φ = π + , φ = − . (6.4) i,f i 2 f 2 By angular momentum conservation,r ˙ = −bu−2u0φ˙ = −(l/b)u0, i.e.,

Gmb cos(∆φ/2) l 2Gmb r˙ = × ' = v (6.5) f l2 sin(∆φ/2) b b2v∆φ for weak interactions with small ∆φ and relatively minor velocity variations along the trajectory, so that bφ˙ ' −v, so that l = r2φ˙ = −bv. It follows that 2Gm ∆φ ' . (6.6) bv2 As a star crosses a globular cluster, it will experience numerous small deflections. The sum will be zero, on average, but the dispersion, σ, of the net deflections will be nonzero as defined by the theory of random walks. Simplifying the interaction with the stars in the cluster by the interactions with its projection onto a disk with density Σ = N/(πR2), we have

N 2Gm2 δσ2 = Σ(2πbδb)(∆φ)2 = (2πbδb) (6.7) πR2 bv2 56 Lectures in Astrophysics (c)2013 van Putten we ﬁnd 2 2 8N Gm R σ = 2 2 log , (6.8) R v bmin where we used bmax = R. A natural scale for bmin obtains by putting the maximal (in absolute magnitude) binding energy on par with the kinetic energy in circular motion i.e., Gm = v2. (6.9) bmin

2 GM 2 By the Virial Theorem, 2Ek + U = 0, we also have Nmv = R , i.e., 2 GmN v = R , so that

R Gm −1 = N, 2 = N . (6.10) bmin Rv Our expression for the variance in the net deﬂection angles (6.8) becomes 2 log N σ = 8 N after one crossing time

3 − 1 2R 2R 2 N 2 tc = = √ . (6.11) v Gm How long does it take for a major deflection to occur? The variance σ2 scales linearly with time. We define one relaxation time trelax to be the time for a large variance of deflection angles to develop, t relax σ2 = 1, (6.12) tc that is

3 1 R 2 N 2 trelax = √ . (6.13) 4 Gm log N In the literature4, you will ﬁnd various estimates for (6.13), generally diﬀering within a factor of two, depending on the approximations used. For leading order estimates, (6.13) is more than accurate.

4e.g. Aarseth, S.J., 2003, Gravitational N-Body Simulations (Cambridge: Cambridge University Press) Globular clusters 57

B. Stars within the globular cluster are gravitationally bound to the cluster, i.e., their total energy is negative when taking into account the binding energy to all other stars in the cluster,

2 pi Gmimj Hi = − Σj6=i < 0. (6.14) 2mi |ri − rj| Due to frequent collisionless interactions, globular clusters preserve near- thermal equilibrium by relaxation. Exchange of energy and momentum in the random gravitational scatterings produces some stars with velocities beyond the escape velocity, i.e., their total energy Hi becomes positive. Slowly, the globular cluster evaporates by ejection of high velocity (HV) stars. Let 2 Gm 1 2 U = −Σj>i ,Ek = Σi mvi (6.15) |rj − ri| 2 denote the total binding and kinetic energy of all stars, subject to the Virial Theorem 2Ek + U = 0. The average binding and kinetic energy per star will be denoted by

2 2 2 ¯ 1 Gm 1 Gm Gm N ¯ 1 U = Σj>i = Σj6=i ' − , Ek = Ek, (6.16) N |rj − ri| 2N |rj − ri| 2R N

Then the kinetic energy k of an escaping star i is such that it exceeds to binding energy Gm2 ui = −Σj6=i (6.17) |rj − ri| to all other stars. Letu ¯ denote the average of the ui over all stars i = 1, 2, ··· N, i.e.,

u¯ = 2U¯ (6.18) by (10.23-10.24). Then by (10.23), the condition for escape is ¯ ¯ k > −u¯ = −2U = 4Ek. (6.19) We thus obtain Ambartsumian’s result that the escape velocity is twice the mean velocity of the stars in the cluster,

C. A general frame work for the evolution of the globular cluster is the system of two ordinary diﬀerential equations, describing the time rate of change of the number of stars N and the total energy H, dN dH = −Nτ −1, = Hτ −1 (6.21) dt AS dt KH with the Ambartsumian-Spitzer time scale τAS and the Kelvin-Helmholtz time scale τKH deﬁned as multiples of the relaxation time,

−1 −1 τAS = fN trelax, τKH = fH trelax. (6.22)

Here, fN and fH denote the fractional changes in N and H, respectively, per unit of relaxation time, due to the escaping stars.

6.2 Coeﬃcients of relaxation

5 What can we say about fN and fH ? Following Ambartsumian , we approximate the distribution of the energies in the escapers by the tail in the Boltzmann distribution (6.23), defined by velocities larger than the escape velocity. Let us specialize to an idealized globular cluster with equal mass stars. Admittedly, this is an approximation, but only moderately so for the relatively old globular clusters of the Milky Way. The kinetic energy of the stars (below the escape velocity) effectively satisfies the Boltzmann distribution. The number density of stars n, dN = nd3v, satisfies

e − k 1 2 2 2 3 n(e ) ∝ e kB T , e = m(v + v + v ), k T =e ¯ . (6.23) k k 2 x y z 2 B k

An estimate for the fN and fH now follows for an assumed shape of the tail of the velocity distribution. If we consider, following Ambartsumian, that the tail follows the Boltzmann distribution, at least for some energy range of 1 2 order kT beyond 2 meve , then

∞ 3 2 ∞ 3 2 R − 2 s 2 R 2 − 2 s 2 2 e 4πs ds 1 2 s e 4πs ds 1 fN = ∞ 3 2 ' , fE = ∞ 3 2 ' , (6.24) R − 2 s 2 135 R 2 − 2 s 2 29 0 e 4πs ds 0 s e 4πs ds where fE denotes the fraction of the total kinetic energy that exceeds the energy threshold for escape. Let us now look at the following two processes.

5Ambartsumian, V., 1938, Uch. Zap. LGU, 22, 19; in Dynamics of Star Clusters, Princeton, 1984, eds. J. Goodman & P. Hut (IAU Symposium, No. 113), 1985, p. 521 Globular clusters 59

1. Core-Collapse. Focusing on the core of the cluster, fE signiﬁes the production of excess heating that cannot be retained, i.e., it will be transported to the outer layers of the cluster. It leads to Core-Collapse (CC) known as the gravitothermal catastrophe. Considered as heat ﬂow within the cluster, leaving the potential energy essentially unchanged, we have

fH ' fE. (6.25)

2. Evaporation. If, however, we consider the implication to the cluster as a whole due to escapers, then the net energy ﬂux to inﬁnity is given by the excess of the mean kinetic energy of escapers at birth minus the required kinetic energy for escape. That is, fE fH = − 4 fN ' 0.71fN . (6.26) fN

By comparing the fH in (6.25-6.26) with fN , it follows that CC and total energy loss of the globular cluster (GC) as a whole - a net luminosity in kinetic energy to inﬁnity - satisfy, respectively,

CC GC τKH < τAS, τKH > τAS (6.27)

The uncertainties in the inequalities (6.27) stem from the degree to which our collisionless interactions are indeed modeled by weak two-body scattering. Following this assumption and the Virial Theorem and (10.23),

GN 2m H = − , (6.28) 4R so that the coefficients (6.24) define a closed system (6.21-6.22) of ordinary differential equations for (R,N), that is readily integrated. The result is an time to core-collapse, which is relatively short to the total evaporation time of the cluster, given by

tcc = 13.85trelax, tcc = 20.53trh, (6.29) where trelax is the relaxation time derived from R and trh is the relaxation time derived from rh (R = 1.3rh). This result (6.29) is in good agreement 60 with numerical simulations, supporting our ansatz for the tail of the distribution and our conclusion (6.27).

Exercise

1. Based on (6.15), derive (6.25) based on the notion that the potential energy U attains a minimum in the core of the cluster, whereby ∂U ' 0 (6.30) ∂ri for all particles i in the core, that satisfy ri ' 0. As a result, displacement of particles away from the core aﬀects N and Ek, but not U.

2. M80 and NGC 6752 are two well studied globular clusters of the Milky 5 5 Way. We have N = 3 × 10 , rh = 0.61 pc for M80 and N = 2 × 10 and 6 rh = 1.91 pc for NGC 6752. . Here, the half-light radius rh is related to the virial radius by Rvir = 1.3rh. • To estimate the relaxation time of the cluster as a whole, we may choose Rvir or rh, giving trh = ktrelax. Following trelax in (6.13), what is the conversion factor k?

• Calculate trelax following (6.13) for M80 and NGC 6752. • The time-to-core collapse is given by the relaxation-limited evaporative lifetime of a globular cluster according to (6.29). Calculate tcc for M80 and NGC 6752. How do the results compare with the classiﬁcation of M80 and NGC 6752 having KM and, respectively, PCC morphology? You may assume each to have the age of the Universe.

6Harris, W.E., 2010, Catalogue of parameters for Milky Way Globular Clusters, http://physwww.mcmaster.ca/ harris/mwgc.dat Chapter 7

Angular momentum vector

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7.1 Introduction

Angular momentum is interesting concept that appears in numerous problems with rotation. For freely rotating objects, angular momentum is a conserved quantity in proportion to angular velocity, where the latter is rep- resented by a vector. The length of this vector denotes the rate of rotation, and the direction of the vector denote the orientation of the rotation. For periodic motion with period P , the angular velocity satisﬁes 2π Ω = . (7.1) P If this motion is about a ﬁxed axis of rotation along a unit vector n, the angular velocity vector is Ω = Ω n. (7.2) A point mass in orbital motion at a separation r about to the axis of rotation has an instantaneous velocity, that is a tangent to the circle of radius r about this axis of rotation, dr v = = Ω × r. (7.3) dt The associated linear momentum is the vector d p = mr = mv, (7.4) dt

61 62 when the mass m is time-independent. The angular momentum (about the origin) of a particle with linear momentum at a distance r (away from the origin) is defined by J = r × p. (7.5) Since r and p are two vectors, J is a vector also. Hence, the transformation of J follows the transformation rules for vectors, such as when considering a translation or rotation of a coordinate system. In circular motion, J is a vector with the same orientation as the angular velocity Ω. Using the vector identity a × (b × c) = b(a · c) − c(a · b) (7.6) between any three vectors a, b, c, circular motion gives a specific angular momentum (angular momentum per unit mass) j = r × v = r × (Ω × r) = r2Ω = r2Ωn, (7.7) since r · r = r2 and r · Ω = 0. With (7.4-7.5), our model problem of circular motion, therefore, implies 2π πr2 dA j = r2 = 2 = 2 n, (7.8) P P dt that is, the specific angular momentum represents twice the rate-of-change of surface area traced out by the radius r in the motion of the particle. Based on (7.4-7.8), this is a geometrical identify, not restricted to circular motion. Hence, conservation of angular momentum J = mj (about the origin) implies a constant rate at which surface area is traced out by the position vector of the particle. The latter is Kepler’s third law, discovered in planetary motion.

7.2 Recent experiments on Mach’s principle

Following (7.5) and (7.9), circular particle motion satisﬁes J = I n,I = mΩr2, (7.9) where I denotes the moment of inertia about n.1 Evidently, (7.9) implies that J = 0 whenever Ω = 0 and visa-versa. In an astronomical context, we

1 It is formally denoted as Inn, since I is generally a two-index tensor. Angular momentum 63 may follow Mach2 and deﬁne the angular velocity as the rate of change of angles measured relative to the distant stars. Does (7.9) hold in general?

It turns out that angular momentum is sensitive to matter in the universe anywhere. While (7.9) holds true to great precision under ordinary circumstances when Ω is deﬁned relative to the distant stars, deviations appear in the proximity of massive rotating bodies. This can be detected in tracking the orientation n of a freely suspended gyroscope relative to a distant star. Recently, the NASA satellite Gravity Probe B 3 did just that, and measured an angular velocity in n at a minute rate of

ω = −39 mas yr−1 = −6 × 10−16 rad s−1. (7.10)

It agrees within a 20% window of uncertainty with the frame-dragging angular velocity of space-time around the earth, induced by the Earth’s angular momentum according to the theory of general relativity. A similar result was observed earlier in a complementary experiment by tracking the orientation of the orbits of the LAGEOS II satellites.4 According to the exact solution of rotating black holes in general relativity,5 (7.10) is the frame-dragging angular velocity at about 5 million Schwarzschild radii around a maximally spinning black hole with the same angular as the Earth (and 27 times its mass).

