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Dictionary of Geophysics, Astrophysic, and Astronomy.pdf

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tidal formation of solar system

from the source of the gravitational ﬁeld. The tidal acceleration is a relative acceleration (locally) proportional to the gradient of the gravitational acceleration, equivalent to the second derivative of the potential, φ,ij , and is also proportional to the separation !i between the points considered. Here the subscript “,” denotes partial derivative.

Hence

(δatidal)j = −!iφ,ij .

A similar expression, involving the Riemannian tensor, is found in general relativity (so φ,ij is called the Newtonian (analog of the) Reimannian tensor).

Because they are proportional to the gradient of the gravitational acceleration, tidal forces are proportional to r−3, and so rise sharply at small distances but become negligible at large distances. Typically, tidal forces deform planets and moons. On the Earth, both the solid body of the Earth and, more noticeably, the surface of water in the oceans reacts to these relative accelerations, in the form of tides, because of interactions with both the sun and the moon.

In fact, because it is closer, the moon generates higher tides on the Earth than does the sun. Far from the ocean shores, the height of the lunar tidal wave is 65 cm, and the height of the solar tidal wave is 35 cm. These waves travel around the Earth because the Earth rotates. The tides become much higher at the shores because of local topography (when the tidal wave goes into a funnel-shaped bay, its width is decreased, and so the height must increase) or because of resonances. The latter happens in the Bay of Fundy in Canada, where tidal waves reach the height of 16 m.

Because tidal forces deform a solid body (a planet or moon), tides dissipate energy and act to decrease rotation. As a result, the rotation is decelerated until it becomes synchronous with the orbital motion — which means that afterwards the planet or moon faces its companion body always with the same side. This has happened to our moon in its orbit around the Earth.

tidal formation of solar system A theory attributing the formation of the solar system to a close tidal encounter with another star, which

drew material out of the sun to condense into planets. This theory is now out of favor because it suggests solar systems are rare, since such encounters are rare, while recent observations provide evidence for planetary systems around a number of local stars, and even around neutron stars.

tidal friction As a result of the anelastic response of the Earth, the peak of its tidal bulge lags the maximum tidal force by about 12 min. Because the Earth rotates faster than the moon orbits the Earth, the tidal bulge leads the Earthmoon axis by about 3◦. This lag angle causes a torque acting on the Earth to slow down its rotation, resulting in a length-of-day increase of over 2 msec per century. This effect is called tidal friction. The same torque acting on the moon increases its orbiting speed and hence the Earth–moon distance. There is also tidal friction between the Earth and the sun, about 1/5 of the strength of the lunar tidal friction.

tidal heating The process by which a body’s interior heat is generated by the gravitational tidal forces of external bodies. Tidal deformation leads to friction, which in turn creates internal heat. For the Earth’s moon, tidal heating occurred for only a short period, until its orbit became circularized and synchronous (when the rotation period exactly equals the orbital period). However, if the moon is in a resonance with another moon, as in the case of several moons of Jupiter, the rotation can fail to become locked, and the tidal heating stage can last much longer. This is the situation with Jupiter’s moons of Io and Europa. The combination of Io’s proximity to massive Jupiter and its orbital resonance with Europa causes tremendous tidal heating and gives rise to the active volcanism seen on Io’s surface. Europa is slightly further from Jupiter than Io, so it feels less of Jupiter’s tidal forces, but it also is in a resonance (with Ganymede), and this situation causes some tidal heating on Europa, which may allow for a liquid water ocean to exist under its icy crust.

tidal inlet An opening between the sea and a sheltered estuary or river.

time dilatation

tidal period The time between two points of equal phase in a tidal curve. Commonly measured between two high or two low tides.

tidal prism The volume of water that is exchanged between a tidal estuary and the sea in one tidal period. Equivalent to the average tide range within the estuary times the area of the estuary. Since the range will vary in space and time, the volume is not simply a horizontal slice, but a complicated function.

tidal radius The radius within which all the luminous matter of a cluster or a galaxy is contained. The tidal radius can be measured for globular clusters and for galaxies belonging to clusters, which are found to have well-deﬁnite outer limits (in contrast with brightness proﬁles of isolated elliptical galaxies, described by de Vaucouleurs or Hubble’s–Reynolds law). The name arises from the understanding that in the case of a cluster galaxy, repeated encounters with nearby galaxies can lead to tidal stripping of the outer stars, which are loosely gravitationally bound, and to the evaporation of the outer envelope, leaving only stars which are inside the tidal radius.

tidal stripping The escape of gas and stars gravitationally bound to a system, such as a galaxy or a globular cluster, due to tidal forces exerted by an object external to the system. For example, in a cluster of galaxies, tidal stripping may remove loosely bound stars from the galaxy outer envelope; in a close encounter between galaxies, stars and gas can be transferred from one galaxy to the other.

tidal tail A highly elongated feature produced by tidal forces exerted on a spiral galaxy by a companion galaxy. A most notable example of tidal tails is observed in the “Antennae” pair of galaxies (NGC 4038 and NGC 4039), where the tidal tail extends for a projected linear size of ≈100 kpc, much larger than the size of the galaxies themselves. Computational models show that very extended tails, like the ones in the Antennae, are produced by a prograde encounter between galaxies, i.e., an encounter between a spiral galaxy and an approaching com-

panion galaxy which moves in the same sense of the spiral rotation.

tide The response of the solid or ﬂuid components of a planet or other astronomical body, under the inﬂuence of tidal forces.

tide range The vertical distance between high tide and low tide at a point. Will vary in time because of temporal variation in the tide signal.

tilt angle In solar magnetohydrodynamics, on the source surface, a neutral line separates the two hemispheres of the sun with opposing magnetic polarity. The maximum excursion of this neutral line with respect to the heliographic equator is called tilt angle. Since this neutral line is carried outwards as the heliospheric current sheet, the tilt angle also deﬁnes the waviness of the current sheet and, therefore, is an important parameter in the modulation of the galactic cosmic radiation.

