n

of scatterers. Calling

N

a total number, as the PDF does, leaves the displayed expression dimensionally wrong:

σ^{3} / (G^{2} m^{2} N)

has units of time per volume. The intended local expression must use

n

, or

N / V

for a homogeneous sample of volume

V

. We shall also make the normalization of the stellar relaxation time explicit, since an unspecified representative one-dimensional velocity dispersion does not fix an exact numerical coefficient.

In a small-angle gravitational encounter, approximate the relative trajectory by a straight line of speed

u

and impact parameter

b

. The perpendicular acceleration of one star due to the other is

G mb / (b^{2} + u^{2} t^{2})^{3/2}

. Its impulse is therefore

∣Δ v_{⊥} ∣ = G mb \int_{- \infty}^{\infty} \frac{d t}{( b ^{2} + u ^{2} t ^{2} ) ^{3/2}} = \frac{2 G m}{b u} .

(1)

A star encounters

2 πb d b n u

scatterers per unit time in this interval of impact parameters. Independent deflections have random directions, so their variances add as in a random walk. Hence the accumulated squared velocity change has rate

D_{v} \equiv \frac{d ⟨ ∣Δ v ∣ ^{2} ⟩}{d t} = \int_{b_{m i n}}^{b_{m a x}} (\frac{2 G m}{b u})^{2} 2 πbn u d b = \frac{8 π G ^{2} m ^{2} n}{u} ln Λ, Λ = \frac{b _{m a x}}{b _{m i n}} .

(2)

The Coulomb logarithm in stellar dynamics arises because each logarithmic interval of impact parameter contributes equally. The lower cutoff is approximately the strong-deflection scale,

b_{m i n} ≃ G (2 m) / u^{2}

for equal masses, or the physical collision scale if larger. The upper cutoff is a scale on which the local density and straight-line encounter approximation cease to apply, such as the local disk thickness. Thus

Λ

is dimensionless; it is not a number density or an arbitrary dimensional argument of a logarithm.

To display the normalization issue, use the representative relative speed

u = 2 σ

and define

T (κ)

D_{v} T = κ σ^{2}

. This gives

T (κ) = \frac{κ}{4 π 2} \frac{σ ^{3}}{n G ^{2} m ^{2} l n Λ} .

(3)

Choosing

κ = 3/4

recovers precisely the printed coefficient after repairing

N

n

T_{R}^{printed convention} = \frac{3}{16 π 2} \frac{σ ^{3}}{n G ^{2} m ^{2} ln Λ} .

(4)

This is an explicitly defined convention, not a uniquely derivable normalization from the supplied data. If instead relaxation means a mean-square change equal to the full initial random-speed variance,

3 σ^{2}

, the same encounter approximation gives

T_{R} = 4 T_{R}^{printed convention}

. Averaging the encounter speed distribution instead of using one representative

u

changes the coefficient again. The robust result is the scaling

T_{R} \propto σ^{3} / (n G^{2} m^{2} ln Λ)

For a transparent solar-neighbourhood estimate, take

n = 0.1 pc^{- 3}

m = M_{⊙}

σ = 30 km s^{- 1}

and

b_{m a x} = 300 pc

. With

G = 4.3009 \times 1 0^{- 3} pc M_{⊙}^{- 1} (km s^{- 1})^{2}

, one obtains

b_{m i n} = 4.78 \times 1 0^{- 6} pc

ln Λ = 17.96

, and

T_{R}^{printed convention} ≃ 3.35 \times 1 0^{13} yr, T_{R}^{full variance} ≃ 1.34 \times 1 0^{14} yr .

(5)

These illustrative parameters give an order

1 0^{13}

1 0^{14}

year stellar relaxation time, much longer than a galactic lifetime of order

1 0^{10}

years. The solar neighbourhood is consequently a collisionless stellar system with respect to encounters between individual stars. Scattering by massive clouds and collective structure is a different process; the estimate does not assert that every mechanism of disk heating is negligible.

ii

 0  0

Solution

 0  0

Write

μ_{g} = γm

for the gravitational parameter of the relative Kepler orbit, and use specific orbital energy and momenta per unit reduced mass. The Hamiltonian is

E = \frac{1}{2} (p_{r}^{2} + \frac{p _{θ}^{2}}{r ^{2}} + \frac{p _{ϕ}^{2}}{r ^{2} s i n ^{2} θ}) - \frac{μ _{g}}{r} = - \frac{μ _{g}}{2 a} .

(1)

If physical momenta rather than these specific momenta are used, multiply the actions by the reduced mass. This only introduces a constant factor into the phase space measure and leaves the eccentricity fractions unchanged. The formulas in the question adopt the specific-action convention.

