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perfect gas

four-velocity ua is normalized using the metric, uaubgab = −1. See tensor.

perfect gas A gas which at the molecular level can be taken to have pointlike noninteracting molecules; and which is thus described by the perfect gas law: pV = nkT , where p is the pressure, V is the volume, T is the (absolute) temperature, k is Boltzmann’s constant, and n is the number of gram-moles of gas involved. Real gases always have some intermolecular interaction, but perfect gas behavior can be approached by dilution. Among other properties, an ideal gas has internal energy which is a function of temperature alone, and constant values for its speciﬁc heats (e.g., CV at constant volume and Cp at constant pressure).

periapsis The point in an elliptical orbit where the orbiting body is the closest distance to the body being orbited. (The apoapsis is the point of the farthest distance.) When the sun is the central body, the point of periapsis is called the perihelion.

periastron In planetary motion, the closest distance achieved to the gravitating central star. Generically one says periapse. Speciﬁc applications are perihelion when referring to the motion of planets in our solar system; perigee, when referring to orbits around the Earth. Similar constructions are sometimes invented for orbits about the moon or about other planets.

perihelion The point in an elliptical orbit around the sun that is closest to the sun. (The aphelion is the point farthest from the sun.) The time of perihelion passage for the Earth is typically around January 3.

perihelion shift The parameter of an orbit of an astronomical object (usually a planet) that tells, by what angle the planet’s perihelion is displaced during a unit of time (the preferred unit is different for different situations, for Mercury it is seconds of arc per century, for the binary pulsar PSR 1913+16 it is degrees per year). If the central body (e.g., the sun) were perfectly spherical and if it had only one companion (planet) on an orbit around it, then Newtonian

mechanics and gravitation theory predict that the orbit would be an ellipse, with the central body placed in one of the foci of the ellipse. In reality, the central body is not exactly spherical, and there are several planets and moons in the planetary system that perturb each other’s orbit. Because of those perturbations, the real orbits are not ellipses even according to Newtonian mechanics. The orbit is then a curve that is close to an ellipse during one revolution of the planet around the star, but the axes of the ellipse rotate during each revolution by a small angle in the same sense as the planet’s motion. The angle by which the perihelion is rotated in a unit of time is the perihelion shift. The nonsphericity of the sun gives a negligible effect, but the perturbations from other planets sum up to a perihelion shift of 530 sec of arc per century in the case of Mercury. On top of that comes the additional perihelion shift predicted by Einstein’s general relativity description of gravity. According to this description, even with a perfectly spherical central star and a single planet orbiting it, the orbit would not be an ellipse, but a rotating ellipse, like a perturbed Newtonian orbit. For Mercury, relativity predicts this additional perihelion shift to be 42.95 sec of arc per century. This additional component in Mercury’s orbital motion was detected in 1859 by LeVerrier, and called an “anomalous perihelion motion of Mercury”. It was a major problem for 19th-century astronomy that several astronomers tried (unsuccessfully) to solve by methods of Newtonian mechanics, e.g., by postulating that the additional shift is caused by interplanetary dust, the solar wind, the nonsphericity of the sun, or even by a thus far unobserved additional planet. Explanation of this anomaly was the ﬁrst, and quite unexpected, triumph of Einstein’s general relativity theory. For other planets, the relativistic perihelion motion is small and became measurable only with modern technology; it is 8.62 for Venus, 3.84 for the Earth, and 1.35 for Mars. (All numbers are in seconds of arc per century. Values observed differ from these by 0.01 to 0.03.) The fastest perihelion motion observed thus far is the periapse shift of the binary pulsar PSR1913+16; it amounts to 4.2◦ per year. Its shift has been observed via the phase of pulses from this binary pulsar.

perturbative solution

period The amount of time for some motion to return to its original state and to repeat its motion.

period-luminosity relation The relation between the luminosity of Cepheid variable stars and the variability period of these stars. The basic trend is that those stars with longer periods are brighter. By measuring the period of Cepheid variables, astronomers can use this relationship to deduce the intrinsic luminosity of these stars. Combined with a measurement of the apparent luminosity, the distance of Cepheid variables can be estimated. The Hubble Space Telescope has been used to detect Cepheid variables out to the Virgo cluster, measuring the distance of the Virgo cluster to roughly a 10% accuracy. See Cepheid variable.

permafrost Soil or subsoil that is permanently frozen for two or more years, typical of arctic regions, or in other arctic climates (such as high altitude) where the temperature remains below 0◦C for two or more years.

