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Carter–Peter model

and its density 2.8 g cm−3. Its geometric albedo is not well determined, and it orbits Jupiter (retrograde) once every 692 Earth days.

Carnot cycle An ideal thermodynamic reversible cycle over a substance which consists of the following four processes: 1. isothermal expansion, in which the substance does work and there is inflow of heat Q2 at a constant temperature T2; 2. adiabatic expansion, in which the substance is thermally insulated and does work, there is no heat flow, and the temperature of the substance decreases; 3. isothermal compression, work is done on the system and there is outflow of heat Q1 at a constant temperature T1; 4. adiabatic compression, the substance is thermally insulated, and work is done on the substance, there is no heat flow, temperature of the substance increases, and at the end of this process the substance returns to its initial state. By applying the second law of thermodynamics to the Carnot cycle, one shows that the ratio of heat inflow and heat outflow Q2/Q1 for a substance undergoing a Carnot cycle depends only on the temperatures T2 and T1 and is independent of the substance. Applying the Carnot cycle to an ideal gas, one further shows that the ratio of heat outflow and heat inflow is equal to the ratio of the corresponding temperatures, i.e., Q1/Q2 = T1/T2. One can then represent an arbitrary reversible cyclic process by a series of Carnot cycles and show that rev dQ/T = 0 for any reversible process. This in turn leads to the definition (up to an arbitrary constant) of entropy S = rev dQ/T as a thermodynamic function of state. Named after Nicolas Léonard Sadi Carnot (1796–1832).

Carnot efficiency The ratio between the work done and the amount of heat introduced into a system going through a Carnot cycle. The Carnot efficiency is equal to the difference between the two temperatures of the isothermal steps of the cycle divided by the higher of the two temperatures.

Carnot engine An ideal heat engine whose working substance goes through the Carnot cycle. A heat engine receives heat at a given temperature, does work, and gives out heat at a lower temperature. The efficiency of a Carnot engine

or Carnot efficiency is the maximum efficiency possible for a heat engine working between two given temperatures.

Carrington longitude A fixed meridian on the sun as measured from a specified standard meridian. Measured from east to west (0◦ to 360◦) along the sun’s equator. Carrington longitude rotates with the sun and is a particularly useful coordinate when studying long-lived features on the sun. When combined with a Carrington rotation number, the Carrington longitude is commonly used as an alternative to specifying a time.

Carrington rotation The period of time covering 360◦ of Carrington longitude. Used to provide a temporal reference frame where the time unit is a solar rotation period. For example, Carrington rotation 1917 corresponds to the time period 9 December 1996 to 5 January 1997, while Carrington rotation 1642 relates to the rotation between 28 May 1976 and 23 June 1976.

Carter–Peter model The dynamics of a superconducting cosmic string is macroscopically describable by means of the duality formalism. This formalism only requires the knowledge of a Lagrangian function L depending on a state parameter w, the latter being interpretable as the squared gradient of a phase φ, namely

w = κ0γ ab∂aφ∂bφ ,

with κ0 a normalization coefficient, γ ab the inverse of the induced metric on the string surface, and the superscripts a, b representing coordinates on the worldsheet. In the Carter–Peter (1995) model for describing a conducting cosmic string of the Witten kind, the Lagrangian function involves two separate mass scales, m and m0 say, respectively describing the energy scale of cosmic string formation and that of current condensation; it takes the form

L = −m

2

−

m02

ln 1 +

w

.

 

2

m02

This model implies the existence of a first order pole in the current, as is the case in realistic conducting string models taking into account the microscopic field structure. The figure shows

© 2001 by CRC Press LLC

Cartesian coordinates

5.4

 

 

 

5.2

 

 

 

5.0

 

 

 

4.8

0.000

0.010

0.020

0.010

Energy per unit length U (upper curves) and tension

T (lower curves) as functions of the sign-preserving square root of the state parameter: the full lines represent the actual values derived from the Witten microscopic model, the dashed line being the values obtained with the Carter–Peter macroscopic model. The fit is almost perfect up to the point where U and T both increase for positive values of the state parameter. This is satisfying since the macroscopic model there ceases to be valid because of instabilities.

the shape of the energy per unit length and ten-

√

sion as function of the state parameter w together with a comparison with the same functions in the Witten conducting string model. It is clear from the figure that the fit is valid in most of the parameter space except where the string itself is unstable. The string equation of state is different according to the timelike or spacelike character of the current. For the former it is

U = T + m02 exp 2

m2 − T /m02 − 1 ,

while for positive w we have

T = U −m02 1 − exp −2

U − m2 /m02 .

