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Prandtl number

grasses with few trees except along riverbanks. Prairies covered much of central United States and Canada prior to the development of farming and ﬁre control methods in these areas in the late 19th century.

Prandtl number A dimensionless number (Pr ) given by the ratio of the kinematic viscosity to the diffusivity. It expresses the ratio of the diffusivity of momentum to that of temperature through a ﬂuid.

Pr = ν

where ν is kinematic viscosity, χ is heat diffusivity coefﬁcient, χ = ρCKp , in which, K, ρ, and Cp are thermal conductivity, density, and speciﬁc heat at constant pressure, respectively. For air, Pr ≈ 1.4. Pr is strongly temperaturedependent. The turbulent Prandtl number is deﬁned analogously by P rt = νt /χt where νt is turbulent viscosity and χt is turbulent diffusivity. For very intense turbulence P rt → 1, whereas for very weak turbulence (laminar),

P rt → P r.

Prasad–Sommerﬁeld limit A limiting case of parameters describing (magnetic) monopoles so that they have mass equal to a minimum value called the Bogomol’nyi bound. For this set of parameters, for some theoretical models also including a scalar ﬁeld (called a Higgs ﬁeld) it becomes possible analytically to solve the ﬁeld equations, to ﬁnd that the force between two equal (magnetically repelling) monopoles exactly vanishes because this force gets compensated by the long-range Higgs ﬁeld attractive interaction. See Bogomol’nyi bound, cosmic topological defect, monopole, t’Hooft– Polyakov monopole.

Pratt compensation In Pratt compensation, the density ρP varies above a depth of compensation W in order to balance the mass of elevated topography.

Pratt isostasy An idealized mechanism of isostatic equilibrium proposed by J.H. Pratt in 1854, in which the crust consists of vertical rock columns of different densities with a common compensation depth independently ﬂoating on

a ﬂuid mantle. A column of lower density has a higher surface elevation. See Airy isostasy.

precession The slow, conical movement of the rotation axis of a rotating body. In the case of solar system objects (planets), precession is caused by the gravitational torques of other nearby objects. The Earth’s orbit around the sun deﬁnes a plane (the plane of the ecliptic) and the Earth’s axis of rotation is at an angle of 66.5◦ to this plane. The Earth’s “obliquity” is the angle between the rotation axis and a normal to this plane, i.e., 23.5◦. The Earth’s rotation creates a bulge around the equator, which itself deﬁnes a plane at 23.5◦ to the ecliptic. Both the sun and the moon exert a tidal couple on the Earth’s bulge, which results in the forced precession both of the Earth’s rotational axis and of the material orientation of the Earth itself. This is analogous to the precession of a rapidly spinning gyroscope pivoted at one end in a gravitational ﬁeld: The material orientation of the gyroscope’s symmetry axis rotates so as to form a cone, while the axis of rotation of the gyroscope (which lies very close to the symmetry axis) follows. Similarly, the Earth’s pole swivels around the normal to the ecliptic so as to describe cones. The period of this precession is 25,730 years. Superimposed on the simple rotation of the Earth’s pole are irregularities termed nutations, as well as the Chandler wobble and a longer term variation in the obliquity to the ecliptic. A notable effect of precession is the changing of the North Star (today the north celestial pole is near the star Polaris, but around 2000 BC it was near the star Thuban) as the Earth’s rotation axis points toward different locations in space.

precession of the equinoxes Slow shift in the celestial coordinates of astronomical objects because of the precession of the Earth’s pole direction. Celestial positions in published ephemerides are typically correct at the epoch 2000.0 (i.e., correct for noon UT, January 1, 2000), and correction tables and formulae correct the coordinates to any particular epoch. (Older tables give coordinates correct at the epoch 1950.) See precession.

primordial black holes

precipitation Any form of liquid or solid water particles that fall from the atmosphere and reach the Earth’s surface. Liquid precipitation includes drizzle and rain. Solid precipitation can include snow and agglomerated snow ﬂakes, frozen rain (sleet), or hailstones.

precursor In solar physics, a precursor, sometimes also called pre-ﬂare phase, can be observed prior to very large ﬂares as a weak brightening of the ﬂare site in soft X-rays and Hα, indicating a heating of the ﬂare site. The precursor can last for some minutes to even 10 minutes. See ﬂare electromagnetic radiation.