permeability In electromagnetism, the relation between the microscopic magnetic ﬁeld B and the macroscopic ﬁeld (counting material polarizability) H. In ﬂuid ﬂow, permeability, also called the intrinsic permeability, characterizes the ease with which ﬂuids ﬂow through a porous medium. Theoretically, permeability is the intrinsic property of a porous medium, independent of the ﬂuids involved. In reality, the permeabilities of some rocks or soils are affected by the ﬂuid. Permeability in ﬂuid ﬂow has the dimension of area. For an anisotropic porous medium, the permeability is a tensor. See Darcy’s law.

permeability coefﬁcient In electromagnetism, the coefﬁcient µ in the relation B = µH between the microscopic magnetic ﬁeld B and the macroscopic ﬁeld H. In general, µ may be a 3 × 3 linear function of position, with time hysteresis, though it is often taken to be a scalar constant. In ﬂuid ﬂow, the coefﬁcient measuring how easily water molecules can cross the surface ﬁlm. For clear water (Davies and Rideal, 1963), it is equal to 5 ms−1.

persistent current	See current generation
(cosmic string).
perturbation theory	A tool that can be ap-

plied to questions ranging from classical mechanics to quantum theory. Independent of the actual question, the basic idea is to ﬁnd an approximate solution of the equations for a complex system by ﬁrst solving the equations of a physically similar system chosen so that its solution is relatively easy. Then the effects of small changes or perturbations on this solution are studied. In classical mechanics, for instance, the motion of a planet around the sun is studied ﬁrst and the inﬂuences of the other planets are added later.

Formally, a complex system is described by a set of coupled non-linear partial differential equations. If only small-amplitude disturbances are considered, such a system can be simpliﬁed by linearization of the equations: whenever two oscillating quantities are multiplied, since both are small, their product is a higher order term and can be ignored. If these results are to be applied to a real situation, however, one always has to take one step back and justify whether the amplitudes calculated in the real situation are small enough so that the non-linear terms actually are negligible compared with the linear

ones.
perturbative solution	An approximate so-

lution, usually found to a problem that is too difﬁcult to be solved by an exact calculation. In a perturbative solution there exists at least one small dimensionless parameter (ε) that must be distinctly smaller than 1. Then, ε2 ε. The equations to be solved are expanded in power series with respect to ε and in the ﬁrst step all terms proportional to εn where n > 1 are assumed to be equal to zero. In the next step, the already found ﬁrst approximation is substituted for the terms proportional to ε, the terms proportional to εn where n > 2 are assumed equal to zero both in the equation and in the solution, and the coefﬁcient of ε2 is determined. The procedure can, in principle, go to an arbitrarily high degree of approximation, but in practice the calculations often become prohibitively complicated in the second step (this happens, e.g., with Einstein’s equations in almost every

Peru current

case). Perturbative solutions are ubiquitous in astrophysics. Planetary orbits and the positions of planets on the orbits are calculated by such perturbative methods where the small parameter is the eccentricity of the orbit. In cosmology, the most commonly used description of the formation of galaxies is by solving Einstein’s equations perturbatively, with the small parameter being the size of the matter deviation from the spatially homogeneous background. Perturbative solutions are often possible where an exact solution seems prohibitively difﬁcult, but they have their own difﬁculties. In applications in general relativity care must be taken to distinguish between genuine perturbations and spurious perturbations.

Peru current A cold ocean current running up the west coast of the South American continent, feeding into the South Equatorial current.

Petchek reconnection Fast reconnection at an X-point or in a small localized region. About three-ﬁfths of the inﬂowing magnetic energy is converted into kinetic energy behind the shock waves, the remaining two-ﬁfths heat the plasma. With vA,in being the Alfvén speed in the inﬂowing plasma, L being the length scale of the reconnection region, σ being the conductivity, and c being the speed of light, the reconnection rate Rp can be written as

Rp =	8	ln		LσvA,in4π	.
	π			c

Petchek reconnection varies only weakly with the conductivity σ and is very efﬁcient in mixing ﬁelds and plasmas. It probably does not play a role in magnetospheric plasmas but might be important in solar ﬂares.

Petchek reconnection is stationary reconnection: the onset of reconnection does not destroy the general ﬁeld conﬁguration. Stationary reconnection results in an equilibrium between inﬂowing mass and magnetic ﬂux, magnetic diffusion, and outﬂowing mass and magnetic ﬂux.

Petrov types Types in a classiﬁcation of gravitational ﬁelds according to the algebraic structure of the Weyl tensor (a tensor algebraically constructed from the Riemann tensor to be completely traceless), introduced by Petrov (1966).

