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ridge push


ridge push	See ridge slide.
ridge slide (ridge push)		A mid-ocean ridge
is elevated compared to the surrounding ocean
basin. The oceanic plate is driven to move away
from the ridge by the component of gravitational
force parallel to the plate in the down slope di-
rection. This plate driving force is called ridge
slide or ridge push. See seaﬂoor spreading.

Riemann-Christoffel tensor	See Riemann
tensor.

Riemann tensor In geometry, the curvature tensor of an afﬁne space. In terms of a 1-form basis {ωa} and its second covariant derivatives, the Riemann tensor Rabcd is deﬁned as

a bωc − b aωc = −Rabcd ωd .

In a coordinate basis, the components Rbcda may be written as

Rbcda = Obd,ca − Obc,da + Ofa cObdf − Ofa d Ocbf

where Obca is the connection, and the “,” denotes a partial derivative. Note that when Obca is symmetric (no torsion) Rabcd = −Rabdc and Rabcd = Rcdab. See afﬁne connection, covariant derivative, sign convention.

Rigel (β Orionis), Spectral type B, RA 05h 14m32s.3, dec −08◦12 06 . The brightest star

in the constellation Orion and the seventh brightest star in the sky (apparent magnitude +0.12). Rigel is a blue (very hot) supergiant (absolute magnitude -8.1), about 100 times bigger in diameter than the sun. It is approximately 430 pc distant.

right ascension An angle coordinate on the celestial sphere corresponding to longitude, and measured in hours thus dividing the circle of 360◦ into 24 hours, measured eastward from the location of the vernal equinox (also called the ﬁrst point of Aries). At the time of the vernal equinox the right ascension of the sun is 0 hours. The direction to the sun at this time is then 0hRA, 0◦ declination, and this direction on the celestial sphere is referred to as the Vernal Equinox.

right-hand coordinate system One in which zˆ = xˆ ×yˆ in which {ˆx , yˆ , zˆ} are the unit vectors in the coordinate directions.

right-hand rule A prescription to determine the direction in a right-hand coordinate system, of a vector cross product A × B: right-hand ﬁngers in the direction of A, curl ﬁngers towards B; thumb points in the direction of product.

right-lateral strike-slip fault In geophysics, a strike-slip fault such that an observer on one side of the fault sees the other side moving to the right. The San Andreas fault in California is a right-lateral strike-slip fault.

rigidity See shear modulus.

rigid lid approximation A boundary condition applied when the surface displacements are small compared with interface displacements. In the continuity equation of ﬂow motion, one neglects the time change in the surface displacements, but retains the time change in the interface displacements.

rille Channel formed by ﬂowing lava. Rilles are old lava tubes where the roof has collapsed along part or all of the channel length. Lava tubes (hence rilles) are formed by very low viscosity lavas. On the moon, rilles have widths from a few tens of meters to a few kilometers and lengths of hundreds of kilometers. Longer channels are seen on Venus, although the composition of the ﬂows that created these channels is still debated. Rilles typically are widest at the upstream end and narrow as they progress downstream. Rilles show no delta-like deposits at their termini — they typically just fade away into a lava plain.

Rindler observer A uniformly accelerated observer in ﬂat Minkowski spacetime. The observer’s 4-dimensional trajectory is given by

x = ca−1 ea ξ cosh(aτ/c) t = c2 a−1 ea ξ sinh(aτ/c) ,

with y, z, constant. Here x, y, z are Cartesian coordinates (x along the direction of motion), t is the time of an inertial observer, τ is the observer’s proper time, ξ is a parameter related to

rip current

the position of the turning point, and c is the speed of light. The constant proper acceleration is given by a e−a ξ .

Because of the form of the trajectory, which approaches the light cones x ± t for t → ±∞, the Rindler observer is prevented from getting signals from events where x < t, and cannot send signals to events where x < −t. Thus x = t is a future causal horizon for the observer and x = −t is a past causal horizon. See future/past causal horizon.

ring Rings of small icy debris orbit around each of the four Jovian planets. The rings around Jupiter, Uranus, and Neptune are composed of very small particles (generally only a few microns in diameter) while the major rings of Saturn contain ice particles ranging in size from dust to boulders a few meters in diameter. Only Saturn’s rings are easily visible through ground-based telescopes. The origin of the different ring systems is still controversial. Saturn’s thick rings (primarily the A and B rings) were probably formed either by tidal disruption of an icy moon which wandered within the planet’s Roche Limit, or by the collision and destruction of two or more icy moons. The thinner rings around Saturn (primarily the E, F, and G rings) and the rings around Jupiter, Uranus, and Neptune, may result from material ejected off the small inner icy moons which orbit these bodies — this material may be ejected either from volcanism (as is proposed for Saturn’s moon Enceladus) or by meteorite impact (suggested for Jupiter’s ring). Resonances with moons may explain some of the gaps seen within rings, and shepherd moons are responsible for the maintenance of some rings, but the structure and narrowness of many rings are still not well understood.