Though small, (7.10) deﬁnes a key result in our views on the relation between rotation and angular momentum, that comes out non-trivially in curved space-time around us as predicted by the theory of general relativity. In particular, it changes our perception of the ballerina eﬀect (Fig. 7.1). In reality, a ballerina standing still with respect to the distant stars experiences a slight lifting of her arms up, due to her non-zero angular momentum imparted by frame-dragging around the Earth. She would have to be co-rotating with an angular velocity (7.10) in order to feel her arms down in a fully relaxed state.6 64

Figure 7.1: (Left) In flat space-time, the ballerina effect a correspondence between zero angular velocity Ω relative to the distant stars and zero angular momentum J (Mach’s principle). (Right.) In curved space-time, the ballerina effect is different. Here, J = (Ω−ω)I, where I denotes the moment of inertia and ω is the frame-dragging angular velocity along the angular momentum JM of a massive object nearby. As a result, Ω = ω for J = 0 and J < 0 when Ω = 0. Mach’s principle is to be generalized include all matter, including massive objects in a local neighborhood. Angular momentum 65

Figure 7.2: Changing the orientation n of the angular momentum of a spinning wheel by a rotation introduces a component in an orthogonal direction, here along the vertical direction. Since angular momentum is conserved, a corresponding negative amount of angular momentum along the vertical direction is imparted by the person holding the wheel. The person will experience a counter-torque along the vertical axis. 7.3 Energy and torque

Angular momentum J = Jn can be changed by application of a torque, deﬁned as d d d T = J = n J + J n. (7.11) dt dt dt The dimension of torque is energy, as follows from [J]=g cm2 s−1 (mass times rate of change of area). Because angular momentum is a vector, (7.11) shows the appearance of a torque already when changing its orientation, even when keeping its magnitude constant. In this case, (7.11) may be due to a rotation, i.e.,

∆T = J (R − I) n (7.12)

2Ernst Mach (1838-1916) 3Everitt, C.W.F., et al, 2011, Phys. Rev. Lett., 106, 221101; www.einstein.stanford.edu 4Ciufolini, I., & Pavlis, E.C., 2004, Nature, 431, 958 5Kerr, R.P., 1963, Phys. Rev. Lett., 11, 237 6A similar result aﬀects her weight by the associated Papapetrou forces. 66 where R is a rotation. For a rotation over an angle ϕ about the x−axis, for example, we have  1 0 0  R =  0 cos ϕ − sin ϕ  (7.13) 0 − sin ϕ cos ϕ Feymnan7 gives an illustrative set-up that can be performed using a bicycle wheel attached freely to a rod. In this event, n is along the y−axis when the rod is initially held horizontally. Attempting to rotate the rod along the x−axis in an eﬀort to move the wheel overhead is described by (7.12), see Fig. 7.2. By (7.13), it introduces a component of ∆T along the z−axis. The person performing the rotation will experience a tendency to start rotating in the opposite direction to the angular momentum of the wheel, by conservation of total angular momentum in all three dimensions (in each of the three components x, y and z), i.e.,

Jwheel + Jperson = 0. (7.14) Since power is a scalar of dimension energy s−1, the power delivered to or extracted from a rotating object is given by the inner product of torque and angular velocity, i.e., P = Ω · T. (7.15)

d d For our circular motion, we have T = dt J = I dt Ω, and hence d 1 P = Ω2 (7.16) dt 2 It follows that the rotational energy in case of J = IΩ satisﬁes 1 1 E = Ω2I = Ω · J. (7.17) rot 2 2 Although (7.17) applies to non-relativistic mechanics such as spinning tops, somewhat remarkably it gives a fairly good approximation also to the rota- −1 2 tional energy Erot = k Ω · J, k = 2 cos (λ/2), of a rotating black hole with non-dimensional angular momentum sin λ, since 1 ≤ k ≤ 0.5858. (7.18) 2 7Feynman, R., 1963, Lectures on Physics, Vol. I (Addison-Wesley Publishing Co.), Ch. 20 Angular momentum 67 7.4 Coriolis forces

Conservation of angular momentum gives rise to apparent forces when moving things around by external forces that leave the angular momentum invariant, as in the absence of any frictional forces. The speciﬁc angular momentum in the presence of an angular velocity ω is

j j = ωσ2 : ω = , (7.19) σ2 where σ denotes the distance to the axis of rotation. Moving a ﬂuid element along the radial direction changes ω, as when the ballerina moves stretched arms inwards, according to δω = −2jσ−3δσ. It comes with a change in azimuthal velocity δvϕ = σδω satisfying

δvϕ = −2ωδσ. (7.20)

In vector form, (7.20) is

d v = −2ω × v . (7.21) dt ϕ σ This result is commonly expressed in terms of the Coriolis force

d F = m v = 2mv × ω (7.22) c dt ϕ Coriolis forces are particularly relevant when working in a rotating frame of reference. In particular, all of us terrestrial inhabitants living with the rotating frame ﬁxed to the Earth’s surface. Air moving to a diﬀerent latitude is subject to (7.22), since it changes the distance σ to the Earth’s axis of rotation, which is approximately polar. Let Ω denote the absolute angular velocity of the Earth (relative to the distant stars), and express the angular velocity of the air ω0 = ω − Ω relative to it, as measured in this rotating frame. Since δω0 = δω, moving air from, say, in the direction of the equator produces a retrograde azimuthal velocity (rotation at an angular velocity ω < Ω). Moving it a constant angular velocity towards the equator produces a curved trajectory in response to the (retrograde) constant Coriolis force (7.22). This may give rise to large scale circulation patterns in combination with pressure gradients. 68

Figure 7.3: Precession of the spinning top causes a velocity n˙ in the orientation n of the angular momentum, such that dJ/dt = Jn˙ absorbs the torque due to the gravitational force Fg applied at its center of mass CM. In the idealized friction-free set-up, this process involves no exchange of energy or dissipation.

7.5 Spinning top

The motion of a spinning top tilted at at angle θ exempliﬁes the interaction of angular momentum as a vector with a torque, T, applied continuously by the Earth’s gravitational force Fg as illustrated in Fig. 7.3. In general, we have the relations d d T = J = r × p = r × F . (7.23) dt dt g For a top that spins with no friction, the magnitude of its angular momentum vector is conserved. By (7.11-7.12), the top precesses at an angular velocity Ωp about the z−axis, Ωp = dφ/dt, satisfying d d J = J n = JΩ × n = Ω × J. (7.24) dt dt p p

By (7.23), T = ΩpJ sin θ = rW sin θ, and hence the angular velocity of precession about the vertical axis satisﬁes

ΩpJ = rW, (7.25) Angular momentum 69 where W denotes the weight of the top and r the distance of its center of mass away from its tip on the table. The Table 1.1 summarizes this discussion.

Table 1.1 Angular momentum 1. Angular momentum is a vector is related to angular velocity by the moment of inertia;

2. It can be changed by a torque and it stores rotational energy;

3. In non-relativistic classical mechanics Erot = (1/2)Ω · J; 4. Angular momentum is conserved in the absence of friction;

5. Angular momentum conservation gives rise to Coriolis forces when moving ﬂuid radially with major applications to weather patterns.

7.6 Exercises

1. Let

 1   0   2  a =  2  , b =  1  , c =  0  . (7.26) 0 2 1

Calculate

1. a × b · c

2. a × (b × c)

3.( a × b) × c

4. a · b × c 70

2. Consider a massive ring of radius R spinning at an angular velocity ω, whereby it attains an angular momentum per unit mass J = ωR2. Suppose it is mounted to one end of a rod, that is suspended at a pivot at the other end. An approximately horizontal rod hereby precesses with an angular velocity ωp about the vertical axis without dropping to a vertical position. Explain this process in terms of (7.24). In linear translation, such spinning ring does not produce any effects different from a non-spinning ring. As a result, precession is due to rotation of mass-elements in the ring about its center of mass (CM). That is, the precession effect is due to motion of each mass-element on a sphere of radius R about the CM, e.g., as if the CM where positioned at the pivot holding the rod. Derive (7.24) as a result of this motion, described by the combined effect of ω and ωp.[Hint. Consider a Cartesian coordinate system (x, y, z) about the CM and express it in spherical coordinates. This is an extension of the same in cylindrical coordinates about the z−axis. For instance, at an angular velocity vector ω = ωiz, the outer product ω × r is the rotational velocity vφ of the end point of a vector r and that vφ = ωσ, where σ denotes the distance to the axis of rotation.]

3. For a rotating black hole of mass M described by the Kerr metric, we can parametrize the angular momentum as J = M 2 sin λ,8 so that Ω · J = 2 2 M sin (λ/2) and Erot = 2M sin (λ/4). Derive (7.18).

4. Consider a neutron star of mass M = pM with uniform mass density ρ and radius R = q10 km.

1. Calculate the moment of inertia from the volume integral Z I = ρr2d3x (7.27) V

over the volume V = (4π/3)R3 of the star and express the result in terms of the scale factors p and q.

2. Held by self-gravity, calculate the maximal angular velocity Ω before break-up.

8van Putten, M.H.P.M., 1999, Science, 284, 115 Accretion disks 71

3. What is the maximal rotational energy Erot of the neutron star? [Hint: 2 The rotational energy satisﬁes Erot = (1/2)IΩ .]

4. What is the maximal ratio Erot/E0, expressed relative to the rest mass 2 energy E0 = pM c . Here, c denotes the velocity of light. 72 Chapter 8

Theory of thin accretion disks

(Quote)

8.1 Some astronomical constants

A few basic astronomical trivia that we commonly encounter are the masses of some familiar objects,

27 • The Earth: M⊕ = 5.7 × 10 g,

1 • The moon: ∼ 80 M⊕, 33 • The Sun: M = 2 × 10 g,

11 • A typical galaxy: 10 M , • The Universe: 1011 galaxies, 1055 g, and those of compact objects,

• White dwarfs: MWD < 1.5M ,

1 • Neutron stars: MNS ∼ 1 − 2M ,

• stellar mass black holes: 4 − 14M , that probably derive mostly as remnants of core-collapse in relatively high mass stars. Conceivably, the very low mass black holes are also produced by accretion induced collapse of neutron stars in interaction with a binary stellar companion,

1Lattimer, J.M., & Prakash, M., 2007, Phys. Rep., 442, 109, Fig. 3

73 74

6 9 • supermassive black holes: 10 − 10 M . Their mass is typically 0.1% of the mass of the host galaxy. Some exceptions apply, such as the 6 ∗ mass of about 3 × 10 M of the supermassive black hole SgrA , which is small in light of the total mass of about 1011 of the Milky Way.

What is a black hole? We still adhere to the original deﬁnition of Michell and Laplace of the late 18th century of black holes being black objects, from which no light can escape. Remarkably, this predicts the correct radius of a non-rotating Schwarzschild black hole following the classical Hamiltonian

1 GM H = v2 − (8.1) 2 r of a particle of unit mass at a distance r from a central potential well given by a gravitating mass M. At radius R, H = 0 gives for the escape velocity r 2GM v = . (8.2) e R

Any particle with a velocity v < ve will remain within r < R. Light, with a velocity c, thus remains trapped within the Schwarzschild radius

GM R = 2R ,R = , (8.3) S g g c2 where Rg is a commonly used deﬁnition for the gravitational radius of an object of mass M. Even though we used the classical Hamiltonian, we ar- rived at the exact result predicted by general relativity. This is is not merely a coincidence, as energy conservation is explicitly incorporated in general relativity by the Bianchi identity. The result is that massive bodies of radius RS are black, at least when neglecting (quantum mechanical) radiation processes. For many astrophysical applications, this interpretation of black holes as black object is useful. However, the surface of a black hole is a trapped surface commonly referred to as the event horizon, that is not hard, like the surface of a neutron star, but eﬀectively permeable. It allows for matter to accrete without a radiative back reaction, such as what ensues when matter falls on the surface of a neutron star. In particular, it allows, under some circumstances, sub-luminous accretion. This property of black hole event horizons is possibly relevant to the sub-luminous state of SgrA∗. Accretion disks 75

How is the mass of black holes measured? Various observational methods exist for determining the mass of black holes. Good results obtain by identification of mass motions in their environment, i.e., of stars, tori of gas or accretion disks, that act as test particles probing the gravitational field produced by the total mass-energy of a central black hole. In the absence of such mass measurements, a powerful constraint derives from time variable emissions from black holes in active nuclei, as when producing outflows in galactic and extragalactic sources alike. For instance, the extragalactic source PKS 2155-304 observed by H.E.S.S. showed short time scale variability on the order of 10 minutes in a > 200 GeV flare. A 10 minute time scale can be realized in a nucleus, provided its linear size is no larger than the corresponding light crossing distance, i.e., c3t R ≤ t c, M ≤ var ∼ 107M . (8.4) S var 2G 8.2 Thin accretion disks

Accretion disks are a natural outcome of the interaction of compact objects with their environment including stars in compact binaries. The strong gravitational field surrounding a compact object attracts matter, derived from nearby gas and perhaps by Roche lobe overflow directly from a stellar companion. The basic physics of an accretion disk is that of transport of mass inwards and angular momentum outwards, the first towards the central compact object such as a black hole, neutron star or perhaps a white dwarf, the second by radial expansion of the disk, radiation in a magnetic disk winds associated with astrophysical jets, or, more extreme, in gravitational radiation. Following our random walk model, molecular diffusion gives rise to a kinematic viscosity r kT ν = λc , c = , (8.5) s s m where λ denotes the mean free path length between two consecutive collisions between atoms in air or protons in a fully ionized accretion disk and cs denotes the isothermal sound speed. We will consider a thin accretion disk, whose scale height H = H(r), where r denotes the radial distance to the central object, is much smaller than 76 r. The density in the disk is vertically stratified, much like the atmosphere on Earth. The vertical component of the gravitational field from the central object with mass M satisfies GM GMz g = − tan θ = − (8.6) z r2 r3 z where tan θ = r denotes the poloidal angle relative to the equatorial plane. In the isothermal approximation, balance of pressure p = ρcs with gz, 1 ∂p = g (8.7) ρ ∂z z gives rise to the vertical density distributen ρ with vertically integrated col- R H(r) umn density Σ(r) = −H(r) ρ(r, z)dz, satisfying √ −z2/H2 ρ(r, z) = ρ0e ,H(r) = 2rcs/vK , (8.8) where H denotes the scale height. For a Keplerian orbital velocity vK = q GM rΩ = r at radius r, the disk will be thin whenever vK >> cs, i.e., when the disk is relatively cold and its orbital motion is supersonic. A Keplerian disk is in a state of differential rotation: matter at inner radii rotate at angular velocities larger than that at outer radii. According to (4.6), viscous transport of angular momentum outwards will ensue by the associated velocity shear between concentric surface elements, here dΩ 3 r = − Ω (8.9) dr 2 representing the v/L in the plane parallel problem (4.6). The associated viscous force on a cylindrical cross section of the disk follows directly, giving rise to an azimuthal torque upon including an additional factor r,

Z H dΩ 3 dΩ τφ = r ρν r 2πrdz = 2πνΣr (8.10) −H dr dr In a stationary state, viscous transport of angular momentum outwards by ˙ τφ and convective angular momentum inwards due to an accretion rate M is constant across the disk:

˙ 2 τφ + MΩr = C, (8.11) Accretion disks 77 where C is some constant. A complete boundary value problem for this ﬁrst order diﬀerential equation obtains by providing one boundary condition. In our present model, we shall use the zero stress boundary condition at the 2 inner radius r = rin, putting C = 0. Integration then gives

1 r 2 3πνΣ = M˙ 1 − in . (8.12) r

At large radii, it shows the asymptotic radial drift velocity 3ν v = − . (8.13) r 2r

8.2.1 Disk luminosity Following (4.5), the energy dissipation per unit volume satisﬁes

1 dΩ2 dΩ2 q˙ = ρνr = ρν r , (8.14) ρν dr dr and hence the dissipation per unit surface area at radius r is

Z H 2 ˙ 1 dΩ 3GMM rin 2 jq = qdz˙ = Σν r = 3 1 − , (8.15) −H dr 4πr r where we used rdΩ/dr = −(3/2)Ω in Keplerian motion. The total luminosity of the disk follows by integration,

Z ∞ ˙ GMM ˙ 2 Rs L = 2π jqrdr = = Mc . (8.16) rin 2rin 4rin In view of (8.12), the disk luminosity (8.16) is proportional to ν. This raises the question: how much luminosity can molecular viscosity (8.5) support?

