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Conrad discontinuity

This implies that the tangent vectors to the congruence form a vector ﬁeld and, further, every smooth vector ﬁeld generates a congruence. See vector ﬁeld.

conical spacetime See deﬁcit angle (cosmic string).

conics Pitch angle distributions whose peak intensity is neither along the ﬁeld line (“cigar”) nor perpendicular to it (“pancake”) but at some intermediate angle. Ion conics (generally of O+ ions) are often observed on magnetic ﬁeld lines above the auroral zone, at altitudes of the order of 5000 km, associated with the aurora. It is believed that they arise from wave-ion interactions which energize the ions at an altitude below the one where the conic is observed. Such an interaction preferentially increases the ion’s velocity components perpendicular to the magnetic ﬁeld, producing a pancake distribution. By the conservation of magnetic moment, the pancake transforms into a cone at higher altitudes.

conic section Any of the plane ﬁgures obtained by intersecting a circular cone with a plane. If the plane is parallel to the base of the cone, the ﬁgure is a circle. As the tilt angle β of the plane increases, 0 < β < α, where α is the apex half angle of the cone, the ﬁgure is an ellipse. If β = α the ﬁgure is a parabola, and if β > α, the ﬁgure is a hyperbola.

In Newtonian physics, gravitational motion is an orbit that is a conic section with the sun at one focus.

conjugate depths The two water depths that appear on either side of a hydraulic jump in open channel ﬂow. Also referred to as sequent depths. They differ from alternate depths in that the speciﬁc energy is not the same for conjugate depths due to energy loss in the hydraulic jump.

conjunction Orientation of planets so that the angle planet-Earth-sun equals zero. For outer planets, the planets are as far from Earth as possible in their orbits, because they are then on opposite sides of the sun. For inner planets, conjunction is either a closest approach to Earth (both planets on the same side of the sun), or the

most distant (the inner planet on the opposite side of the sun from Earth).

connection See afﬁne connection.

connection longitude Heliographic longitude to which a spacecraft is connected magnetically. The connection longitude φconn can be determined from the “offset” ?φ of the footpoint of the archimedian magnetic ﬁeld spiral through the observer with respect to the observer’s heliolongitude φo according to

φ conn = φ o + ?φ = φ o + rω

vsowi

with r being the radial distance of the observer from the sun, ω the sun’s angular speed, and v sowi the solar wind speed. The connection longitude is important in the study of energetic particles because, in contrast to the electromagnetic radiation, particles do not propagate radially but along the magnetic ﬁeld line. For average solar wind conditions, the Earth, or an earth-bound spacecraft, magnetically is connected to a position around W 58 on the sun. Connection longitudes also can be established to shocks. See cobpoint.

Heliolongitude and connection longitude.

Conrad discontinuity Seismic discontinuity at a depth of around 20 km between the upper and the lower crusts beneath a continent and an island arc. The Conrad discontinuity is not a sharp discontinuity like Mohoroviciˇ c´’s discontinuity. The Conrad discontinuity was originally thought to represent a boundary between the lower crust consisting of basaltic rocks and the upper crust consisting of granitic rocks, deducing from velocity of Conrad head waves (P *).

conservation of angular momentum

But now it is thought to be a thermodynamical interface or a rheological boundary.

conservation of angular momentum In the absence of external torques, the total angular momentum L of a system is a constant:

L = L(a) = constant pseudovector.

Here a sums over the subsystems comprising the system.

conservation of energy The observation that energy cannot be created or destroyed but can only be changed in form.

Forms of energy include

1.potential energy (the ability to do work),

2.kinetic energy (the energy of motion),

3.and relativistically, the energy equivalent of mass: E = mc2.

Since an understanding of special relativity implies that mass and energy are interconvertable, the conservation law is often posed as conservation of mass and energy. In normal macroscopic laboratory physics, mass is very closely conserved; the deviations are of order p/ρc2. In astrophysical settings, the interconversion of mass and energy via E = mc2 is very important, describing the internal energy source of stars via nuclear fusion, which releases energy because of an overall conversion of the mass of the constituents to energy.

conservation of momentum In any isolated system, the total momentum P is constant:

P = pa, a constant vector.