In general relativity the traces of the Riemann tensor are pointwise determined by the stressenergy tensor. The trace-free part, the Weyl tensor Cabcd is symmetric in the skew pairs ab and cd of indices. There are six independent components of each skew pair; hence the Weyl tensor can be viewed as a symmetric 6 × 6 matrix. The eigenvalues of this matrix sum to zero. An alternate (calculationally simpler) development can be given in terms of spinors. The eigenvectors determine null vectors; for the general type I there are the four different principal spinors αA , βA , γA and δA which determine different principal null directions. For types II, III, and N, respectively, two, three, and all four of the principal directions coincide. Type D arises when there are two pairs of coincident principal null directions. When the Weyl curvature vanishes, the type is O. See principal spinor, Weyl tensor.

Petzold data A widely used data set containing volume scattering functions for various waters ranging from very clear to very turbid; the scattering phase function is highly peaked at small scattering angles and is somewhat independent of water type.

Pfund series The set of spectral lines in the extreme infrared region of the hydrogen spectrum with frequency obeying

ν = cR 1/n2f − 1/n2i ,

where c is the speed of light, R is the Rydberg constant, and nf and ni are the ﬁnal and initial quantum numbers of the electron orbits, with nf = 5 deﬁning the frequencies of the spectral lines in the Pfund series. This frequency is associated with the energy differences of states in the hydrogen atom with different quantum numbers via ν = =E/h, where h is Planck’s constant, and where the energy levels of the hydrogen atom are:

En = hcR/n2 .

phase In planetary astronomy; variance in the total amount of the visible disk of the moon, or of the inner planets, which appears illuminated at any one time. Phases arise because our vantage point sometimes allows the unlit side of

phase space

the object to be viewed, as with the new or old moon, close to the sun in the sky. Phases can also be observed for Venus and Mercury, though they are behind the sun when at full phase as viewed from the Earth. For the moon, the phases are new (rising very close to the sun after dawn in the sky), waxing crescent (rising after dawn, becoming noticeable just at sunset), ﬁrst quarter (rising at noon), waxing gibbous, full (rising at sunset; near zenith at midnight), waning gibbous, last quarter (rising at midnight), waning crescent.

In physical chemistry, a phase is a homogeneous state of matter that is (conceivably) physically separable from any other phase in a system. A phase is also used loosely to describe a mineral species, that has different solid forms of different structure and density, in a metamorphic mineral assemblage.

In signal physics, in periodic or nearly periodic systems, the point in the waveform measured in degrees or radians from a ﬁducial event such as a positive-going zero crossing. The phase of 360◦ = 2π is assigned to the interval between two such ﬁducial events (one period).

phase angle The angle from the sun to an object to an observer. An object’s phase angle is equal to the angular separation (elongation) between the sun and the observer as viewed from the object.

phase frequency threshold In cosmology, the current trapped in cosmic strings can be timelike as well as spacelike. In the former case, the energy it contains is derivable essentially from the energy of a bound current carrying particle, which, in order for a bound state to exist, must have energy less than the free energy equivalent of the mass of the particle.

One therefore cannot increase the energy indeﬁnitely. In fact, as it increases towards mc2, the state it describes becomes closer and closer to that of a free particle, namely, it starts spreading. Also, as the energy tends to mc2 the integrated current increases indeﬁnitely and it is seen to diverge. This limit is referred to as the phase frequency threshold. It simply reﬂects the fact that no bound state can be obtained with energy greater than the rest mass energy equivalent of the free particle. Spacelike currents are not

subject to this constraint, since they can be set arbitrarily far from a free particle state; however, spacelike currents are limited also by current saturation.

At this phase frequency threshold, the energy per unit length diverges while the string tension decreases also without limit. The interesting consequence is that at some stage the tension vanishes (before eventually becoming inﬁnitely negative) so that springs could be formed and the corresponding strings become violently unstable. See Carter–Peter model, cosmic spring, current saturation, duality in elastic string models, Witten conducting string.

phase function In optics, the change in the brightness of an object as a function of the phase angle. In general, an object gets brighter as the phase angle approaches 180◦ or 0◦. The function is usually fairly smooth except for at small phase angles where there may be a “spike” of increased brightness. The phase function is usually described as the change in magnitude (brightness) per degree of phase angle. In descriptions of wave propagation in scattering media, the ratio of the volume scattering function to the scattering coefﬁcient [sr−1]; the integral of the phase function over all directions is unity.

phase space For Hamiltonian systems in classical mechanics, the even-dimensional space coordinatized by the conﬁguration coordinates qi and their conjugate momenta pi. Here i = 1 · · · N; N is the physical dimension of the system under study, so the dimension of phase space is 2N. The Hamilton equations describe evolution in phase space that proceeds in a way that preserves an antisymmetric differential form (covariant antisymmetric 2-tensor)

N

	dpi dqi
A ≡	dpi dqi
i=1

where the “ ” indicates an antisymmetric tensor product. Thus, in these canonical coordinates