2The state of the inner disk is a key property in astrophysical accretion disks. Already, general relativity introduces a minimum radius, deﬁned by the Inner Most Stable Circular Orbit (ISCO), in genuine departure from Newtonian gravity. Around non-rotating black holes, the radius of the ISCO is 6Rg, while it is smaller for prograde orbits around rotating black holes, down to Rg around an extreme Kerr black hole. In addition, the inner face of the accretion disk may be subject to Maxwell stresses in the presence of magnetic ﬁelds. 78

First, let us express ν relative to some ﬁducial values of our model parameters. Recall the condition λnσc = 1, where σc is the cross section of pp collisions. The typically proximity between two protons that can be attained is determined by a balance between their repulsive Coulomb interaction and 2 the kinetic energy as set by the temperature. That is, σc = πrc , where e2 e2 πe4 = kBT : rc = , σc = 2 . (8.17) rc kBT (kBT ) We thus obtain 2 1 (kBT ) λ = = 4 , (8.18) nσc nπe and hence s 5 kBT (kBT ) 2 ν ∼ λ ∼ 4√ . (8.19) mp nπe mp Let’s look at this result for some characteristic values, e.g., n = 1016 cm−3 6 −10 −16 −1 and T = 10 K. With e = 4.8 × 10 esu and kB = 1.38 × 10 erg K , −18 2 6 −1 σc = 8.75 × 10 cm , λ = 11 cm, cs = 9 × 10 cm s , (8.20) and so 5 T 2 n −1 ν = λc ∼ 108 cm2s−1. (8.21) s 106K 1016cm−3

Next, let us consider (8.12) with Σ ' 2Hρ, ρ = nmp, to estimate

3 2 ˙ 24 3 GM r M ' 3πΣν ' 6πmpcsH/σc = 4 (kBT ) 3 (8.22) e c RS at r >> rin. Scaled to some characteristic values, we have

3 3 2 M˙ = 6 × 103 T r M g s−1 106K RS M 3 (8.23) 3 2 −14 ˙ T r ' few × 10 MEdd 106K RS ˙ 2 17 in terms of the Eddington accretion rate MEdd = LEdd/c = 10 (M/M ) −1 38 g s associated with the Eddington luminosity LEdd = 10 (M/M ) erg s−1. For the active nuclei that we observe - in galactic X-ray binaries and extragalactic AGN - (8.23) is essentially negligible. Accretion disks 79

8.2.2 The α-disk model The observed Eddington luminosities in numerous cases force us to consider a physical process diﬀerent from molecular viscosity. Shakura and Synyaev3 proposed a phenomenological Ansatz in the form of an α-disk model wherein viscosity is mediated by turbulent eddies,

ν = αHcs. (8.24)

Here, α is a dimensionless constant of order unity. If suﬃcient turbulence develops, (8.24) can greatly exceed (4.7) allowing for Eddington limited accretion rates. The onset to turbulence now appears to be facilitated in MHD due to the Magneto-Rotational Instability in poloidal magnetic ﬁelds. 4

8.3 Gravitational waves from binaries and disks

Gravitational waves are increasingly important in a broad trend towards mul- timessenger astronomy and astrophysics focused on the Transient Universe. Non-Newtonian properties of curved space-time may be governing the evolution and radiation processes in the most violent astrophysical processes since the Big Bang, in cosmological gamma-ray bursts, mergers of compact objects, and some hyper-energetic core-collapse supernovae. Gravitational physics is gradually becoming an experimental science. The discovery of the Hulse-Taylor binary pulsar system PSR 1913+16 provided us with a system to successfully test our understanding of gravitational wave emission as predicted by general relativity.5 (The agreement with observations is within 0.1%.6) Recent measurements by the LAGEOS II 7 and Gravity Probe B 8 satellites provided a test for frame dragging around the Earth, again conﬁrming general relativity to within measurement error.

3Shakura, N.I., & Sunyaev, R.A., 1973, A&A, 24, 337. 4Balbus, S.A., & Hawley, J.F., 1991, ApJ, 376, 214; Hawley, J.F., & Balbus, S.A., 1991, ApJ, 376, 223. 5Russel A. Hulse and Joseph H. Taylor received the Nobel Prize in Physics, 1993, for their discovery. 6Weisberg, J. M., & Taylor, H., in Radio Pulsars, eds. M. Bailes, D. J. Nice & S. E. Thorsett, San Francisco: APS 2003 (Conf. Series CS 302), p. 93. 7Ciufolini, I., & Pavlis, E.C., 2004, Nature, 431, 958. 8Everitt, C.W.F., et al., 2011, Phys. Rev. Lett., 106, 221101. 80

Our next step is to develop and utilize the window of gravitational wave observations to probe some of the greatest mysteries in the Transient Universe: the inner engines of the aforementioned and most relativistic catastrophic events in the sky. To this end, various large gravitational wave detectors are currently being built in the US, Europe and Japan. The Hulse-Taylor binary is a system of two neutron stars with

• Mass M1,2 of the two neutron stars: MNS ' 1.44M ,

• Semimajor axis: a = 2 × 1011 cm,

• period: P = 7.75 hr,

• ellipticity: e = 0.617.

The angular velocity ω = 2π/P satisﬁes

G(M + M ) 2GM R 2GM ω2 = 1 2 ' NS , ω2 = S ,R = NS . (8.25) (2a)3 8a3 4a3 S c2

In the second expression, we converted to geometrical units, wherein mass (MNS → RS) and time (t → ct) are expressed in cm. The associated binding energy can similarly be expressed in geometrical units,

GM M R2 U = − 1 2 → U = − S . (8.26) 2a 2a By the Virial Theorem, binding energy is about twice the kinetic energy (equal in their time-averaged values),

2 2 −U ' 2Ek = RSv ' RS(ωa) . (8.27)

According to general relativity, energy produces a strain amplitude in space time, here in the form of a gravitational wave describing the transverse periodic deformation of circles into ellipses (linear polarization, as seen in the plane of the binary) or rotation of ellipsoidal deformations (circular polariza- tions, as seen along the axis of rotation of the binary). These strains reflect the rotation of the gravitational tidal field along with the binary motion. The dimensionless strain amplitude h measured at a distance r from the source will be proportional to the dimensionless ratio defined by the energy in the Accretion disks 81 tidal field (in geometrical units) relative to r, much like the calculation of a (very small) angle. By (A.1), we have the wave amplitude E R hˆ(r) ∼ k ' S (ωa)2. (8.28) r 2r The luminosity in h(r, t) = hreˆ ikr−iωt follows the standard expressions for a transverse wave,

∂h2 L ∼ 4πr2 ∼ R2 a4ω6. (8.29) GW ∂t S This result should be compared with the exact quadrupole formula for gravitational wave emission from two masses M1 and M2 in a circular binary (e = 0),

3 3 5 5 32 32 10 M M 2 4 6 3 1 2 LGW = µ a ω = (ωµ) , µ = 1 (8.30) 5 5 (M1 + M2) 5 in units of c5 L = = 3.6 × 1059erg s−1 = 1.8M c2s−1 (8.31) 0 G

− 6 For an equal mass binary, µ = 2 5 RS, so that (8.29-8.30) agree within a 32 12 − 5 factor of 5 × 2 = 1.21, which is better than anticipated. However, the Hulse-Taylor binary is highly elliptical. As a result, radiation also appears in higher harmonics, which signiﬁcantly increases its gravitational wave luminosity beyond that predicted by circular motion. This enhancement is described by an additional factor in the Peters & Mathews formula, 9

73 2 37 4 32 2 4 6 1 + 24 e + 96 e LGW = µ a ω F (e),F (e) = 7 . (8.32) 5 (1 − e2) 2 For the Hulse-Taylor binary, the ellipticity e = 0.617 increases the luminosity by an order of magnitude to

31 −1 LGW = 7.35 × 10 erg s (8.33)

9Peters, P. C., & Mathews, J. ,1963, Phys. Rev., 131, 435. 82 beyond the luminosity (8.30) for circular motion. The quadrupole formula (8.30) can immediately be applied also to a mass inhomogeneity δm in an accretion disk around a compact object of mass. In this event M1 = M and M2 = δm gives

32 M 5 δm2 4M 5 δm 2 L = ∼ few × 1051 erg s−1(8.34) GW 5 R M R 10−3M for an inhomogeneity in an accretion disk at a radius R. Here, the scaling on the right hand side refers to an estimate for δm ' 10%MD for a disk MD ' 1%M around a rotating black hole of mass M. Rotating black holes are believed to form in core-collapse of relatively high mass stars. This opens the prospect that some of the cosmological gamma-ray bursts and hyper-energetic supernovae are potentially loud in gravitational waves.10 This question may be investigated by future observations with LIGO, Virgo and KAGRA.

8.4 Exercises

1. For Euler’s experiment dropping a sphere into a bottle of viscous fluid, derive the terminal velocity. Take into account all three forces: the gravitational force downwards, the viscous force upwards and the buoyancy force due to a discrepancy between the density of the sphere and the fluid. 2. Reynolds numbers larger than a few thousand, the flow past a sphere be- 1 2 comes turbulent due to instabilities. The resulting drag force F = 2 CDρv A 2 is described by a dimensionless drag coefficient CD, where A = πR denotes the cross-sectional area of a sphere of radius R. CD = 0.47 for a fully turbulent flow. Describe the consequences for the timing behavior of the pendulum, starting with a relatively large θ0 for which the flow past m is turbulent. 3. Calculate the luminosity in gravitational waves for the orbital motion of 30 13 Jupiter around the Sun. Here, MJ = 2 × 10 g and RJ = 7.8 × 10 cm. p 4. In §1.1, estimate the adiabatic sound speed cs = γkBT/mp for the Sun, using the adiabatic index γ = 5/3 of fully ionized hydrogen at the mean temperature (4.25). In (4.31), verify ts = R /cs.

10van Putten, M.H.P.M., & Levinson, A., 2012, Relativistic Astrophysics of the Transient Universe (Cambridge: Cambridge University Press). Chapter 9

Binary evolution

(Quote)

Since Galileo’s discovery of sunspots on the Sun, we have come to see the Universe as a system evolving in time. Following its inception at the Big Bang, large scale structure formed out of clustering of matter, traced by galaxies and galaxy clusters. Galaxies are visible across a broad range of the electromagnetic spectrum due to the presence of interstellar medium and stars. These represent a relatively small fraction of the total mass, mostly in dark matter whose presence we infer gravitationally from velocity rotation curves and gravitational lensing. Binaries are an interesting class of time dependent sources, by tidal interactions, mass transfer in stellar winds and Roche lobe overﬂow. The latter gives rise to X-ray binaries, when one of the members is a compact object, i.e., a white dwarf, neutron star or stellar mass black hole. These are remnants of stars, at the end of their evolution, that includes supernovae for relatively massive stars. Here, we introduce some of the basic processes involved.

9.1 Stars and their lifetimes

Stars are born in regions of molecular clouds (Fig. 9.1) that are unstable to gravitational collapse, as described by Jeans’ instability criterion. It typically involves lumps of mass much larger than the mass of the stars produced. Stars are hereby believed to be born mostly in multiples, wherein orbital motion serves to absorbs most of the excess in angular momentum. Thus, binaries are a natural outcome of stellar formation in molecular clouds.