PREM (Preliminary Reference Earth Model)

A standard but preliminary model concerning interior of the Earth proposed by A.M. Dziewonski et al. (1981). On the basis that the Earth is anisotropic to a depth of 220 km, depth distributions for depths of seismic discontinuities, seismic velocity, density, physical properties, and pressure of the Earth’s interior are given, using many observation data concerning earthquakes such as Earth’s free oscillation and travel times.

pressure A scalar quantity indicating force per unit area.

pressure altitude The altitude calculated from the barometric height formula, under the assumption of standard atmospheric conditions and setting 1013.25 Pa as the pressure at zero altitude. This is the basis of the pressure altimeter.

pressure anisotropy	See plasma stress ten-
sor.
pressure coordinate	See isobaric coordi-
nate.
pressure head (p/ρg)	The pressure head

has units of length and is a component of the hydraulic head that may be thought of as the “ﬂow work” or the work due to pressure per unit ﬂuid weight, p/ρg, where p is the ﬂuid pressure, ρ is the ﬂuid density, and g is the acceleration of gravity. Pressure in hydrology is conventionally measured as gage pressure, which is deﬁned as zero at the piezometric surface (psurf ace = 0),

thus p ≥ 0 for groundwater (saturated) systems. In unsaturated ﬂow, the pressure head is negative and is equal in magnitude to the tension head ψ.

primary production The amount of organic matter produced from inorganic matter by photosynthesis, e.g., in [g C (Carbon) m−3] or, for a water column, in [g C (Carbon) m−2].

primary productivity The rate of production of organic matter from inorganic matter by photosynthesis, e.g., in [g C (Carbon) m−3 h−1]

or, for a water column, in [g C (Carbon) m−2 h−1].

primary rainbow The bright rainbow produced when light undergoes one internal reﬂection in water droplets. The colors range from blue on the inner part of the arc at about 40◦ from the center of the arc, to red on the outside of the arc at about 42◦ from the center. If a secondary rainbow is seen, it lies outside the primary rainbow.

prime focus The location where rays reﬂected from the primary mirror of a reﬂecting telescope meet, with no intervening secondary reﬂections. In very large telescopes, detectors are placed at prime focus.

primordial black holes Black holes produced at the early stage of the cosmological expansion of the universe. Study of the gravitational collapse of stars indicates that only sufﬁciently heavy stars can produce a black hole, and black holes of masses much less than the solar mass cannot form at the present stage of the evolution of the universe. However, the density of matter a short time after the Big Bang was enormous, and inhomogeneities in expansion could have produced small black holes. When the quantity L = cT , where c is the speed of light and T is the time elapsed since the Big Bang, became of the order of the size of a typical gravitational inhomogeneity, formation of primordial black holes became possible. Primordial black holes are a candidate constituent of cosmological dark matter, though it is not clear if they could be a major constituent.

principal null directions

principal null directions In pseudoRiemannian geometry and general relativity, one of the null (i.e., lightlike) eigenvectors obtained from the Weyl (conformal) tensor, or from an equivalent spinor representation. The behavior (and possible coincidence of one or more of these directions) is used in the Petrov classiﬁcation. See Petrov types.

principal spinors The spinors αA, βB , · · · , determined up to a complex multiplying factor, in the symmetrized decomposition

ψAB...P = α(AβB , . . . πP )

of a symmetric p-index SL(2, C) spinor ψ.

principle of equivalence See equivalence principle.

probable maximum hurricane A parameterized storm used for design purposes or to predict hurricane damage. See also standard project hurricane.

progressive vector diagram A diagram that shows the displacement a particle would have if it had the velocity observed at the ﬁxed position of the current meter at all times. For inﬁnitesimal motion, this coincides with the particle trajectory.

progressive wave A (water) wave that exhibits a net horizontal translation, such as waves in the open ocean, as opposed to a standing wave.

Prometheus Moon of Saturn, also designated SXVI. It was discovered by S. Collins and others in Voyager photos in 1980. Its orbit has an eccentricity of 0.003, an inclination of 0◦, and a semimajor axis of 1.39 × 105 km. Its size is 73 × 43 × 31 km, its mass is estimated at 2.7×1017 kg, and its density at 0.7 g cm−3. This low density indicates that Prometheus is probably a porous body. It has a geometric albedo of 0.6, and orbits Saturn once every 0.613 Earth days. It is the inner shepherd satellite of Saturn’s F ring.

prominence A structure in the solar corona consisting of cool dense plasma supported by

magnetic ﬁelds. Prominences are bright structures when seen over the solar limb, but appear dark when seen against the bright solar disk (see ﬁlament). Prominences are characterized as quiescent or active with the former being very stable, lasting for months, and the latter being mostly associated with solar ﬂares and having lifetimes of minutes or hours.

proper time The time measured by an observer with the clock that the observer carries with him/herself, i.e., time measured in a reference frame in which the clock or process is at rest. One of the effects predicted by special relativity is that if an observer moves with respect to another, with a velocity comparable to the velocity of light, then the clock of the moving observer goes more slowly than the observer at rest measures. The effect is relative: each of the observers sees that the other one’s clock slows down. However, if one of the observers moves noninertially (e.g., going to a distant star, reversing and returning), then on comparison at return, the moving clock shows less elapsed time. (See time dilatation.) The increment dτ of the proper time τ is related to the coordinate time and the velocity dxdti , i = 1, 2, 3 by

c2dτ2 = c2d2t − dx2 + dy2 + dz2 ,

where {dxα} = {dt, dx, dy, dz} are the change in the event coordinates during the interval dτ.