See conducting string, cosmic string, current carrier (cosmic string), current generation (cosmic string), current instability (cosmic string), phase frequency threshold, summation convention, Witten conducting string, worldsheet geometry.

Cartesian coordinates A coordinate system in any number of spatial dimensions, where the coordinates {xi} define orthogonal coordinate lines. In such a system, Pythagoris’ theorem holds in its simplest form:

ds2 = δij dxidxj ,

where the summation convention is assumed for i and j over their range.

Or, in nonflat spaces or spacetimes, a system in which the coordinates have many of the properties of rectangular coordinates, but Pythagoris’ theorem must be written as:

ds2 = gij (xk)dxidxj ,

where gij (xk) is the coordinate dependent metric tensor. In this case the description is reserved for coordinates that all have an infinite range, and/or where the metric coefficients gij (xk) are “near” δij everywhere.

Cartesian coordinates [in a plane] A relationship between the points of the plane and pairs of ordered numbers called coordinates. The pair of numbers corresponding to each point of the plane is determined by the projection of the point on each of two straight lines or axes which are perpendicular to each other. Cartesian coordinates thus establish a nonsingular relationship between pairs of numbers and points in a plane. The point in which the two axes intersect is called the origin. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. Named after Rene Descartes (1596– 1650).

Cartesian coordinates [in space] A relationship that is established between the points of space and trios of ordered numbers called coordinates. Cartesian coordinates are a coordinate system in which the trio of numbers corresponding to each point of space is determined by the projection of the point on each of three straight lines or axes which are perpendicular to each other and intersect in a single point. The positive direction of the z-axis is generally set, such that the vectorial product of a non-null vector along the positive x-axis times a non-null vector along the positive y-axis generates a vector along the positive z-axis; this is called a right-handed coordinate system. Named after Rene Descartes (1596–1650).

Casagrande size classification A classification of sediment by particle size (diameter). The basis for the Unified Soils Classification commonly used by engineers.

© 2001 by CRC Press LLC

cataclysmic variable (cataclysmic binary)

case 1 water Water whose optical properties are determined primarily by phytoplankton and co-varying colored dissolved organic matter and detritus; not a synonym for open ocean waters.

case 2 water Water whose optical properties are significantly influenced by colored dissolved organic matter, detritus, mineral particles, bubbles, or other substances whose concentrations do not co-vary with the phytoplankton concentration; not a synonym for coastal waters.

Cassegrainian A type of reflecting telescope invented by G. Cassegrain, with a small convex secondary mirror mounted in front of the primary mirror, to reflect rays approaching a focus back through a hole in the primary mirror, where they are viewed using a magnifying lens (eyepiece) from behind the telescope.

Cassini A spacecraft destined for Saturn that was launched on October 15, 1997, and is expected to arrive in July 2004. At this time, it will orbit Saturn for four years. It is a joint mission of NASA, the European Space Agency (ESA), and the Italian Space Agency. The spacecraft consists of an orbiter and ESA’s Huygens Titan probe. The latter will be dropped through the atmosphere to the surface of Saturn’s largest moon, Titan.

In total the spacecraft weighs 5650 kg. In order to get to Saturn in the nominal 6 years and 9 months, it was initially launched inward, not outward, and aimed toward Venus rather than Saturn, to provide a “gravity-assisted” trajectory. This consists of two Venus flybys, a flyby of Earth, and a flyby of Jupiter.

The mission is named in honor of the seventeenth-century, French-Italian astronomer Jean Dominique Cassini, who discovered the prominent gap in Saturn’s main rings, as well as four icy moons. The Titan probe is named in honor of the Dutch scientist Christiaan Huygens, who discovered Titan in 1655, and realized that the strange Saturn “moons” seen by Galileo in 1610 were a ring system surrounding the planet.

Cassini’s division A gap, detectable by small telescope observation, between the A and B rings of Saturn, discovered by Cassini.

Cassiopeia A A discrete strong radio source at RA 23h21m10s, dec +58o32 05 emitting 21cm radiation. Cas A was apparently created in a supernova explosion in 1667. The constellation Cassiopeia is located almost directly opposite the Big Dipper across the north celestial pole. Five bright stars in the constellation form a rough W (or M) in the sky. (Tycho’s supernova also appeared in the constellation in 1572 and disappeared in 1574.)

Castor Double star (alpha Geminorum A, B). A is an A1 type 1.94 magnitude star located at RA 07h34.4m, dec +31◦54 ; B is a type A2, 2.92 magnitude star located at RA 07h34.5m, dec +31◦54 .