Here is an explicit local canonical transformation. A separated complete integral of the Hamilton-Jacobi equation is

S (r, θ, ϕ; L, G, H) = H ϕ + \int^{θ} G^{2} - \frac{H ^{2}}{s i n ^{2} ϑ} d ϑ + \int^{r} 2 E (L) + \frac{2 μ _{g}}{s} - \frac{G ^{2}}{s ^{2}} d s, E (L) = - \frac{μ _{g}^{2}}{2 L ^{2}} .

(2)

Choose branches along an orbit and take

p_{j} = \partial S / \partial q_{j}

l = \partial S / \partial L

g = \partial S / \partial G

h = \partial S / \partial H

. Then

d S = \sum_{j} p_{j} d q_{j} + l d L + g d G + h d H,

(3)

whose exterior derivative proves preservation of the canonical symplectic form. Thus these variables are canonical, and the absolute Jacobian determinant is one. The angular separation gives

G = ∣ r \times v ∣

and

H = G cos i

, including the sign of the angular momentum along the reference axis.

For clarity, the radial action can be evaluated rather than assumed. Let

r = a (1 - e cos u)

, with

u

the eccentric anomaly. Then

p_{r} = μ_{g} / a e sin u / (1 - e cos u)

, and

J_{r} = \frac{1}{2 π} \oint p_{r} d r = \frac{μ _{g} a}{2 π} \int_{0}^{2 π} \frac{e ^{2} s i n ^{2} u}{1 - e c o s u} d u = μ_{g} a (1 - 1 - e^{2}) = L - G .

(4)

In the integral, use

e^{2} sin^{2} u / (1 - e cos u) = 1 + e cos u - (1 - e^{2}) / (1 - e cos u)

and

\int_{0}^{2 π} (1 - e cos u)^{- 1} d u = 2 π / 1 - e^{2}

. The polar action is

J_{θ} = G - ∣ H ∣

, while the signed azimuthal action is

J_{ϕ} = H

. The separated angular coordinates have the usual Delaunay variables interpretation:

g

is the argument of periapsis, and

h

is the longitude of ascending node. In particular,

l = \frac{d E}{d L} \int \frac{d r}{p _{r}} = \frac{μ _{g}^{2}}{L ^{3}} \frac{a ^{3/2}}{μ _{g}} \int (1 - e cos u) d u = u - e sin u

(5)

with a suitable origin at periapsis, so

l

is the mean anomaly. We have therefore recovered

L = μ_{g} a, G = L 1 - e^{2}, H = G cos i,

(6)

not merely assigned names to the new coordinates.

For nondegenerate bound Kepler orbits, the ranges are

0 < L < \infty, 0 < G < L, - G < H < G, l, g, h \in R / (2 π Z) .

(7)

The boundaries describe circular, coplanar or radial degeneracies, where some angles cease to be defined; they have zero measure in the present calculation. The actions are canonical coordinates with nested bounds, rather than three independently unrestricted Cartesian coordinates. The square of

H

alone would miss the distinction between prograde and retrograde inclinations.

By the canonical transformation, the population measure becomes

d N = F (- \frac{μ _{g}^{2}}{2 L ^{2}}) d L d G d H d l d g d h .

(8)

At fixed

L

, the condition

e^{'} < e

L 1 - e^{2} < G < L

. Integrating all three angles and then

H

gives

N (e^{'} < e) = (2 π)^{3} \int_{0}^{\infty} F (E (L)) d L \int_{L 1 - e^{2}}^{L} 2 G d G = (2 π)^{3} e^{2} \int_{0}^{\infty} F (E (L)) L^{2} d L .

(9)

Consequently the Delaunay phase-volume proof of the thermal eccentricity distribution yields

\frac{N ( e ^{'} < e )}{N _{bound}} = e^{2}, \frac{1}{N _{bound}} \frac{d N}{d e} = 2 e (0 \leq e < 1) .

(10)

The PDF's

e^{2}

dependence must be interpreted as a cumulative distribution function, not an eccentricity probability density function or the value of the original six-dimensional phase-space distribution function. A finite value of

F (E)

at every energy also does not suffice to make the total population finite:

F = 1

on all bound energies gives the divergent integral

\int_{0}^{\infty} L^{2} d L

. The fractions are well defined at each fixed energy, over a finite allowed energy range, or for an energy distribution with a finite total integral.

A Boltzmann distribution is therefore unnecessary. Any normalizable energy-only distribution has the same eccentricity marginal, although such a distribution is a collisionless steady state by Jeans theorem. The marginal alone does not even prove stationarity: for