General relativity is locally equivalent to a ﬂat space (because of the equivalence principle). In general, for any coordinate frame, even in general relativity, one can compute: c2dτ2 = −gαβ dxαdxβ , where gαβ are components of the metric assumed to have signature (−+++). See summation convention.

proplyd A proto-star or young stellar object whose residual gas reservoir is being ionized away by ultraviolet radiation from nearby more massive stars.

Proteus Moon of Neptune, also designated NIII. It was discovered in 1989 on Voyager photos. Its orbit has an eccentricity of 0.00044, an inclination of 0.04◦, and a semimajor axis of 1.18 × 105 km. Its size is 218 × 208 × 201 km, but its mass is unknown. It has a geometric

PSF

albedo of 0.064, and orbits Neptune once every 1.122 Earth days.

proton A stable elementary particle (a baryon) of unit positive charge and spin h/¯ 2. Protons and neutrons, which are collectively called nucleons, are the constituents of the nucleus. See Planck constant.

proton ﬂare Any solar ﬂare producing signiﬁcant ﬂuxes of 10 MeV protons in the vicinity of the Earth.

proton-proton chain The set of nuclear reactions for hydrogen burning that begins with two protons interacting directly via the weak interaction to form a deuteron. This occurs at somewhat lower temperatures than the CNO cycle and so dominates energy production in stars of less than about 1.5 solar masses (including the sun). It must also be the mechanism of hydrogen fusion in stars of the ﬁrst generation, before any heavy elements are present. The main chain of reactions (see nuclear reactions for an explanation of the notation) is

1H p, e+νe 2H(p, γ )3He 3He, 2p 4He .

The chain can be completed in other ways that are much more strongly dependent on temperature. These other closures are also the main sources of high energy neutrinos (the ones to which the ﬁrst experiment was sensitive; see solar neutrinos). The additional reactions are

3He(α, γ )7Be e− , νe 7Li(p, α)4He 7Be(p, γ )8B , e+νe 8Be(, 2α)4He .

See CNO cycle, deuterium, hydrogen burning, nuclear reactions, solar neutrinos, weak interaction.

protostar A condensation of interstellar gas and dust that will become a star after accreting enough matter to start nuclear reactions at its center; the stage of stellar evolution immediately preceding the main sequence. The time scale is the Kelvin–Helmholtz one. Many protostars are surrounded by accretion disks, in which planets can or may form. They also typically display mass loss along their rotation axes, in

a bipolar pattern. The most massive protostars are completely shielded from optical study; less massive ones can be studied at infrared and optical wavelengths and evolve into T Tauri stars. The bipolar outﬂow often ionizes gas clouds at some distance from the proto-star. These are called Herbig–Haro objects.

protosun The early sun, forming from a molecular cloud, before or just after it began to generate internal energy by nuclear reactions.

protuberance Bright arc extending high above the photosphere on the solar limb, covering angular distances up to some 10◦. When viewed against the photosphere, the protuberance becomes visible as a dark strip and is called a ﬁlament. See ﬁlament.

Proxima Centauri The nearest star from the sun at 1.31 pc (4.2 light-years) and a member of the ternary α Centauri system. Proxima Cen is a cool M5.5 main sequence dwarf, about 20,000 times intrinsically fainter than our sun, of apparent magnitude 11.2, and undergoes irregular ﬂares when it brightens by as much as one magnitude but for only several minutes. Its mass is 0.1 solar with a radius 0.093 times that of our sun. Its orbital period within the α Centauri system is estimated to be about one million years.

pseudobreakup Rapid motion and brightening of aurora in the midnight sector, resembling the ones that characterize the breakup phase of substorms. Pseudobreakups often precede the breakup phase and differ primarily in failing to develop into a full substorm.

pseudovector A 3-dimensional quantity that behaves like a vector, except that its deﬁnition

is such that it does not change sign under an inversion {x , y , z} −→ {−x , −y , −z} of the

coordinate system. An example is the angular momentum L = r × p where both r and p do change sign under an inversion; hence L does not; L is a pseudovector. See vector cross product.

PSF See point spread function.