DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

a precession of 2.0 ◦ yr −1 , and a semimajor axis
of 4.36 × 10 5 km. Its radius is 789 km, its mass
3.49 × 10 21 kg, and its density 1.66 g cm −3 .It
has a geometric albedo of 0.27 and orbits Uranus
once every 8.707 Earth days.
Tolman model Also called the Tolman–
Bondi model and the Lemaître–Tolman model.
An inhomogeneous cosmological model containing
pressureless fluid (dust). The most
completely researched of the inhomogeneous
models of the universe. It results from Einstein’s
equations if it is assumed that the spacetime
is spherically symmetric, the matter in
it is dust, and that at any given moment different
spherical shells of matter have different
radii. (If the radii are the same, then a complementary
model results whose spaces of constant
time have the geometry of a deformed 3dimensional
cylinder; in the Lemaître–Tolman
model the spaces are curved deformations of the
Euclidean space or of a 3-dimensional sphere.)
The model was first derived in 1933 by G.
Lemaître, but today it is better known under
the name of the Tolman, or the Tolman–Bondi
model. The Oppenheimer–Snyder model is a
specialization of such models to homogeneous
dust sources. See Oppenheimer–Snyder model,
comoving frame.
tombolo A sand spit in the lee of an island
that forms a bridge between the island and the
mainland. Referred to as a salient if the connection
is not complete.
Tomimatsu–Sato metrics (1973) An infinite
series of metrics describing the exterior gravitational
field of stationary spinning sources. The
δth Tomimatsu–Sato metric has the form
ds 2 =
B
δ 2 p 2δ−2 (a − b) δ2 −1
+ gik dx i dx k
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dy 2
b
dx2
−
a
Toomre’s stability parameter Q (1964)
where (the other two coordinates are x 3 ,x 4 )
and
g33 = bD
δ 2 B
g34 = 2q bC
B
g44 = A
B
a = x 2−1 , b = 1 − y 2 , p 2 + q 2 = 1 .
The polynomials A, B, C, and D are all constructed
from the order-δ Hankel matrix elements
Mik = f(i+ k − 1)
where f(k) = p 2 a k + q 2 b k . The first, δ = 1,
member is the Kerr metric. The δ = 2, 3, and
4 members were found by A. Tomimatsu and
H. Sato in 1973. The full theory was established
by S. Hori (1978) and M. Yamazaki (1982). The
singularities of the δth Tomimatsu–Sato spacetime
are located at δ concentric rings in the equatorial
plane.
Toomre’s stability parameter Q (1964) A
numerical parameter describing the stability (or
lack of stability) of a system of self-gravitating
stars with a Maxwellian velocity distribution.
By consideration of the dispersion relation
for small perturbations in an infinite, uniformly
rotating gaseous disk of zero thickness and constant
surface density , at angular velocity
one finds the stability criterion
Qg = v · 2
> 1
πG
where G is Newton’s constant, and v is the speed
of sound in the gas.
A generalization of the above statement to a
differentially rotating gaseous disk in the tight
winding approximation leads to a local stability
condition at radius r: Qg(r) ≡ v(r)κ(r)/πG
(r) > 1, where κ(r) is the local epicyclic
frequency at radius r. The single unstable
wavelength in the Qg(r) = 1 gaseous disk is
λ∗(r) = 2π 2 G(r)/κ 2 (r). [These gaseous
and stellar disks are shown to be stable to
all local non-axisymmetric perturbations (Julian
and Toomre, 1966).] A similar local
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