83 84

Figure 9.1: (Left.) A composite image of the Orion Nebula (M42, NGC 1976) is a nearby region of massive star formation (M ' 2000M , D = 1344 ± 20 ly, 85 × 60 arcmin) here captured by the Wide Field Imager on the MPG/ESO 2.2-metre telescope at the La Silla Observatory, Chile. (red: red light, hydrogen gas; green: yellow-green light; blue: blue light; purple: UV-light.) (ESO/Igor Chekalin (2011)). (Right.) An IR composite image of the Orion ”bullets” (blue: Fe II; orange: H2 1-0in the wakes), representing 50 arcs across and structure on 0.1 arcs (2 pixel). (Gemini Observatory, 2007)

Isolated stars or stars in wide binaries evolve in their hydrogen buying phase as main sequence stars with a lifetime

M −5/2 TMS = 13 Gyr. (9.1) M

Thus, TMS of stars born with M ≤ 1M is on the order of the age of the Universe of about 13.7 Gyr or longer. Those with higher mass, however, can have remarkably short lifetimes, e.g., 400 Myr for M = 4M or 40 Myr for M = 10M . Following (9.1), evolution proceeds on relatively short time scales with the burning of heavier elements, starting with helium. Ultimately, core burning ceases with the formation of an iron core. The ﬁnal fate of the star depends on the mass of the iron core thus formed, which will be a function of the initial mass of the star at birth. In the absence of nuclear burning, the core is supported against gravitational collapse by electron-degenerate pressure up to a certain point. While stars with M ≤ 4M are believed to end as white dwarfs, relatively more massive cores produced by M > 4M will experience continued collapse to a (proto-)neutron star. Core-collapse Binary evolution 85 of massive stars is believed to trigger supernovae, ultimately leading to the formation of neutron stars (4M < M < 10M ) and stellar mass black holes (M > 10M ). Based on Salpeter’s initial mass function for new born stars,

ψ(m)dm = m−2.35dm pc−3 yr−1 (9.2) we readily obtain estimates for the population density of white dwarfs, neutron stars and stellar black holes as remnants of stellar formation within a Hubble time (M ≥ 1M ),

−3 −3 −3 nWD = 0.1 pc , nNS = 0.02 pc , nBH = 0.0008 pc (9.3) obtained by integration of (9.2) over the intervals M ≤ M ≤ 4M , 4M ≤ M ≤ 10M and, respectively, M ≥ 10M . When core-collapse and supernovae are not exactly spherically symmetric, the newly formed neutron star or black hole may receive a substantial kick velocity on the order of one hundred km/s. This is believed to account for the large number of high velocity neutron stars detected in the Milky way. When stars are born in close binaries, the evolution will be different from those on the main sequence due to interactions. Overall, we distinguish three classes of compact binaries: detached, semi-detached and contact binaries. The photospheres are then, respectively, inside their Roche lobes, one inside and one coincident with its Roche lobe, and both coincident or exceeding their Roche lobes forming a common stellar envelope. For circular orbits, the two Roche lobes in a binary system meet a fixed first Lagrange point L1 as seen in a co-rotating frame of the binary.

9.2 Roche lobes

The Roche lobes are defined as equipotential surfaces passing through the first (inner) Lagrange point L1 (Fig. 9.2). It is defined as a saddle point in the Roche potential

GM GM 1 Φ(r) = − 1 − 2 − (ω × r)2 , (9.4) |r − r1| |r − r2| 2 where the ﬁrst two terms give the potential binding energies to the stars of mass M1 and M2 at positions r1 and r2, respectively. The third term 86

Figure 9.2: (Left.) Shown are the equipotential surfaces of Φ(r) and the location of the Lagrange point L1 at the saddle point between the two masses M1 and M2 for various mass ratios M1/M2.(Right.) The locus of L1 moves towards the relatively less massive star, shown as a function of the mass ratio M1/M2. contributes the centrifugal force at the distance σ to the axis of rotation in the force

F = −∇Φ(r). (9.5)

In the full dynamical equations of motion of a mass-less test particle (in the restricted three-body problem), F is balanced by inertia of the two stars and Coriolis forces,

d2r = F − 2ω × v (9.6) dt2 where v denotes the velocity as measured in the co-rotating frame. We commonly choose the angular velocity of the latter to be constant, deﬁned by the orbital period P in Kepler’s law:

2π G(M + M ) |ω| = Ω = , Ω2 = 1 2 . (9.7) P a3 Binary evolution 87

Here, p = a(1 − e2) denotes the latus semi-rectum of an orbit with ellipticity e and semi-major axis a. The motion in the orbital plane is conveniently described in polar coordinates (r, θ) by

a(1 − e2) r = . (9.8) 1 + e cos θ Thus, 1/p equals the orbital mean of 1/r when e < 1. For circular orbits, e = 0 and a = a1 + a2 equals the separation distance between the two stars, equal to the sum of the distances of each to the center of mass (the baryonic center of the binary). For these orbits, the Roche lobes are time-independent. Because L1 is a saddle point, the Roche lobes have the shape of droplets at the moment of pinching off a water jet. In describing their role to containing stars, we are interested in the effective radius RL of a spherical star with the 1 same volume. The effective radius of M1 satisfies

1 R 2 M 3 L ' 1 (9.9) a 34/3 M and to a better approximation2

2/3 RL 0.49q M1 = rL, rL = 2/3 1/3 , q = . (9.10) a 0.6q + ln (1 + q ) M2

The ratio RL/a increases with the mass ratio q. For instance, RL = 0.3789a for an equal mass binary (M1 = M2), increasing to RL = 0.5782a when M1/M2 = 10 and decreasing to RL = 0.2068a when M1/M2 = 0.1. Binaries may be observable as visual binaries, (single or double-line) spec- troscopic binaries, photometric binaries by periodic variability of ﬂuxes or colors, or (partial or full) eclipsing binaries.

9.3 Kepler orbits

The theory of orbits describes the motion of two bodies of mass M1 and M2 in the presence of a Newtonian gravitational attraction. Newton’s force is

1Paczyn´nski,B., 1967, Acta Astron., 17, 287 2Eggleton, P.P., 1983, ApJ, 268, 368; Benacquista, M., 2013, An Introduction to the Evolution of Single and Binary Stars (Springer-Verlag) 88

conservative. The orbital motion hereby has a Hamiltonian H = Ek + U, formed by the sum of the kinetic energy Ek and the potential energy U, 1 1 GM M H = M v2 + M v2 − 1 2 , (9.11) 2 1 1 2 2 2 r where r = |r1 − r2| denotes the separation distance between the two masses with velocities v1,2 = r˙ 1,2. It will be convenient to choose the origin of our polar coordinate system at the center of mass,

M1r1 = M2r2, r = r1 + r2, p1 + p2 = 0, pi = Mivi (i = 1, 2), (9.12) where the pi refer to the momenta of the stars. Thus, we may write

p2 p2 GM M p2 GM M H = 1 + 2 − 1 2 = 1 − 1 2 , (9.13) 2M1 2M2 r 2µ r where we introducing the reduced mass µ = M1M2 . Let v denote the mag- M1+M2 nitude |v1 − v2| of the velocity diﬀerence between the two stars, satisfying p v2 = v2 + v2 − 2v · v = v2 + v2 + 2v v = 1 (9.14) 1 2 1 2 1 2 1 2 µ2 in view of (9.12). With M = M1 + M2, we now have the equivalent one-body Hamiltonian p2 p GM M 1 GM H = 1 + 2 − 1 2 = µ v2 − . (9.15) 2M1 2M2 r 2 r

The form (9.15) describes the separation between M1 and M2 on terms of the equivalent motion of a test particle (of unit mass) around a centre with the total mass M = M1 + M2. Because M and v are invariant with respect to a Galilean transformation of the system, the sign of H is a Galilean invariant. This will be useful when discussing the explosive end of one of its members in a supernova. The associated Lagrangian L = Ek−U is formed by the diﬀerence between Ek and U, 1 GM L = µ v2 + . (9.16) 2 r Binary evolution 89

Using polar coordinates (r, θ) in the orbital plane to describe the position r of the test particle about the origin with mass M in the reduced one-body problem, the equations of motion follow from the Euler-Lagrange equations, d ∂L ∂L d ∂L ∂L − = 0, − = 0. (9.17) dt ∂r˙ ∂r dt ∂θ˙ ∂θ Evidently, the problem is invariant with respect to overall rotations with respect to θ about the axis of rotation, i.e., ∂L/∂θ = 0. (It is a so-called gauge transformation.) The second equation in (9.17) hereby expresses conservation of specific angular momentum: ∂L j = = r2θ.˙ (9.18) ∂θ˙ The first of (9.17) expresses the radial acceleration GM r¨ − rθ˙2 + = 0. (9.19) r The Möbistransformation u = 1/r takes (9.19) into the form

d2u GM + u = . (9.20) dθ2 j2 By (9.20), the solution is harmonic in the polar angle (polar form relative to focus), GM u = (1 + e cos θ) , (9.21) j2 where e ≥ 0 denotes the ellipticity. Elliptical orbits (bound states) exist for e < 1, whereas hyperbolic (unbound states) exist for e > 1. The corresponding trajectory in (r, θ) is an ellipse with maximum and minimum radius

j2 j2 r = = a(1 + e), r = = a(1 − e), (9.22) max GM(1 − e) min GM(1 + e) expressed in terms of the major semi-axis a (with θ = 0, π, rmax + rmin = 2a) with corresponding speciﬁc angular momentum

j = pGMa(1 − e2). (9.23) 90

The orbital period P follows from (9.18), when seen in the geometric form of tracing out a constant amount of area per unit time, 1 dA A j = = . (9.24) 2 dt P Here, A = πab denotes the total surface area enclosed by the ellipse, where b denotes the minor semi-axis. Note that ae is the distance of the center of the ellipse to the origin of our polar coordinate system. The projection r sin θ coincides with the minor semi-axis when r cos(π − θ) = ae, i.e., − cos θ = e, which recovers the familiar expression √ b = a 1 − e2. (9.25) √ Combining (9.24-9.25), we have A = πa2 1 − e2, and so P = 2A/j gives Kepler’s law GM 2π Ω2 = , Ω = . (9.26) a3 P We are now in a position the further express the total energy in our one- body formulation (9.15). At the points of maximal (perihelion) or minimal (aphelion) distance, we have v = rθ˙ = uj with kinetic energy 1 1 E = v2 = j2u2. (9.27) k 2 2 Since H is conserved, we can evaluate it at either of these extrema θ = 0, π,

1 µG2M 2 GMµ H = µ j2u2 − GMu = − (1 − e2) = − , (9.28) 2 ± ± 2j2 2a where we used the short hands u+ = u(0) and u− = u(π). Since H is constant, we have along the orbit the Vis-viva equation

2GM 2 1 v2 = 2µ−1H + = GM − . (9.29) r r a

Of particular interest is the combination

GM 1 1 e2 + e cos θ 2E + U = v2 − = GM − = GM , (9.30) k r r a a(1 − e2) Binary evolution 91 averaged over one orbit:

GMe2 2E¯ + U¯ = ≥ 0. (9.31) k a(1 − e2)

For bound orbits, the orbital average (9.31) is non-negative. When it vanishes, we say the binary is virialized: circular orbits are virialized orbits, whereas elliptical have an excess in kinetic energy in the sense of (9.31). More precisely, we may consider H = H(e, J) for constant J, and ask how the total energy changes as a function of ellipticity when preserving angular momentum. Keeping j2 = GMa(1 − e2) ﬁxed, we see from

GMµ G2M 2(1 − e2) H = − = − (9.32) 2a 2j2 that a more negative total energy corresponds to a smaller ellipticity. A circular (virialized) orbit is a minimum energy state for a given angular momentum. In practical terms, orbits may circularize by angular momentum preserving dissipative interactions. Dissipation processes induced by tidal interactions may be one example at work in the past for some of the observed binaries, provided the time scale of dissipation is short enough to be com- patible with their ages. The highly circular moon-Earth system may be one such example. A number of binaries of compact objects show a residual ellipticity away from zero, even though the time scale of circularization would predict essentially circular orbits. The residual ellipticity may be understood to result from the dissipation-ﬂuctuation theorem applied to the tidal interaction (as a function of orbital separation). 3

9.4 Mass transfer in binaries

The evolution of stars in binaries is generally diﬀerent from isolated stars due to gravitational tidal forces as well as mass transfer by stellar winds. If semi- detached, mass transfer can proceed directly through the inner Lagrange point L1 by Roche lobe overﬂow. It may proceed in a natural way from the more massive to the less massive star, by lowering the gravitational binding

3Phinney, E.S., 1992, Phil. Trans. R. Soc. Lond. A, 341, 39; Lanza, A., & Rodon`o, M., 2001, A&A, 376, 165 92 energy in the binary. In contrast, mass transfer in the opposite direction requires energy input from one of the two stars. As the binary becomes more tight or when both stars expand, a common stellar envelope phase may follow. Let us look at the case of conservative mass transfer in Roche lobe overﬂow in a sami-detached binary, keeping the total mass M = M1 + M2 and the total orbital angular momentum J constant, ignoring exchanges of orbital angular momentum with stellar angular momentum by tidal interactions. In this event,

M M m(M − m) J = µj = 1 2 pGMa(1 − e2) = √ pGa(1 − e2) (9.33) M M is constant as the mass m = M1 changes in time. Note that m(M − m) is a parabola, which assumes a maximum at m = M/2. If the orbit does not change shape (e remains constant), e.g., for circular orbits, mass transfer in the direction of equalizing stellar masses (from the more massive to the less massive member) causes a tightening of the binary, i.e., the major semi-axis a decreases and the binary becomes more compact. This process is natural, in that the total energy (9.32)

Gm(M − m) H = − (9.34) 2a is hereby decreasing. Note that increasing e may ameliorate this energy loss. Opposite conclusions hold mass transfer in the opposite direction, from the less massive to the more massive. This process requires forcing by one of the stellar members and/or circularization. For circular orbits, (9.33) can be written as

a M 2 2 (1 + q)4 J 2 = = 2 , a∗ = 3 , (9.35) a∗ M1M2 q GM where a0 = 16a∗ denotes the minimum binary separation when M1 = M2. Diﬀerentiating (9.33) with respect to time keeping J constant (J/J˙ = 0) gives

˙ a˙ 2(M1 − M2) ˙ M1 M1 = M1 = 2 (q − 1) , q = . (9.36) a M1M2 M1 M2 Binary evolution 93

Given the rate of change of the eﬀective radius according to Eggleton (9.10) (with q = Q−1), we have

R˙ r0 q˙ a˙ 1 + q r0 M˙ L = L + = 2(q − 1) L + 1 1 . (9.37) RL rL q a 2 rL M1

0 Here rL = drL/dq > 0 and hence RL of the more massive star will be decreasing with M1 when q > 1. The stability of conservative mass transfer depends on the rate at which ˙ the radius of the donor star changes relative to RL. Evidently, mass transfer is stable near mass equilibrium q = 1. However, stability is not ensured when q > 1. If RL decreases faster than the radius of the donor star, mass transfer proceeds increasingly faster and a run-away may occur, at least for a while. It does not preclude recovering stability as q approaches 1. The governing physical parameter of the donor star is the logarithmic derivative 4 d log R ξ = d log M . Generally, we distinguish different parameter regimes of ˙ mass transfer, M1 = M1/τ according to different time scales τ. We may have thermal stability, as in slow mass transfer, with evolution proceeding on the nuclear burning time scale; thermal instability with stability on the dynamical time scale of the binary; and dynamical instability, when nothing stops the system from rapid shrinking as in the formation of a common envelope state. A detailed consideration of falls outside the scope of our present discussion. Stellar winds offer an additional, robust channel of mass transfer that operates also in detached binaries.