ring current The dipole-like topology and the inhomogeneity of the Earth’s magnetic ﬁeld cause the energetic particles to be trapped in the magnetosphere in the “inner” and “outer” radiation belts. The magnetic dipole ﬁeld decreases with increasing distance from the Earth’s center by the inverse cube law, hence the particles will not follow exactly circular orbits while gyrating about a line of force. While gyrating, the magnetic ﬁeld will be stronger when the particle is

closer to the dipole than when it is farther away from it. Therefore, the radius of curvature is smaller at lower altitudes than in the higher and hence after gyration the particle will have drifted longitudinally. Electrons will drift eastward and protons will drift westward, thus producing a net westward electric current known as ring current. It is situated normally in the 4 to 6 Earth radii region and overlaps parts of the outer radiation belt. During geomagnetically disturbed periods, particularly during the “main” phase, the ring current is intensiﬁed. The activity of the ring current during disturbed periods is measured by an hourly Dst index, which is the magnitude of the horizontal component of the disturbed magnetic ﬁeld. High and low latitudes are avoided in the Dst determination to avoid the effects of the auroral and equatorial electrojets. The intensity of the ring current varies from about 1 to 2 nA/m2 in quiet times to up to 10 nA/m2 in disturbed periods.

ring galaxy A galaxy showing a bright, prominent outer ring encircling the whole galactic body. Ring galaxies are thought to be produced by the head-on encounter, nearly perpendicular, between a disk galaxy and a second galaxy: After the collision, an outwardmoving density wave within the disk gives rise to the ring, which appears to surround a central area of much lower surface brightness. Ring galaxies are of rare occurrence (only a few tens are presently known), consistent with their formation requiring a special orbit orientation for the encounter. A prototypical example of ring galaxies is the Cartwheel galaxy, believed to have been crossed by a smaller companion galaxy visible in its vicinity.

riometer For “Relative Ionospheric Opacity Meter”, an instrument measuring the amount of ionization below the usual height of the main ionospheric layers, caused by energetic particles that precipitate into the Earth’s atmosphere. It detects a decrease in the radio noise observed on the ground and coming from the distant universe. See polar cap absorption.

rip current A shore-normal current at a beach that carries water from a region landward

riprap

of a sand bar through a gap in the sand bar and into the offshore region.

riprap A term used to refer to the heavy material used to armor a shoreline. May be comprised of rock, concrete elements or debris, or some other heavy material.

Robertson–Walker cosmological models

A class of spacetimes describing homogeneous isotropic expanding universes. The cosmic matter is assumed to be a perfect ﬂuid. The metric can be written in spherical coordinates (t, r, θ, φ) as

ds2 = −dt2 + R2		dr2	+ r	2	dθ2 + sin2 θ dφ2	.
	1	k r2
		−

special space-time in which the repeated principal null direction diverges but has no shear or rotation. It is thought to describe the gravitational ﬁelds of radiative sources. The vacuum metric, discussed by I. Robinson and A. Trautman, has the form

ds2

2 dr du

−

2r2P

H du2

−2dζ dζ

where

H =

2P 2

∂2 log P

∂ log P

−

∂ζ

− 2r

∂u

r .

∂ζ

The function P satisﬁes

∂ log P

= k++ log P,

+ =

∂2

∂u

∂ζ∂ζ

An alternate equivalent formulation is:

and ζ is a complex stereographic spherical coor-

dinate. No physically well-behaved Robinson–

dχ

+ sinn

dθ

+ sin−

Trautman solutions other than the (nonradiative)

θdϕ

Schwarzschild space-time are known.

where

Roche limit

The closest orbital distance, R,

sinnχ

sin χ, k = +1

at which an orbiting satellite can hold itself to-

gether gravitationally, in spite of tidal forces

χ, k = 0

from its primary. Thus, the gravitational ac-

sinh χ, k = −1 .

celeration holding a particle on the surface of

The curvature parameter k = 0, ±1 for, respec-

a body of mass, mb of radius Rb is:

tively, ﬂat, spherical, and hyperbolic 3-space.

ag =

Gmb

In all cases, R(t) is an arbitrary function of

Rb2

the time-coordinate called the scale factor. The

Einstein equations equate the Einstein tensor

On the other hand, the relative tidal acceleration

(calculated from derivatives of the metric coef-

between the center of the satellite and a point

ﬁcients) to the stress-energy tensor, which must

radially towards or away from the primary is:

have the same symmetry as the geometry, and

2GM

so contains only spatially homogeneous matter-

at =

−

≈

Rb .

density ρ and pressure p. The time-evolution

(R + Rb)2

of the model is undetermined until an equation

If the ratio at /ag exceeds unity, the satellite will

of state is imposed on ρ and p; the equation of

be destroyed due to the tidal forces:

state results in determining the function R(t).