In terrestrial magnetospheric research, it is the angle between the z-axis in GSM coordinates and the dipole axis of the Earth. The GSM x-y plane can be viewed as providing an approximate north–south symmetry plane of the magnetosphere (see equatorial surface). A tilt angle

ψ= 00, therefore, signiﬁes a magnetosphere with the dipole axis perpendicular to the equator, and the larger ψ, the more the axis departs from the perpendicular. Sunward inclination of the dipole gives ψ > 0, tailward inclinations

ψ< 0.

In the Earth’s magnetosphere, ψ can vary within ±35◦, while for Uranus and Neptune all values are possible. For Earth the tilt shifts the location of the polar cusps (both near the magnetopause and in the ionosphere) and causes warping of the plasma sheet. See modulation of galactic cosmic rays, source surface.

time dilatation The slowing-down of clocks in a system moving with respect to a given observer. Suppose a system S (for example, an interstellar spacecraft) moves with respect to observer O with the velocity v along the straight line OS.

Suppose an observer in the system S measures the time-interval T between two events,

time dilation

say two ticks (1 sec apart) of his clock, which is at rest in his frame. Then the observer O will ﬁnd that the time-interval T between the same two events on O’s own clock is T =

T / 1 − v2/c2, where c is the velocity of light. Hence, T > T , and the observer O will decide that the clock in S runs slow. This formula follows from the experimentally veriﬁed fact that the velocity of light is the same in every inertial reference frame. The time dilatation is a relative effect: the observer in S will also observe that O’s clock goes more slowly. This is one of the most famous predictions of the Special Relativity Theory and is very directly or accurately veriﬁed. Many elementary particles observed in laboratories and cosmic rays are unstable; they decay into other particles. For instance, the mean decay-time for µ-mesons is 2.3·10−6sec. Such µ-mesons, formed at the top of the Earth’s atmosphere by cosmic ray collisions, would move close to the velocity of light c = 3 · 1010cm/s. Thus, in the absence of time dilatation they would be able to ﬂy, on average, the distance of only 690 m before decaying, and only a minute fraction of the initial number would survive the journey to the surface of the Earth. In fact, µ-mesons created in high layers of Earth’s atmosphere reach detectors on the surface of the Earth in copious quantities. They are seen to live longer because they move with a large velocity with respect to us. See special relativity.

time dilation See time dilitation.

time-distance helioseismology The study of the solar interior using the direct measurement of travel times and distances of individual acoustic waves. Carried out using temporal crosscorrelations of the intensity ﬂuctuations on the solar surface.

time in semiclassical gravity Current approaches to quantization of gravity do not produce an obvious time variable. However, one can introduce time if the semiclassical approximation is assumed to hold for (at least some of) the gravitational degrees of freedom, i.e., if these degrees of freedom behave classically, while the rest of the system is quantized. The introduced time is associated with the change of such ob-

servables; gravity is treated as a clock. In a more reﬁned model, one could imagine some macroscopic apparatus which couples to the speciﬁed gravitational degrees of freedom and gives the value of the time. In this sense, the rotation of the Earth or any other geodesic motion in a curved space-time is a good and well-known example. See Wheeler–DeWitt equation.

time-like inﬁnity (i±). The distant future and distant past limits of timelike curves. See conformal inﬁnity.

time-like vector An element t of a linear

space with a Lorentzian metric g of signature

(−, +, +, +), such that g(t, t) = gabtatb < 0. In the theory of general relativity, a time-like four-vector represents the velocity of propagation of a particle with rest-mass. See metric, signature.

time zone One of 24 zones approximately 15◦ (1 h) of longitude in width and centered in multiples of 15◦ from Greenwich, England, throughout which the standard time is constant and one hour earlier than the zone immediately to the east, except at the International Date Line at longitude 180◦, when 24 h is subtracted from the date in moving westward.

Titan Moon of Saturn, also designated SVI. It was discovered by Huygens in 1655. Its orbit has an eccentricity of 0.029, an inclination of 0.33◦, a semimajor axis of 1.22 × 106 km, and a precession of 0.521◦ yr−1. The radius of its solid body is 2575 km, but its thick atmosphere extends more than 100 km above this surface. Its mass is 1.35 × 1023 kg, and its density is 1.89 g cm−3. Its geometric albedo in the visible is about 0.2, and it orbits Saturn once every 15.95 Earth days. Titan is the only solar system satellite with a signiﬁcant atmosphere.

Titania Moon of Uranus, also designated UIII. Discovered by Herschel in 1787, its surface is a mixture of craters and interconnected valleys. Titania shows signs of resurfacing and may have once undergone melting. Possibly the resultant volume change upon cooling caused the observed cracks and valleys. Its orbit has an eccentricity of 0.0022, an inclination of 0.14◦,

Toomre’s stability parameter Q (1964)

a precession of 2.0◦ yr−1, and a semimajor axis

of 4.36 × 105 km. Its radius is 789 km, its mass 3.49 × 1021 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 3- dimensional 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

ds2	=	B	dy2	−	dx2
ds2	=	δ2p2δ−2(a − b)δ2−1	b	−	a
		+ gik dxi dxk

where (the other two coordinates are x3, x4)

bD g33 = δ2B

g34 = 2q

g44 = A B

and

a = x2−1 , b = 1 − y2 , p2 + q2 = 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) = p2ak + q2bk. 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) = 2π2G (r)/κ2(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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