9.5 Supernovae in binaries

Since (9.15) is expressed in terms of the velocity diﬀerence v and the separation distance r between the two members of the binary, it is Galilean invariant. If one member goes supernova, mass is lost eﬀectively instanta- neously. If the supernova is spherically symmetric, v is the same just before and after the explosion. If the binary is also circular (before the explosion), then r = a and v2 = GM/a. For H to remain negative after the explosion, M the mass lost must satisfy ∆M < 2 , i.e., no more than 50% of the total mass may be ejected.

4Hjellming, M. S. & Webbink, R.F. 1987, ApJ, 318, 794; Soberman, G.E., Phinney, E.S., & van den Heuvel, E.P.J. 1997, A&A, 327, 620 94

Table 1.1 shows some of the highlights of binary evolution.

Table 1.1. Overview of binary theory Keplerian orbits:

GM 1 GM GMµ Ω2 = ,H = µ v2 − = − (9.38) a3 2 r 2a J = µpGMa(1 − e2) (9.39)

where a denotes the major semi-axis, e the ellipticity of the orbit, M = M1 + M2 and µ = M1M2/M denotes the reduced mass. H is invariant under a Galilean transformation. Orbits can virialize by circularization: GMe2 2E¯ + U¯ = ≥ 0 (9.40) k a(1 − e2)

will approach zero when e → 0. If the orbital angular momentum J is constant, then a shrinks and H becomes more negative (dissipation). Following virialization, the orbital separation may continue to evolve by continuing mass transfer with

J 2 M 2 2 a = 3 . (9.41) GM M1M2

In a circular binary, if one member ejects a mass Mej in a supernova explosion, then the binary remains intact if Mej < 50%M.

9.6 Exercises

1. Based on the data given in Fig. 9.1, derive an estimate of the mean density of the Orion molecular cloud.

2. Compute the following time scales

• Consider the radial free fall of a test particle to a central mass M by Binary evolution 95

gravitational attraction, starting at t = 0 at rest from a distance R,

d2r GM = − , r(0) = R, r˙(0) = 0. (9.42) dt2 r2 Derive the free-fall time π R3/2 tff = √ . (9.43) 2 2GM The free fall time of a cold (zero pressure) cloud with mean density ρ, therefore, is κ t = √ . (9.44) ff Gρ Derive the coeﬃcient κ.

• Consider a virialized self-gravitating mass M of radius R. Calculate the Kelvin-Helmholtz time.

• Consider a main sequence star of mass 1M . Estimate the total amount of hydrogen burning over its lifetime of about 13 Gyr. 96 Chapter 10

Bondi-Hoyle-Lyttleton accretion

(Quote)

Binary evolution of stars has a natural tendency for circularization, due to tidal interactions, and contraction, due to mass transfer from the more to the less massive member, that may lead to a semi-detached binary. If so, accretion sets in that will dramatically aﬀect stellar evolution (of both members). In some cases, a supernova may ensue, that may leave a compact remnant. If mass ejection and the kick velocity of the newborn neutron star or black hole are limited, the binary may survive. Alternatively, one of the stars may form a white dwarf, if the progenitor is relatively light. Either way, accretion on a white dwarf, neutron star or black hole may result. The former may be pushed towards accretion induced collapse to a neutron star by suﬃciently rapid accretion.1 Accretion onto compact objects is appealing for a broad range of multi- wavelength phenomena observed in X-ray binaries. Accretion is a complex process that may entail mass transfer directly onto the compact object or indirectly through the formation of an accretion disk, when direct matter infall encounters an angular momentum barrier. Here, we discuss the problem of mass transfer onto compact objects without the formation of accretion disks. It applies to some of the accretion processes in binaries, as well as accretion onto isolated neutron stars or black holes moving through the interstellar medium or a molecular cloud.

1However, observational evidence for this scenario remains elusive.

97 98 10.1 Bondi accretion

Bondi accretion2 describes the capture of a surrounding wind by gravitational focusing. In a binary, it applies to the capture of stellar winds from a binary companion. It extends to an earlier analysis of the capture of gas when a star moves through the interstellar medium or a molecular cloud.3. The latter is particularly applicable to neutron stars or black holes in response to a kick velocity received at birth in a preceding supernova. To study this, let us first review the problem analyzed by Bondi of spherical accretion. It shows the remarkable result is the existence of smooth solutions (with no shocks) for spherical accretion onto a black hole for special values of the accretion rate, given specific values for the density and sound speed at infinity, and a bound on the allowed adiabatic index 1 ≤ γ ≤ 5/3. Starting point is the specific enthalpy of adiabatic flow along streamlines,

1 a2 GM h = v2 + − . (10.1) 2 γ − 1 r It follows by integration of the time-independent Euler equations of motion 1 ∂P GM v∂ v = − − (10.2) r ρ ∂r r2 for a ﬂuid with a polytropic equation of state

P = Kργ (10.3) described by the adiabatic index γ (with corresponding polytropic index 1 + 1/n = γ). By conservation of baryon number, the accretion ﬂow has a second conserved quantity

M˙ = 4πr2ρv (10.4)

The existence of smooth solutions from inﬁnity to the accreting object depends on boundary conditions. In what follows, we consider accretion onto

2Bondi, H., 1952, MNRAS, 112, 195; Shapiro, S.L., & Teukolsky, S.A., 1983, Black Holes, White Dwarfs and Neutron Stars (John Wiley & Sons); Ryden, B.S., Lecture notes AST825, http://www.astronomy.ohio-state.edu/ ryden/ast825/ch8.pdf 3Hoyle, F. & Lyttleton, R. A.: 1939, Proc. Cam. Phil. Soc. 35, 405; Bondi, H. & Hoyle, F., 1944, MNRAS, 104, 273 Binary evolution 99 a black hole with ingoing boundary conditions on the event horizon. Smooth solutions imply the existence of derivatives. To this end, we consider (10.2) and the logarithmic derivate of (10.4), giving

a2 GM ρ 2 v0 + ρ0 + = 0, v0 + ρ0 + ρ = 0. (10.5) vρ vr2 v r As a system of equations for two unknowns, its solutions are

v0 v2 a2 −1 2ρ = vρ r , (10.6) ρ0 2 2 ρ GM v − a −1 v vr2 that is 2a2 − GM GM − 2v2 v0 = r r−1v, ρ0 = r r−1ρ. (10.7) v2 − a2 v2 − a2 The vanishing of the determinant in (10.6) defines a critical point, which is the sonic point where the inflow velocity equals the sound velocity. For smooth solutions to exist, the numerators of (10.7) will have to vanish simul- taneously, i.e., 2 2 2 2 5 − 3γ GM vs = as = a0, rs = 2 , (10.8) 5 − 3γ 4 a0 where a0 denotes the sound velocity at infinity, where we assume v = 0. In meeting these conditions (10.8), we fix the accretion rate, i.e., the accretion rate assumes critical values for smooth solutions satisfying

2 ˙ 2 GM −3 M = 4πrs ρsvs = 4πλ 2 ρ0a0 ∝ a0 , (10.9) a0 where ρ0 denotes the density at inﬁnity and

1 " # 2(γ−1) 1γ+1 5 − 3γ −(5−3γ) λ = . (10.10) 2 4

5 The coeﬃcient λ is real when 1 ≤ γ ≤ 3 , and assumes values of order unit as shown in Fig. 1. Note that the accretion rate decays with the third power of the sound speed at inﬁnity: accretion prefers cold over hot gas. 100

10 1.3

8 1.2

6 1.1 F(w) λ 1 4

0.9 2 0 1 2 3 w 0.8

20 0.7

15 0.6 Bondi accretion scale factor

10 0.5 G(u)

5 0.4

0 0.3 0 10 20 30 40 50 1 1.2 1.4 1.6 1.8 u γ

Figure 10.1: (Left. Shown are the functions F (w) and G(u) for the values γ = 1.16, 1.31, 1.61 (top to bottom). The minima of F (w) at the sonic point (Mach number equal to one) satisfy w = 1. The corresponding locus of the minima (bold line) of G(w) are at us = 4/(5 − 3γ). (Right.) The ratio of these minima deﬁnes the Bondi accretion scale factor λ, which serves as a bound on the accretion rate through a smooth ﬂow (with no shocks).

As Bondi (1952) showed, it turns out that (10.9) is the maximal attainable accretion rate (without encountering shocks). It follows from the dimensionless implicit formulation for the Mach number w = v/a as a function of the 2 dimensionless distance u = GM/(ra0), given by F (w) = kG(u). (10.11) With α = (γ − 1)/(γ + 1), we have 1 1 1 F (w) = w−2α w2 + ,G(u) = u−4α u + . (10.12) 2 γ − 1 γ − 1 Here, k = λ−2α is a constant set by the accretion rate. In view of 1 < γ < 5/3, 1 0 < α < 4 . The function F (w) satisﬁes 1 F (w) ≥ F = , (10.13) s 2α where the minimum is attained at the sonic point ws = 1. For (10.11) to have a solution for all radii r > 0, i.e., for all u, kG(u) must exceed the lower Binary evolution 101

2.5

transonic k>k (supersonic) 2 s (outflow)

1.5

two−valued solution 1 (unphysical, k

0.5

k>k (subsonic) transonic s (inflow) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 r/r s

Figure 10.2: Shown are the different solution branches to the Bondi’s equations of spherical accretion. They include the two in- and outflow transonic solutions as critical solutions, above and below which are two physical super- and subsonic solutions. To the left and right are two unphysical two-valued velocity solutions. The results shown are calculated by numerical continuation for γ = 1.2. bound (10.13). The function G(u) similarly satisfies

− 5−3γ 1 5 − 3γ γ+1 G(u) ≥ G = (10.14) s 4α 4 at us = 4/(5 − 3γ). This puts an upper bound on the accretion rate scale factor λ, by

5−3γ γ+1 Fs 5 − 3γ k ≥ ks = = 2 . (10.15) Gs 4 Fig. 10.2 is shows the (real) solution branches of w = w(u) (0 < u < ∞) and u = u(w) (0 < w < ∞) to (10.11) for γ = 1.2. Neither F (w) or G(u) 5 has no analytic inverse for 1 < γ < 3 (an inverse is easily seen for γ = 1 when α = 0), so we calculated these branches by numerical continuation 4.