Castor, John I. Astrophysicist. In 1976, in collaboration with David Abbott and Richard Klein, developed the theory of winds in early type stars (CAK theory). Through hydrodynamic models and atomic data, they showed that the total line-radiation pressure is the probable mechanism that drives the wind in these systems, being able to account for the observed wind speeds, wind mass-loss rates, and general form of the ultraviolet P-Cygni line profiles through which the wind was originally detected.

cataclysmic variable (cataclysmic binary)

A star that suddenly and unpredictably brightens by several magnitudes, most likely by a transfer of stellar material from one star to its close companion. A few weeks or so after the eruptive event, the star returns to its original brightness, indicating that a permanent transformation or evolution of the system has not taken place. Stars exhibit some or all of the following characteristics: flat or blue optical spectral distribution, broad emission or absorption lines of hydrogen and helium, rapid variability, marked aperiodical changes in optical brightness, and low X-ray luminosity. Binary star in which one component is a white dwarf and one is a main sequence star or red giant, where the stars are close enough together that material flows onto the white dwarf from its companion, either through Roche Lobe overflow or in a stellar wind. Pairs without such transfer are called V471 Tauri stars. Mass transfer at a rate of 10−7−8 solar masses per year is relatively sta-

© 2001 by CRC Press LLC

cataclysmic variable [binary models of]

ble, leading to systems whose brightness varies rather little. These are nova-like variables and symbiotic stars. Another kind of cataclysmic variable star is the flare star, whose prototype is UV Ceti. Flare stars are intrinsically cool red stars (of type M or less commonly K) on the main sequence that will unpredictably brighten by up to two magnitudes over the course of a few seconds and then fade back to normal in 20 or 30 minutes. Most of these stars are known to have close companions. Mass transfer at 10−9 solar masses per year is unstable, with an accretion disk building up for a while and then dumping material onto the white dwarf rapidly. These events are the outbursts of the dwarf novae. In most cases, hydrogen gas builds up on the surface of the white dwarf until it is somewhat degenerate, at which point it burns explosively, producing a nova. If this happens frequently enough to have been seen twice or more in historic times, the system is called a recurrent nova. It is sometimes possible for similar accretion (perhaps of helium or heavier elements) to trigger degenerate ignition of the carbon-oxygen core of the white dwarf. This leads to a supernova explosion of Type Ia.

cataclysmic variable [binary models of]

Thought to be binary systems composed of a white dwarf (primary), a main sequence star (secondary), and characterized by accreting mass flowing from the secondary towards the primary. Typically the accreting mass forms a disk or accretion disk around the primary (with the exception of magnetic variables).

cataclysmic variable [galactic distribution]

Due to interstellar absorption, cataclysmic variables (which have luminosities on the order of solar luminosity) cannot be observed at a distance greater than 1 kpc. In spite of this, 380 CVs had been detected by 1976. Thus, it seems that CVs are relatively common astrophysical objects.

cataclysmic variable [outbursts] Some CVs present “outbursts” or periodical increases in luminosity with respect to their usual quiescent state. Classical novae are cataclysmic variables that have been known to present only one outburst; recurrent novae are cataclysmic vari-

ables that present outbursts periodically every 10 years or more; dwarf novae are cataclysmic variables that present outbursts every few weeks or months.

cataclysmic variables [phenomenological classification of] Cataclysmic variables are classified into five groups. Classical Novae: cataclysmic variables that have been known to present only one outburst. Recurrent Novae: cataclysmic variables that present outbursts every 10 years or more. Dwarf Novae: cataclysmic variables that present outbursts every few weeks or months. Nova-like: cataclysmic variables that have not been known to present outbursts but have the same spectroscopic characteristics as other cataclysmic variables when they are quiescent. Magnetic Variables: cataclysmic variables that present relatively strong magnetic fields.

cataclysmic variables [physical parameters]

The luminosities of cataclysmic variables vary between 0.001 and 10 solar luminosities. Binary orbital periods are typically between 0.7 hours and 1 day. Mass accretion rates vary from 10−10 to 10−7 solar masses per year. The primary star (white dwarf) typically has a mass of 0.6 solar masses, varying between 0.4 and 0.9 solar masses, and a radius of 0.01 solar radii. The secondary is a main sequence star with a mass lower than its companion star since the primary has evolved through the main sequence faster.

catastrophic formation of solar system A theory attributing the formation of the solar system to a collision of another massive object (presumably another star) with the sun, which threw material out of the sun, or to a close encounter with another star — a tidal encounter. 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.

Cauchy singularity Any region of spacetime where violations of causality can occur because the deterministic evolution of physical systems from initial data is not preserved. See naked singularity, white hole.

© 2001 by CRC Press LLC

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