In situations where constituents are separated “before” and “after” (in a collision, for instance), Newton’s second law guarantees that each individual p(a) will be constant except during the collision. Conservation of momentum then provides (three) relations between the “before” and “after” momenta.

conservative system In mechanics, a system in which there is a potential energy function

that depends only on position, and such that the sum of the kinetic energy and the potential energy, for any particular particle motion, remains constant. Simple examples are mass point motion in a ﬁxed Newtonian gravity ﬁeld, motion of charged particles in electrostatic ﬁelds (neglecting radiation reaction), and the exchange between kinetic and potential energy in the motion of a mass-on-spring, harmonic oscillator or in the motion of simple pendulum is a uniform gravitational ﬁeld. In all of these cases, the total energy E is unchanged during the motion, and E = kinetic energy plus potential energy, so computing one determines the other. In Lagrangian mechanics, a conservative system is one in which the Lagrangian L = T − V , the kinetic energy T is quadratic in velocities, the potential V is a function only of coordinates, and the Lagrangian has no explicit time dependence. Such a Lagrangian is related through a Legendre transformation to a Hamiltonian H , which is a function of the momenta and the velocities, is constant in time, and H = T + V . The idea may be extended more generally to a time independent Hamiltonian, even if H is not simply T + V .

constellation An apparent association of stars in the sky which are given symbolic or mythic signiﬁcance by associating ﬁgures to their pattern on the sky. According to current practice, the sky is divided into 88 such areas of association.

constitutive law of frictional sliding	See
fault constitutive law.

constraint equations In general relativity, an ADM formulation separates the spacetime into a 3 + 1 decomposition (3 space plus 1 time). Einstein’s equation can then be written with the 3-metric gij and the extrinsic curvature kij (both symmetric 3-dimensional 2-tensors) as the fundamental variables. Kij appears as a momentum conjugate to gij , and six of the ten 2-order Einstein equations become a hyperbolic set of equations giving the time derivative of gij and of Kij (twelve ﬁrst-order equations). However, there are in addition four equations (the Gσ 0 and the Gσ i components of the Einstein tensor), which are equated to the corresponding component of

continental margin

the matter stress-energy tensor, or to zero for an empty (vacuum) solution. Those four equations, the constraint equations, contain no time derivative of either gij or of Kij ; hence, they must be satisﬁed at the particular time in question. They can be written as one equation, which is a nonlinear generalization of Newton’s gravitational equation including the contribution of the gravitational ﬁeld as a source; and three components of a tensor equation which is a transversality condition on the momentum. It is easy to show that these four equations put real restrictions on the gij and the Kij that can be chosen for initial data (hence constraints). However, it can further be shown that the evolution via the hyperbolic equations preserves the solution of the constraint equations after their solution at the initial time. These equations are elliptic, and a standard procedure has been developed to solve this. See ADM form of the Einstein–Hilbert action.

contact binary A binary star in which the two components are touching. Main-sequence contact binaries are also called W Ursa Majoris (abbreviated as W UMa) and β Lyrae binaries, depending on the mass of the primary.

continent Regions of substantial area on the surface of the Earth that rise above sea level. The continents are made up of thick, light continental crust. The granitic continental crust has a typical thickness of 35 km, whereas the basaltic oceanic crust has a typical thickness of 6 km. The continents do not participate in the plate tectonic cycle but move and ﬂoat on the surface of the mantle so that the continents are much older than the oceanic crust, about 2 billion years vs. 100 million years.

continental climate The climatic type describing the climate character within a great land. Comparing to the marine climate, the main features of continental climate are strong annual temperature variations, which cause colder winters and hotter summers, and strong diurnal variations.

continental collision When subduction in an ocean basin dominates over sea-ﬂoor spreading, the ocean closes and the continents on the two

sides of the ocean collide. This is happening today in the Himalayas where the Indian subcontinent has collided with Eurasia.

continental drift Refers to the movement of continents with respect to each other. The remarkable similarity between the east coast of the Americas and the west coast of Europe and Africa has been recognized for several centuries and led to the hypothesis that these continents were once attached into a single supercontinent, which then broke apart and created the distribution of land masses we see today, called continental drift. The idea was popularized by German meteorologist Alfred Wegener in 1922, although many other people from 1620 onward contributed to the idea. The remarkable similarity extends beyond the similarity of continent boundaries to the distribution of geologic features, such as mountain ranges and glacial remains and fossil types, and the divergence in the evolution of ﬂora and fauna after a certain point in time. It was initially rejected by geophysicists who could not envision that the solid mantle of the earth could ﬂow.