4Keller, B., 1986, Tata Lectures on Numerical Methods in Bifurcation Problems (Springer-Verlag) 102

Consider the roots of Z(w, u) = 0, (10.16) where Z(w, u) = F (w) − kG(u). (10.17) For a given k satisfying (10.15), roots of (10.17) define flow solutions of interest that approach the transonic point in the limit as k approaches ks. Root finding is efficiently accomplished using Newton’s method. At an upstream position u < us, e.g., some small value like u = 0.1, we choose an initial guess for w and iterate

Z(wn, u) ∂Z(w, u) wn+1 = wn − ,Zw(w, u) = . (10.18) Zw(wn, u) ∂w

Newton’s method converges quadratically, so that wn converges rapidly to a ∗ limit point w = w∗(u) with a few iterations. For instance, for u = 1.25 with k = 1.04ks, we ﬁnd

wn : 0.5000 0.1217 0.2093 0.2592 0.2677 0.2679, −4 −7(10.19) Zn : −0.5851 0.9451 0.2752 0.0358 7.66 × 10 3.6582 × 10 , For a choice of step size h, we next choose a neighboring point u+h with initial guess w∗(u) from the previous iteration. A few iterations of (10.18) now produces w∗(u + h). Repeating this procedure step-by-step obtains w∗(u + nh), n = 1, 2, ··· ., thus producing a continuous branch. For the initial choice of initially subsonic value w = 0.5, a subsonic branch obtains. Alternatively, for an initial choice of supersonic value w = 1.5, a supersonic branch obtains. The sonic point is numerically unstable as a bifurcation point across which continuation can choose between a subsonic or supersonic branch on r < rs (u > us). The result is physically determined by the boundary condition at r = 0, i.e., a hard surface or a surface of maximal inﬂow in case of a black hole event horizon. In case of k < ks, numerical continuation can similarly be applied to the inverse of (10.18), i.e.,

Z(w, un) ∂Z(w, u) un+1 = un − ,Zu(w, u) = . (10.20) Zu(w, un) ∂u It produces two additional “forbidden” branches: the two-valued velocities show that these solutions are unphysical. Binary evolution 103 10.2 Hoyle-Lyttleton accretion

An interesting variation to the spherical Bondi accretion is the accretion of gas from the environment onto a moving star or compact object, a white dwarf (WD), neutron star (NS) or black hole (BH).5 The characteristic quantities in this problem are ρ : density of the ISM or molecular cloud [ρ] = gcm−3 R : radius of the star (WD, NS or BH) [R] = cm −1 v0 : velocity of the star (”) [v] = cms (10.21) b : impact parameter [b] = cm G : Newton’s constant [G] = cm3g−1s−2. In the frame of the star, flow appears, say, from the left with asymptotic velocity v0 from infinity. We may choose a Cartesian coordinate system with the z−axis along the direction of motion with the star at the center. Because of spherical symmetry, consider a poloidal cross-section with polar coordinate system (r, θ). Streamlines passing the star and approaching infinity will have θ → 0. Away from the θ = 0, the flow may be treated in the limit of zero pressure, i.e., as if it consists of dust particles. Conservation of enthalpy hereby reduces to 1 1 GM h = r˙2 + r2θ˙2 − . (10.22) 2 2 r The streamlines hereby follow Keplerian motion away from θ = 0, i.e., GM v u = (1 + cos θ) − 0 sin θ (10.23) j2 j taking into account the boundary condition u = 1/r → 0 as θ approaches π, where

0 r˙ = −ju → −v0 (10.24) as gas is coming from the left as seen in the frame of the star. Here, 2 ˙ j = r θ = bv0 (10.25) denotes the speciﬁc angular angular momentum associated with the impact parameter b.

5Historically, this problem was treated before Bondi’s spherical accretion. 104

By gravitational focusing, the streamlines will cross the semi-infinite line θ = 0 to the right of the star at some distance r. At this point, our approximation (10.22) breaks down, and a fluid dynamical treatment with internal energy due to pressure induced by compression of the gas will have to be included. However, we model this as being local to the θ = 0, where the angular momentum of the fluid elements is reduced to zero in a small angle column about θ = 0. This column will consist of an outflow, of fluid elements with positive enthalpy and of an inflow, of fluid elements with negative enthalpy, i.e., those that are unbound (h > 0) and, respectively, gravitationally bound (h < 0) to the star. At θ = 0, the angular momentum of the gas is reduced to zero (by collision of particles from two streamlines reflected about θ = 0). The poloidal velocity rθ˙ effectively drops to zero discontinuously in approaching θ = 0. The line θ = 0 hereby contains a stagnation point at some radius r∗ from the star, that may be derived from the reduced specific energy 1 GM h∗ = r˙2 − (θ = 0). (10.26) 2 r

According to (10.23),r ˙ = v0, and hence 2GM r∗ = 2 . (10.27) v0

Gas escapes coming from streamlines crossing θ = 0 at r > r∗, while gas is accreted coming from streamlines crossing θ = 0 at r < r∗. 1 2 Since h is conserved up to the crossing θ = 0, h = 2 v0. In view ofr ˙ = v0 at θ = 0, the kinetic energy associated with the poloidal momentum equals GM/r at the point of crossing. Removing this energy by conversion to heat by ﬂuid dynamical compression leaves us with the sign of (10.26) to determine the possibility of escape or accretion. The approximation in (10.27), therefore, is in ignoring the possible role of hydrodynamical expansion in any subsequent motion along θ = 0. From (10.23), we have the quadratic relation j2 r = ∗ (10.28) ∗ 2GM between r∗ and j∗. The associated critical angular momentum satisﬁes 2 2 2 4G M j∗ = 2GMr∗ = 2 . (10.29) v0 Binary evolution 105

The speciﬁc angular momentum j∗ of the streamlines entering the stagna- ˙ tion point is given by r∗ times vθ = rθ = ju just before the hydrodynamical interactions set in. From (10.25), we have j∗ = b∗v0, where b∗ denotes the critical impact parameter, that may be measured by the distance to the z−axis at inﬁnity (at θ = π). That is, we have

j∗ b∗ = (10.30) v0

2 The mass rate of accretion now follows from the capture area πb∗:

2 2 ˙ 2 G M −3 MHL = πb∗v0ρ0 = 4π 3 ρ0 ∝ v0 . (10.31) v0

You will notice the similar asymptotic behavior as a function of velocity in (10.9) and (10.31). It let Bondi to suggest the general result

2 2 ˙ 2 G M MBHL = πb∗v0ρ0 = 4π 2 2 3/2 ρ0 (10.32) (v0 + a0)

to apply to bodies moving with velocity v0 in media with asymptotic density and sound speed ρ0 and a0, respectively. The case of bodies moving supersonically is different, in that a shock front will form in the accretion flow. It is interesting to see, that it effectively combines the free stream Hoyle-Littleton accretion flow upstream and the Bondi accretion flow downstream.6 Table 1.1 highlights some of the main issues.

6Hunt, R., 1971, MNRAS, 154, 141 106

Table 1.1. Overview of axisymmetric accretion 2 (A) Let RS = 2GM/c denote the Schwarzschild radius of a star of mass M. The Bondi-Hoyle-Lyttleton mass accretion rate onto a compact object with velocity v0 in a medium with density ρ0 and sound velocity a0 satisﬁes

3 ˙ 2 2 c MBHL = πb∗v0ρ0 = 4πRScρ0 2 2 3/2 . (10.33) (v0 + a0)

˙ −3 −3 Since M ∝ v0 , a0 , the accretion rate is suppressed in the limit of large velocities or high temperature environments. Conversely, compact objects will be relatively bright when moving slowly in low temperature media. (B) Accretion ﬂows satisfy nonlinear equations of conservation of energy-momentum and mass. Formulated implicitly as

Z(x, y) = 0, (10.34)

solution branches can be found by numerical continuation using New- ton’s method. For a choice of x and an initial guess y0, consider the sequence

Z(x, yn) yn+1 = yn − . (10.35) Zy(x, yn)

If convergent, the limit y∗ = y∗(x) is a solution to (10.34). It can be used as an initial guess to (10.35) to calculate y∗(x + h). Repeating this procedure obtains a branch y∗(xm)(m = 1, 2, ···) away from bifurcation points, where Zy = 0. Similarly, branches x = x(y) can be calculated. Fluid dynamics 107 10.3 Exercises

1. Derive the following estimate for the orbital period of a star of mass M1, radius R1 in a semi-detached binary with mass ratio q = M1/M2:

1 0.2 R3 2 2 P ' 0.35 1 day. (10.36) M1 1 + q

2. Consider an isotropic stellar wind from M1 with a companion binary of ˙ ˙ mass M2, i.e., M1 < 0 and M2 = 0. Compute ! J˙ α = . (10.37) J wind Derive an expression fora/a ˙ . How does the binary period evolve?

3. Show that a3 P 2 = (10.38) M1 + M2 expresses the period in years of a binary with major semi-axis a in A.U. and masses M1,2 expressed in units of M . 108 Chapter 11

Fluid dynamics

(Quote)

Since Galileo’s discovery of Sun spots moving along with the Sun’s rotation about 400 hundred years ago (Fig. 1), we learned that the Universe is time dependent.1 Modern optical-radio surveys now cover the evolution of the entire Universe in increasing detail from the time of the last scattering surface as seen in the Cosmic Microwave Background (CMB), the era of re-ionization that gave birth to the ﬁrst stars and galaxies, and the cosmological evolution of galaxies. The latter includes the discovery of active galactic nuclei, (AGN) producing some of the brightest objects in the sky. Most of the dissipation and hence entropy creation in the Universe appears to occur in galaxies, in their nuclei as well as in high energy transient sources involving stellar mass compact objects, i.e., white dwarfs, neutron stars and black holes. Perhaps the best known are supernovae, where the Type Ia supernovae gained recent notoriety as a tool for precision cosmology leading to the measurement of a small but discernibly positive cosmological constant.2 In the last few decades, dedicated high energy satellite missions

1“Seeing is a thing that is learned, and what one sees is what one has learned how to see. In Galileos own day, the Jesuit astronomer Christopher Scheiner observed sunspots at the same time as Galileo, but instead of seeing them as being on the surface of the sun, he thought they were small planets orbiting around it, largely because, in keeping with church doctrine, he knew they could not be on the sun. Perhaps Galileos friendship with artists allowed him to trust what his eyes actually saw.” Jim Long, 2005, in The Brooklyn Rail. 2The discoverers S. Perlmutter, S. Schmidt and A.G. Riess received the Nobel Prize of Physics 2011. For an instructive account, see

109 110

Figure 11.1: Shown are three of Galileo’s drawings of sunspots on June 21, 22 and 23, 1613 (left to right). The spots vary in size as a function of time, and their motion clearly reveals the rotation of the Sun. (Source: http://galileo.rice.edu/sci/observations/sunspot−drawings.html, animated in http://galileo.rice.edu/sci/observations/ssm−fast.mpg) furthermore show a Transient Universe marked by unimaginably bright explosive events: cosmological gamma-ray bursts. GRBs are probably the outcome of catastrophic events involving neutron stars and stellar mass black holes. A common language in all these phenomena is ﬂuid dynamics, often at very high Reynolds numbers. In the following, we review some basic concepts in ﬂuid dynamics that underly some of this phenomenology, in transporting energy and (linear and angular) momentum, as well in mediating dissipation and associated radiative processes.

11.1 Navier-Stokes equations

In the motion of a fluid, energy and momentum are transported. This may happen at some cost, in viscous flows or flows that are radiative in electromagnetic, neutrino and or gravitational waves. In setting up a mathematical model, we commonly use either one of two choices:

1. A Lagrangian description, labeling each of N ﬂuid elements by a po- http://www.nobelprize.org/nobel−prizes/physics/laureates/2011/advanced- physicsprize2011.pdf; Perlmutter, S., et al., 1999, ApJ, 517, 565, and Riess, A.G., 1998, et al., Astron. J., 116, 1009. Fluid dynamics 111

sition vector ξi (i = 1, 2, ··· N), and tracking its evolution in time. In particular, we have the velocities ui = dξi/dt for the individual particle velocities. A Lagrangian description is natural in describing the dynamics of stars in a globular cluster, and can be advantageous in a reduced description of gas dynamics over relatively large regions in space; 2. An Eulerian description using a velocity field u = u(x, y, z, t), describing the velocity of fluid elements at a coordinate position (x, y, z, t). It is a mean field description, in that u describes the mean velocity of all the individual fluid elements in a small box centered at (x, y, z, t). It is widely used in studying detailed properties of fluid dynamics in engineering and astrophysics. The acceleration of fluid elements - individual particles or small boxes with lots of them - is described quite differently in these two alternative descrip- tions. This is the main reason, why they are generally applied to different problems. In the Lagrangian description, we have the acceleration vector d2ξ a = i . (11.1) i dt2 It denotes the rate of change of momentum of each individual particle due to some force, by interactions with neighboring particles, radiation, or due to some electric or gravitational field. In the Eulerian description, the acceleration follows by tracking the velocity u across a displacement uδt along the physical trajectory of a given box of particles. Doing so over a small time interval δt and taking the limit in which δt approaches zero gives for the acceleration a the Lagrangian or convective derivative u(x + uδt, t + δt) − u(x, t) Dtu ≡ lim = ∂tu + (u · ∇)u, (11.2) δt→0 δt where x = (x, y, z). We can convert (11.1) and (11.2) to force per unit volume by multiplication with the mass density, ρ. In the Eulerian description, we thus obtain

ρDtu = −∇p + ··· (11.3) in the presence of a pressure p. The dots refer to other terms that may come into the picture, such as −∇Φ from and external potential such as the gravitational binding energy to the Earth and viscous forces. 112

In our deﬁnition of dynamical viscosity, µ we considered the force

V F = µA (11.4) h we must apply to a plate, to move it with constant velocity V parallel to a wall at a separation h. In a stationary state, the moving plate at height y = h and the wall at height y = 0 experience the exact same force F in (11.4), leaving the ﬂuid in between with a linear velocity proﬁle u = y/h (0 ≤ y ≤ h). A more local description of the force (11.4) is

du(y) F (y) = µA ,F = F (L),F = −F (0), (11.5) dy 2 1 where the minus sign is included to preserve our definition of force, to be applied externally to keep the wall in place (e.g. by its foundations). Applied to the moving plate and the wall in steady state, we have F2 = F1. Suppose we start with zero initial velocity and suddenly change to V > 0 + + + at t = 0 . In this event, F2(0 ) > 0, while F1(0 ) = 0, i.e., the forces are initially out-of-balance. The result is a net force F2 − F1 > 0 on the fluid in between. For a fluid with density ρ, it produces a mean acceleration satisfying

u0(h) − u0(0) µ Z h ρa = µ ' u00(y0)dy0, (11.6) h h 0 that subsequently decays to zero in approaching (11.4) over some relaxation time τ = V/a. The expression (11.6) shows the appearance of a force density due to a curvature in the velocity. We can now consider the steady state Poisseuille flow between two plates at rest, in response to an external forcing. For instance, pushing a fluid between two plates by applying some constant pressure p at the inlet gives the velocity distribution y u = cα(1 − α), α = , (11.7) h where c is some constant. The quadratic relation (11.7) satisfies ∆u = −2 −2ch , which refers to a constant pressure p(y) = p0 at the inlet, assuming zero pressure at the outlet. Following (11.5), the net force F = phW on the Fluid dynamics 113