With p = 0 (“dust”),

the Robertson–Walker

2M mb

models become the Friedmann–Lemaître cos-

at /ag =

(1)

mological models; the dust solutions also apply

= 1 is the Roche limit:

to the interior of a sphere of homogeneous dust.

The R for which at /ag

See comoving frame, Friedmann–Lemaître cos-

mological models, maximally symmetric space,

Oppenheimer–Snyder model, Tolman model.

RRoche =

Rb3

(2)

Robinson–Trautman space-time (1962)

where M is the mass of the central body and mb

general relativity, an algebraically matter-free

and R3 refer to the satellite. Note that according

Rossby wave

to (1), the ratio of accelerations is twice the ratio of densities (in the case of the central body, as if it were spread out to the orbital radius).

Small solid bodies such as rocks or artiﬁcial satellites are held together by molecular bonds and hence can survive the tidal stresses, but bodies with little internal strength (such as “rubble pile” asteroids) are easily pulled apart. The destruction of bodies with low internal strength by movement within the Roche limit may explain the formation of some ring systems. The accretion of very small satellites into a larger moon is also inhibited in this region.

Roche lobe (Also inner Lagrangian surface.) In a binary system, in a corotating frame, the lowest-lying equipotential surface which envelopes both members. If one of the stars ﬁlls its Roche lobe, then mass will ﬂow from it towards the companion star. Stars that form or move so close together that both stars ﬁll their Roche lobes are contact binaries or W Ursa Majoris stars. In the more common case, an evolved star which has already shed a good deal of matter, continues to transfer material from the smaller side of the lobe through a gas stream to the more massive companion, perhaps through an accretion disk. If the accretor is a main sequence star, we would see an Algol system. If it is a white dwarf, we would see a cataclysmic variable.

Rocket effect (string loop) One way of diminishing the mass-energy density of a network of cosmic strings is by the continuous generation of loops. These will form through string interactions, like intercommutation.

Loops will oscillate with characteristic frequencies given by the inverse of their size (in appropriate units). This allows energy to be radiated away in the form of gravity waves. These waves carry not only energy but also momentum and therefore highly asymmetric loops might be propelled like rockets, reaching velocities as high as a tenth of the velocity of light. See cosmic string, intercommutation (cosmic string), scaling solution (cosmic string).

Rosalind Moon of Uranus also designated UXIII. Discovered by Voyager 2 in 1986, it is a small, irregular body, approximately 27 km in radius. Its orbit has an eccentricity of 0, an

inclination of 0.3◦, a precession of 167◦ yr−1, and a semimajor axis of 6.99 × 104 km. Its surface is very dark, with a geometric albedo of less than 0.1. Its mass has not been measured. It orbits Uranus once every 0.559 Earth days.

Rossby, Carl-Gustav (1898–1957) SwedishAmerican meteorologist. He founded the ﬁrst meteorology department in the U.S. at the Massachusetts Institute of Technology in 1928. He was among the ﬁrst to recognize the important role of transient mid-latitude wave disturbances in the atmospheric general circulation. He established many of the basic principles of modern meteorology and physical oceanography during the 1930s. Some of the theoretical ideas that he developed during that time were instrumental in the development of numerical prediction models during the following decades.

Rossby number The Rossby number I is a non-dimensional parameter deﬁned as the ratio between the horizontal advection scale and the Coriolis scale in the horizontal momentum equation, that is, I = fUL where U and L are the velocity scale and length scale of motion and f is the Coriolis parameter. When this number is much less than unity, the ﬂow is said to be geostrophic.

Rossby radius	See Rossby radius of defor-
mation.
Rossby radius of deformation		(Rossby,

1938) This is the horizontal scale at which the Earth’s rotation effects become as important as buoyancy effects and is deﬁned by LR = C/|f | where C is the gravity wave speed and f is the Coriolis parameter. This has been called the

Rossby radius or the radius of deformation. In baroclinic ﬂow, C is the wave speed of each baroclinic mode in a non-rotating system. It describes the radius Ro = u/f[m] of an inertial circle which a freely moving water parcel would follow with a speed u under the effect of the Coriolis parameter f.

Rossby wave A wave in a rotating ﬂuid body, involving horizontal motion, for which the sole restoring force is the Coriolis effect. In neutron stars the retrograde mode, which rotates