All the continents were combined into a single supercontinent called Pangea about 300 million years ago. Pangea broke apart into a northern continent (Laurasia, composed of North America, Europe, Asia, and Greenland) and a southern continent (Gondwana, composed of South America, Africa, Australia, Antarctica, and India) about 250 million years ago, and those two continents subsequently broke apart into the continents we see today. The original idea of continental drift suggested the continents moved over the ocean ﬂoors. In the 1960s, studies of the magnetic polarity of ocean crust revealed that the seaﬂoor was also moving. Combining the ideas of continental drift and seaﬂoor spreading led to the development of the theory of plate tectonics to explain the geologic activity on the Earth’s surface. Direct ranging to geodetic satellites accurately measures the rate of continental drift, with speeds of up to 15 cm/yr.

continental margin The boundary between the continents, including continental shelves, and the ocean basins. There is no unique definition of the margin, it is usually taken as an arbitrary water depth, say 1 km. There are two

continental plate

types of continental margins, active and passive. An active margin is also a plate boundary, usually a subduction zone. An example is the west coast of South America. A passive margin is not an active plate boundary. An example is the east coast of the United States.

continental plate In geophysics, the large pieces of material comprising the outer surface of the lithosphere. Plates are about 50 km in thickness and support the continents, which ﬂoat on the denser viscous-ﬂuid mantle below them. Current knowledge deﬁnes 15 major plates: Paciﬁc, Philippine, Juan De Fuca, Cocos, Nazca, Antarctic, Scotia, South American, North American, Caribbean, Arabian, Indian, African, Eurasian, and Australian.

continental shelf Shallow oceanic margins underlain by continental crust. The water depth is usually taken to be less than 1 km.

continental shelf waves The waves that are generated by the sloping continental shelf topography. In the Northern Hemisphere, the continental shelf wave propagates along a constant topography line with the coast on the right.

continuum In mechanics, a description that ignores the granular or quantum nature of matter. In spectroscopy, the part of a spectrum without apparent lines, arising from solid, liquid, or gaseous sources, in which atomic lifetimes are too short to produce speciﬁc line emission or absorption.

contravariant vector

See vector.

convection Large-scale ﬂow of gases (or other ﬂuids) in stars (or elsewhere) that carries heat energy from one place to another. In the stellar context, hot currents rise, cool ones fall, and the solar granulation is direct evidence for the occurrence of convection near the surface of the sun. Convection sets in whenever a stellar temperature gradient is steeper than the adiabatic one:

−	dT	> −	1 −	1	T dp

	dr			γ	p dr

where γ is the ratio of speciﬁc heats. This can happen either when gas is quite opaque to radiation (as it is near the surfaces of cool stars, including the sun, where hydrogen is in the process of being ionized) or when a nuclear reaction depends on a very high power of temperature, as does the CNO cycle at the cores of massive stars. Stars are fully convective during early phases of their formation (guaranteeing initial chemical homogeneity) and throughout their lives for stars of less than 0.3 solar masses. Convection is the primary way that material inside stars is mixed from one zone to another. In relatively dense regions of stars, convection will carry all the available energy, and the temperature gradient will be very nearly adiabatic. At lower densities, convection becomes inefﬁcient, radiation carries much of the energy, and the temperature gradient can be much steeper.

In the absence of any adequate theory of convection (or other turbulent processes in ﬂuids), convection is often treated in the Mixing Length Approximation, in which gas is assumed to rise or fall through a ﬁxed fraction of the pressure scale height (half is typical) and then come into temperature equilibrium with its surroundings, depositing its extra heat or soaking up its deﬁcit. The approximation is more than 50 years old, and modern numerical computations of gas ﬂow processes are just beginning to replace it in standard computations of stellar structure and evolution. The absence of an adequate theory of convective energy transport is one of the major remaining uncertainties in our understanding of stellar physics. See CNO cycle, solar granula- tion.

convection zone A layer in a star in which convection currents provide the main energy transport mechanism. In the sun, a convection zone extends from just below the photosphere down to about 0.7 R .

convective adjustment One way to parameterize the physical process of convection in climate modeling. For example, in a simple version, one ﬁrst examines the relative humidity and lapse rate in each grid column at the end of each time step of integration; if the lapse rate is superadiabatic, the temperature proﬁle is adjusted to dry static neutrality in a manner that