ﬂuid suspended between the plates across some horizontal width W equals the sum of the two forces that the ﬂuid applies to each plate,

phW = −µAu0(h) + µAu0(0),A = W L, (11.8) over some length L along the x−direction of motion. It follows that dp p h = −h 0 = µhu00(y) = −2µch−1, (11.9) dx L that is, h2p c = 0 (11.10) 2µL in (11.7). It has the dimension of velocity. Note that the dimension of pressure over dynamical viscosity is inverse second: p 0 = s−1, (11.11) µ so that (11.10) has dimensions of velocity. Indeed, the mean velocity across the flow satisfies 1 Z h Z 1 h2p < u >= u(y)dy = c α(1 − α)dα = 0 . (11.12) h 0 0 12µL A similar calculation in cylindrical geometry gives πp r4 m˙ = ρ < u > πr2 = 0 (11.13) 8νL which serves to show the dramatic dependence of the mass flowm ˙ on radius, r. A relatively minor decrease in 15% in radius causes a 50% decrease in flow rate for a given pressure. It explains why accumulation of even minor amounts of fat on the inner walls of our veins can be dangerous. For a Newtonian fluid, the result of velocity curvatures in different directions including the transverse direction z and the direction of propagation x will superimpose. The general result for the force density due to viscous momentum transport, therefore, is ∂2 ∂2 ∂2 f = µ∆u, ∆ = 2 + 2 + 2 , (11.14) ∂x ∂y ∂z 114 where ∆ = ∇ · ∇ is the Laplace operator. The Navier-Stokes equations now follow by including viscous momentum transport (diffusion of momentum) in (11.3), that is

ρ (ut + (u · ∇)u) = −∇p + µ∆u (11.15) supplemented with mass continuity

∂tρ + ∇ · ρu = 0. (11.16) The Navier-Stokes equation (11.15) can be expressed in normalized (dimensionless) variables to explicitly bring about the role of the Reynolds number. To this, we write U u = U u,˜ p = ρ u2p,˜ ρ = ρ ρ,˜ x = Lx˜, t = 0 t˜ (11.17) 0 0 0 0 L where all tilde quantities are now dimensionless. In these variables, we have 1 u˜ + (u˜ · ∇˜ )u˜ = − ∇˜ p˜ + Re−1∆˜ u˜ (11.18) t˜ ρ˜

11.1.1 Large and small Reynolds number limits Let us consider the following two limits of large and small Reynolds numbers. (In what follows, we drop the tildes in (11.18.) 1. In the limit as Re approaches infinity, we may neglect the viscous term on the right hand side of (11.18), leaving Euler’s equations 1 u + (u · ∇)u = − ∇p, (11.19) t ρ which is commonly considered for incompressible flows, ∇ · u = 0. (11.20) Of particular interest is the role of the quadratic nonlinearity defined by the convective term (u · ∇)u on the left hand side of (11.19). Its implications are many. In compressible gas dynamics, including the motion of dust, it can lead to steepening, which is a starting point for the formation of shocks and associated entropy creation. It also provides a mechanism for period doubling, such as the creation of small eddies from large eddies in turbulent flows. We shall discuss some of these further below. Fluid dynamics 115

2. In (11.15), we may focus on balance between pressure (gradients) and viscous forces, i.e., neglect inertia and consider the Stokes equation

µ∆u = ∇p. (11.21)

For incompressible flows (solenoidal flows), ∇·u = 0, the pressure field is harmonic: ∆p = µ∆(∇ · u) = 0. Consequently, the velocity satisfies a biharmonic equation and the vorticity field ω = ∇ × u is harmonic, i.e.:

∆∆u = 0, ∆ω = 0. (11.22)

Exploiting our assumption of incompressibility once more, we may write

u = ∇ × B (11.23)

for some vector field B. Here, we may insist ∇·B = 0, e.g., B = ∇×A for some vector field A. 3 The vorticity hereby expands to ∇ × u ≡ ∇(∇ · B) − ∆B = −∆B. By the second equation of (11.22), it follows that B satisfies the biharmonic equation as well

∆∆B = 0. (11.24)

3. In (11.15), we can alternatively focus on the effect of steepening by the convective term on the left hand side, and neglect the back reaction, if any, from pressure gradients (and dispersion). A familiar example is steeping in traffic flows induced by stopping for traffic lights. In this event, vehicles behind advance on those ahead. Without attention paid to the relative velocity between vehicles, the inevitable result is a collision, i.e., the formation of a shock. In this approximation, (11.15) simplifies to inviscid or viscous Burgers’ equation, given by, respectively,

∂tu + (u · ∇)u = 0, ∂tu + (u · ∇)u = ν∆u. (11.25)

3To see this, recall that (11.23) is invariant under B → B + ∇f for any suﬃciently smooth function f, whereby ∇ · B → ∇ · B + ∆f. The latter expression can be put to zero by a solution f to the Poisson equation ∆f = ρ, ρ = −∇ · B. 116

Some problems are well described by (1), (2) or (3). They may co-exist, at large Reynolds numbers on macroscopic scales of objects and at small Reynolds numbers, in boundary layers and turbulent flows with diffusion of vorticity and dissipation. Generally, (1-3) should be appreciated by exam- ples, their richness being essentially countless,4 further in the presence of, e.g., interfaces between different fluids, (self-)gravity and buoyancy, shocks, heat flux, magnetic fields, ionization and radiation, suspensions in medicine and biology, chemical/nuclear reactions, curved space-time around black holes and in the early Universe - and anisotropy (liquid crystals displays!). Key developments derive from (high-tech) experiments and observations, including numerical simulations.

11.1.2 Vorticity equation In atmospheric problems and in describing transitions between high and low Reynolds number ﬂows near solid boundaries, it is often useful to consider the vorticity ﬁeld

ω = ∇ × u. (11.26)

Based on the Navier-Stokes equation, we can derive the vorticity equation Dω 1 = (ω · ∇)v − (∇ · v)ω + ∇ρ × ∇p + ν∆ω, (11.27) Dt ρ2 where the right hand side brings about variations (ω · ∇)v due to velocity gradients, stretching of vorticity (∇ · v)ω due to compressibility (cf. the ballerina effect), a baroclinic term ρ−2∇ρ × ∇p and diffusion of vorticity in ν∆ω. For a barotropic fluid, p = p(ρ), ∇ρ × ∇p = 0, which also holds true for fluids with constant density. For incompressible fluids, therefore, (11.28) becomes the vorticity transport equation Dω = (ω · ∇)v + ν∆ω. (11.28) Dt In this formulation, explicit appearance of pressure gradients is absent, even though the pressure field is generally non-zero and dynamically relevant in departures from the Stokes flow approximation. For incompressible fluids, however, these pressure gradients do not affect the vorticity distribution.

4Van Dyke, M., 1982, An Album of Fluid Motion (Parabolic Press) Fluid dynamics 117 11.2 Viscous ﬂow past a sphere

An example of the role of viscosity is in providing a drag force which is linear in velocity, in case of a (very) low Reynolds number flow past a sphere, commonly treated by way of (11.21). Let us choose a spherical coordinate system centered at the center of the sphere of radius a. We seek an incompressible velocity perturbation δu, satisfying u = U + δu, u = 0 on r = a, δu → 0 as r → ∞, |δu| ∝ |U| .(11.29) The velocity δu is a vector field, defined everywhere outside the sphere and vanishing on its boundary. At each point in space outside the sphere, we have at hand only two invariant orientations, provided by the vectors U and r. The velocity perturbation is necessarily a linear combination of the two, provided the sphere is non-rotating. These zero-helicity flows have a vorticity

∇ × u = ωiφ (11.30) in the ϕ−direction about the direction of flow U = Uiz. A rotating sphere would produce helical flows also in directions U × r, but we leave this extension as an exercise to the reader. 5 Next, we follow the derivation of Landau & Lifshiftz6, starting with (11.23). By (11.30) and the curl expansion following (11.23), it follows that ∆B is only in the iϕ direction. Then so is B, whereby B = ∇f × U = ∇ × A, A = fU) (11.31) for some scalar function f, where the second expression in A shows that our choice ∇ · B = 0 is satisfied. By (11.24), we have 0 = ∆2 (∇f × U) = ∆2∇f × U = ∇ ∆2f × U, (11.32) since U is a constant vector, denoting the constant flow at infinity. It follows that ∆2f = 0, (11.33)

5Jeffry, G.B., Steady rotation of a solid of revolution in a viscous fluid. Proc. London Math. Soc. 14 (1955) 327; Kanwal, R.P., Slow steady rotation of axially symmetric bodies in a viscous fluid. J. Fluid Mech. 10 (1960) 17; Ram Kissoon, H., A Slip flow problem. J. Math. Sci (Calcutta) 8(1) (1997) 23; Liu, Q., & Prosperetti, A., 2010, Wall effects on a rotating sphere, J. Fluid Mech., 657, 1 6Landau, L.D., & Lifshitz, E.M., 1959, Fluid Mechanics (Oxford, Butterworth- Heinemann) 118 taking into account the asymptotic conditions on u at infinity via (11.23) and (11.31). We are now in a position to complete our analysis. The general solution to (11.33) is readily seen to be f = c1r + c2/r. Tracing back over (11.31) and (11.23), we have r δu = A(r)U + B(r)(U · rˆ)ˆr, rˆ = , r = |r| ,A(a) = −1,B(a) =(11.34) 0, r where the two boundary conditions at r = a and r = ∞ fix c1 and c2 according to

a 3 a2 3a a2 A(r) = − + ,B(r) = − 1 . (11.35) r 4 4r2 4r r2

For the pressure perturbation, also linear in U with a decay to zero at inﬁnity,

p = C(r)U · r,ˆ C(r) → 0 as r → ∞, (11.36) we ﬁnd 3a C(r) = − µ. (11.37) 2r3 Integration of the viscous stresses on the surface of the sphere imparted by (11.34) ﬁnally gives the celebrated result

F = 6πµaU, (11.38) applicable for Re< 0.5.

5 11.3 Kolmogorov k−3 turbulence energy spectrum

With sufficient energy input, intermittent and time-variable flows can develop that are quite common, e.g., water flows in rivers, in air flow around buildings, flows in ducts and pipes, and in outflows from astrophysical systems. These high Reynolds number flows can also be found, perhaps unwittingly, around a cyclist on a fast track, around planes and automobiles and in propulsion Fluid dynamics 119 systems. The first scientific recognition of turbulence is due to Leonardo da Vinci (1452-1519).7 Kolmogorov (1941) realized that the observed turbulent flow represents a response of fluids to a relatively strong energy input at a scale L, that is much larger than a scale l at which viscous dissipation is taking place. 8 In our definition (11.4) for the dynamical viscosity, dissipation is immediate in case of Stokes flows, wherein the scale of energy input and the scale of the dissipative flow structures are exactly the same. Low Reynolds numbers, encountered on small scales, hereby offer a suitable site for dissipation into heat. If the energy input occurs on relatively large scales, however, energy is manifest mostly in vortical motions that are essentially non-dissipative, except for those at the smallest scales. In stationary turbulence, how, then, is the connection established between energy input at some large scale and energy dissipation a much smaller scale? The Lagrangian derivative in the Navier-Stokes equation includes the nonlinear convective term (u · ∇)u. At high velocities, this term provides a self-interaction whereby large scale structures can transform to small scale structures with no loss in energy, first noticed by Lewis F. Richardson. This is most readily expressed in Fourier space, where it can be seen to produce period doubling, that is canonical for quadratic nonlinearities. In turbulent motion, this process has been recognized as an energy cascade from large scale eddies to small scale eddies. 9 The cascade terminates with small eddies approaching low Reynolds number flows, i.e., in the Stokes regime set by a small scale l, where viscous effects dissipate the energy of the smallest eddies into heat. By the qualitative description above, we are led to consider a continuum of scales λ covering the inertial range of turbulent eddies,

l < λ ≤ L. (11.39)

If (k), k = 2π/λ, denotes the energy conversion rate at wave length k associated with the cascade process of continuously creating smaller eddies

7e.g. Ecke, R., 2005, “The Turbulence Problem,” Los Alamos Science, 29, 124. 8Kolmogorov, A. N., 1941, Compt. Rend. Acad. Sci. USSR, 30, 301; ibid. Compt. Rend. Acad. Sci. USSR, 32, 16. 9”... The small eddies are almost numberless, and large things are rotated only by large eddies and not by small ones, and small things are turned by both small eddies and large”, Leonardo Da Vinci, observing water ﬂow past rocks in river beds. 120

(k0 > k) out of larger eddies (k00 < k), then 2π 2π (k) = , k ≤ k ≤ k , k = , k = . (11.40) 0 min max min L max l

Here, (kmax) refers to viscous energy dissipation into heat at the scale of the smallest eddies. Thus, Kolmogorov realized that 0 is the key parameter controlling turbulence, provided that it is large enough for the onset of turbulence. For homogenous, isotropic and stationary turbulence, therefore, a proper starting point is 0 in (11.40) in terms of the energy dissipation per unit mass, that corresponds to the energy dissipationq ˙ per unit volume, encountered earlier, divided by the mass density. Following a Fourier analysis, we may consider the spectral energy density E(k) describing the total energy per unit mass of eddies per wave number k. By the Plancherel formula, 1 Z ∞ u2 = E(k)dk. (11.41) 2 0 By dimensional analysis, we have q˙ 1 [ ] = = cm2 s−2 × , [E(k)] = cm2s−2 k−1 = cm3s−2. (11.42) 0 ρ s

If E(k) is some function of 0 and k only, then

2 b 2 − 5 E(k) = C 3 k = C 3 k 3 (11.43) by selecting b = −5/3 for consistency with (11.42). The Kolmogorov spectrum

− 5 E(k) ∝ k 3 (kmin < k < kmax) (11.44) now follows by Kolmogorov’s insight of a conservative cascade (11.40) in the inertial range above the smallest scale, at which time dissipation sets in. In addition to experiments,10 studies of (11.44) become increasing accessi- ble to detailed Direct Numerical Simulations (DNS) using the Navier-Stokes equations in various numerical implementations.11 10e.g. Fig. 6 in Ecke, R., 2005, “The Turbulence Problem,” Los Alamos Science, 29, 124; Champagne, F. H., 1978, J. Fluid Mech. 86, 67; Grant, L., Stewardt, R., & Moillieta, W., 1962, J. Fluid Mech. 12, 241; Saddouchi, S., & Veeravalli, S.V., 1994, J. Fluid Mech., 268, 333. 11e.g., Foias, C., Holm, D.D., & Titi, E.S.., 2001, Physica D, 152-153, 505 Fluid dynamics 121

The Kolmogorov spectrum (11.44) leads to a two-point correlation function that, in light of the assumed homogeneity, isotropy and stationarity, simpliﬁes to Z ∞ sin kξ Q(ξ) = hu(r + ξ, t) · u(r, t)i = 2 E(k)dk (11.45) 0 kξ that is amenable to experimental observations in turbulent ﬂows in the lab- oratory. Following (11.43), we have

2 2 Q(ξ) = C 3 r 3 . (11.46) This relation is more easily stated than experimentally measured!12 To close the model description above, we next turn to the end of the cascade. Here, the energy input originating at scale L of the smallest wave number kmin = 2π/L is received at the scale l of the largest wave number kmax = 2π/l, U U 3 (k ) = (k ) = U 2 × = . (11.47) max min L L At kmax, the energy input is, ﬁnally, converted into heat due to the molecular viscosity of the ﬂuid. Seeking an expansion of in terms of the kinematic viscosity ν,[ν] =cm2 s−1, and l, we have = ν3l−4, i.e.,

1 ν3 4 l = . (11.48) 0 It follows that L 3 UL = Re 4 , Re = . (11.49) l L L ν 9 4 The eﬀective number of degrees of freedom, therefore, scales with ReL, which can be quite large in realistic turbulent ﬂows for which, typically, ReL >> 103.

11.4 Jeans instability

Stars form in overdense regions in molecular clouds. In its most basic form, it occurs due to gravitational collapse in suﬃciently cold regions, i.e., provided that gravity wins over thermal pressure. Interestingly, the rate of star 12see, e.g., Fig. 14 in Neumann, M., et al., 2009, Flow Meas. Instrum., 20, 252; Benedict, L.H., Gould, R.D., 1999, Exp. Fluids, 26, 3818. 122

Figure 11.2: Shown is a Hubble Space Telescope image of N11B in the LMC showing massive stars (blue and white) on the left with additional star formation in the center and on the right, indicative of a sequential star formation process (http://hubblesite.org/newscenter/archive/releases/2004/22/image/a/). formation in a cloud can be somewhat self-accelerating, as stellar winds from newborn massive stars compress gas in their surroundings. Provided that this process is sufficiently cooled by radiation, this compression can promote additional star formation by gravitational collapse (Fig. 11.2). Star formation can thus be self-regulated in some cases with a dual role of turbulence promoting and inhibiting star formation in overdense regions.13 Here, we focus on the basic process of gravitational collapse in a fluid of finite temperature. Without gravity, a local analysis for small amplitude variations will recover sound waves. What, then, is the impact of self-gravity? By dimensional analysis, we anticipate that gravity will be important at low frequency sound waves at angular frequencies below ω = p4πGρ (11.50) for a mean density ρ describing the local density in the cloud in view of the Poisson equation ∆Φ = 4πGρ (11.51)

13Shetty, R., & Ostriker, E.C., 2012, ApJ, 754, 2; McKee, C.F., & Ostriker, E.C., 2007, ARAA, 45, 565. Fluid dynamics 123 for the gravitational field Φ, where G denotes Newton’s constant. Below this frequency, gravity wins over thermal pressure, thus setting a critical Jeans length defined by the wave length r π λ > c , (11.52) s Gρ p where cs = ∂P/∂ρ denotes the sound speed, using ωλ = 2πcs. Relatively small mass elements are stable by thermal pressure, whereas relatively large masses are unstable to collapse by self-gravity. Since we are working on a relatively large scale, we can derive these results in the approximation of large Reynolds numbers, using Euler’s equations of motion for a compressible fluid. In a local analysis, the governing equations are 1 ∂P 1 ∂Φ ∂2Φ ∂ u + u∂ u = − − , ∂ ρ + ∂ (ρu) = 0, = 4πGρ. (11.53) t x ρ ∂x ρ ∂x t x ∂x2 Linearization to small perturbations about a static (non-moving) background state, we may drop all higher order terms, leaving 1 ∂p ∂δΦ ∂2δΦ ∂ u = − − , ∂ δρ + ρ∂ u = 0, = 4πGδρ, (11.54) t ρ ∂x ∂x t x ∂x2 2 where p ≡ δP , δP = csδρ. It reduces to ∂p ∂2δΦ c−2 = −ρ∂ u, = 4πGc−2p. (11.55) s ∂t x ∂x2 s Equivalently, we have the second order wave equation 2 ptt − cspxx = 4πGp (11.56) with dispersion relation 2 2 2 ω = csk − 4πGρ (11.57) between the eigenfrequency ω and the wave vector k = 2π/λ. Evidently, ω is real provided that k is sufficiently large, and ω becomes complex when k satisfies Jeans’ instability criterion (??). For star formation, regions in molecular clouds, therefore, must be sufficiently large in linear size and/or overdense for gravitational collapse to commence.14 14It may further be extended to a cosmological setting, in which case ρ would be the density contrast relative to the mean density in the universe. 124 11.5 Exercises

1. In light of (11.13), consider (11.7) in cylindrical geometry with coordinates (r, φ), i.e., the flow of coca-cola through a straw, back into a cup. Derive the Poisseuille flow profile u = u(r) in this case as a function of the gravitational acceleration g.

2. Accretion disks around rotating black holes may develop turbulence by forcing provided by input from the black hole.15 In this event, the disk may be driven into instability by induced thermal and magnetic pressures, causing a loss in energy by gravitational radiation by relatively long wave length non-axisymmetric waves. Would this aﬀect the energy spectrum in turbulence? In addition, relatively small mass inhomogeneities develop as a result of turbulent pressure waves, perhaps on the order of 1%. These mass inhomogeneities will appear in eddies at essentially all scales. Will this aﬀect the energy spectrum in turbulence? If so, how?

3. Consider a ﬁnite geometry that is highly anisotropic,16 e.g., the shape of a thin layer such as the atmosphere around Earth or the shape of a torus around a black hole. In such geometry, a turbulent ﬂow spectrum is inevitably anisotropic at wave lengths exceeding the scale at which this anisotropy is apparent (the infrared spectrum), which may be close to equipartition up to a break. Beyond such break, a Kolmogorov spectrum (??) of isotropic turbulence at wave lengths substantially smaller may develop. The turbulent cascade between these two scales will still be inertial (and without energy loss). Assume E(k) ∝ kγ also for the infrared spectrum. How will γ compare to −5/3?

4. Consider the Jeans’ criterion for gravitational collapse in spherical geometry.

15van Putten, M.H.P.M., 1999, Science, 294, 115. 16For atmospheric turbulence, a further distinction in the inertial range can be made between anisotropic large scale turbulence and isotropic small scale turbulence, where the former is sensitive to the large scale geometry. See, e.g., Oboukhov, A.M., 1962, J. Fluid Mech., 13, 77. Similar considerations may apply to deﬁne a break from long to short wave length turbulence in tori around black holes (van Putten, M.H.P.M., 2001, Phys. Rev. Lett., 84, 091101). Fluid dynamics 125

• Show that (11.50) is the frequency of radial oscillations in a stratiﬁed atmosphere, assuming constant background temperature.

• Derive (11.52) by equating the thermal and gravitational energies within a radius λ.

• Derive (11.52) by equating the sound speed to the free fall time from a radius λ.

−1 −24 • Calculate the Jeans length for the ISM, e.g., cs = 10 km s , ρ = 10 −3 g cm . Express the Jeans length as a function of cs and the mass density relative to these ﬁducial values. 126 Appendix A

Some units and constants

For reference to the various electromagnetic spectra discussed in Chapter 1, we here note some conversions of units to facilitate reading multi-wavelength observations. We have the correspondences

−12 14 1 eV = 1.60 × 10 erg = 11, 594 kBK = 2.43 × 10 hHz, (A.1)

−16 −1 −27 where kB = 1.38 × 10 erg K and h = 6.6 × 10 erg s denote the Boltzmann and Plank constants, respectively. The conversions of units (A.1) is distinct from commonly used equivalences to black body radiation using Wien’s displacement law

−1 10 −1 λp = 0.2897 T cm K, νp = 5.879 × 10 T Hz K , (A.2) associated with the location of maxima of the Planck radiation distribution −3 −1 function Pλ [erg cm s ] as a function of wave length λ and, respectively, 2 Pν [erg s ] as a function of frequency ν. Thus, λp = 123 nm (1 eV) in the black body radiation spectrum corresponds to an effective body temperature 14 of 23,553 K (2.03 eV), while νp = 2.43 × 10 Hz (1 eV) corresponds to an effective body temperature of 4,134 K (0.36 eV). The effective surface temperature T = 5778 K of the Sun hereby emits a black body radiation spectrum with λp =500 nm. The effective surface temperature defines the 2 4 33 −1 total luminosity L = 4πR σT = 3.84 × 10 erg s , where R = 6.955 × 1010 cm and σ = 5.67×10−5 erg cm−2 s−1 K−4 denotes the Stefan-Boltzmann constant.

127 128

Physical Constants

Black body constant α = π2k4/15c3h3 = 7.56 × 10−15erg cm−3 oK−4 −4 Stefan-Boltzmann constant σ = π2k4/60~3c2 = 5.67 × 10−5g sec−3 oK 3 49 −2 Bekenstein-Hawking entropy SH /A = kc /4G~ = 1.397 × 10 cm 2 2 −8 Bohr radius a0 = ~ /mee = 0.529 × 10 cm −16 o −1 Boltzman constant kB = 1.38 × 10 erg K o 1/kB = 1160 K/eV 2 3 13 Critical magnetic ﬁeld Bc = mec /e~ = 4.43 × 10 G −11 Compton wavelength λc/2π = ~/mec = 3.86 × 10 cm Velocity of light c = 2.99792458 × 1010cm/s Newton’s constant G = 6.67 × 10−8cm3g−1s−2 κ = (16πG/c4) = 2.04 × 10−24seccm−1/2g−1/2 Planck’s constant ~ = 1.05 × 10−27erg s 4 16 19 Planck energy Ep = lpc /G = 2.0 × 10 erg = 1.3 × 10 GeV −2 2 93 −3 Planck density ρp = lp c /G = 5.2 × 10 g cm 3 1/2 −33 Planck length lp = (G~/c ) = 1.6 × 10 cm 2 −5 Planck mass mp = lpc /G = 2.2 × 10 g 32 Planck temperature Tp = Ep/kB = 1.4 × 10 K −44 Planck time tp = lp/c = 5.4 × 10 s Electron charge e = 4.80 × 10−10esu Electron volt 1eV = 1.60 × 10−12erg −28 Electron mass me = 9.11 × 10 g 2 mec = 0.511 MeV Fine structure constant α = e2/~c ' 1/137 −24 Proton mass mp = 1.67 × 10 g 2 mpc = 938.2592(52) MeV 2 Neutron mass mnc = 939.5527(52) MeV 2 −27 = mpc + 2.31 × 10 g 2 2 = mpc + 1.29 MeV/c 4 2 Rydberg constant mee /2~ = 13.6 eV 4 2 4 −24 2 Thomson cross section 8πe /3mec = 0.665 × 10 cm Fluid dynamics 129

Some astronomical and cosmological constants

second of arc (”) 4.85 × 10−6 radians astronomical unit (A.U.) 1.50 × 1013 cm light year (ly) 0.946 × 1018 cm parsec (pc) 3.26 ly = 3.09 × 1018 cm 33 solar mass (M ) 1.99 × 10 g distance to Virgo 16.5 ± 0.1 Mpc [?] −1 Hubble constant (H0) 70.8 ± 1.6 (km/s) Mpc (WMAP) 3πH2 −30 closure density (ρc) 8πG = 9.4 × 10 g de Sitter temperature ~H0 = 2.7 × 10−30 K. 2πkB 130