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phase speed

the components of the tensor A are

		−1	0
		0	1
Aij	=			1	0
	=			0	1
			−			0	1

						1	0

					−

where the order of the indices is taken as:

q1 , p1 , q2 , p2 · · · · · .

Canonical transformations can be viewed as passive motions of the coordinates past a system point in phase space, or as active motion (dragging) of points in phase space. The ﬂow under Hamilton’s equations can be viewed as such an active dragging. This invariance of A implies the conservation of elementary volume in phase space, Liouville’s theorem, since the volume element is proportional to

A A · · · A

	N
		terms

and each of the terms is separately conserved. Thus, one says the motion in phase is incompressible. Depending on the speciﬁc dynamical system (particularly if the system undergoes chaos) this elementary volume can become distorted into an arbitrarily long, twisted ﬁlament, and “mixed” throughout phase space during the evolutions.

A point in phase space represents the instantaneous state of the system. Hamilton equations are ﬁrst order equations for the time derivative of this state:


q˙i	=		∂H
			∂pi

p˙i	=	−		∂H
					.
				∂qi

Since the Hamiltonian is a function H = H(pi , qi , t), an immediate consequence is that at a particular time, the direction of motion through a particular point in phase space is single valued. Orbits in phase space cannot cross.

phase speed The speed deduced from tracking individual wave crests rather than the group proﬁle in a wave:

vph = ω/k ,

where ω is the angular frequency [sec−1] and k is the wavenumber [m−1] of the wave. Also referred to as celerity.

phase transition Chemical elements or compounds are often capable of being ordered in different crystal structures, or “phases”. The stable phase for ﬁxed temperature and pressure is then the phase which minimizes the Gibbs free energy. In geophysics, because the Earth has gradients in both temperature and pressure, the same rock material may be found to contain different types of crystal at different depths. Lines dividing the areas of stability of different phases may be drawn on a temperature-pressure graph, in which case the slope of a line dividing two phases is given by the Clasius–Clapeyron equation:

dTt = =V dP =S

where Tt is the temperature of the phase transition at some pressure P , =V is the difference in speciﬁc volume between the phases, and =S is the difference in speciﬁc entropy. =S may be either positive or negative for Earth materials, so the slope of the line may be either positive or negative. In the Earth, a positive slope would imply that a phase transition would be found higher where it is locally cool, and lower where hot, which would aid convection (as the phase at higher pressure is more dense). Similarly, a negative Clapeyron slope would impede convection. The 400 km deep transition in the mantle is often identiﬁed with the olivine/spinel transition, and would be an example of the former case, while the 670 km deep transition is identiﬁed with the spinel/perovskite transition, and would be an example of the latter case. The inner core/outer core boundary is an example of a transition between a liquid and a solid phase.

In cosmology, in the early universe, there may have been several phase transitions as the elementary particles ﬁlling the universe cooled and settled into particular low temperature conﬁgurations. Remnants of these transitions may still exist in exotic forms such as cosmic strings or magnetic monopoles.

phase velocity

See phase speed.

photon

pheophytin A phytoplankton pigment that is inert and does not contribute to photosynthesis.

phi unit A measure of sediment size commonly used in geology. Sediment diameter and φ-size are related by D = 2−φ, so that a larger particle has a smaller φ value.

Phobos One of the two moons of Mars, also designated MI. Phobos (meaning fear) was discovered by A. Hall in 1877. Its orbit has an eccentricity of 0.015, an inclination of 1.0◦, a precession of 158.8◦ yr−1, and a semimajor axis of 9378 km. It is one of the three satellites in our solar system whose period (7h 39m) is less than the rotational period of the primary planet (24h 37m for Mars); rises in the west and sets in the east, often twice a day. It is losing orbital energy to the surface tides it raises on Mars; its orbit is decaying at a rate of 1.8 m/century and in about 4 × 107 yr it will either be disrupted by tidal forces (in this way, it may become a ring plane about Mars within the next 50 million years) or it will crash into the surface of

Mars. Its size is 13.5×10.8×9.4 km, its mass is 1.08×1016 kg, and its density is 1.9 g cm−3. Its

geometric albedo is 0.06 and its surface is similar in reﬂectivity to C-type asteroids. It may be a member of that group that was captured in the past. New temperature data and closeup images of Phobos gathered by NASA’s Mars Global Surveyor indicate the surface has been pounded into powder, at least 1 m (3 ft) thick, by eons of meteoroid impacts. Meteorite impacts also initiated landslides along the slopes of some craters. High temperatures were measured at −4◦C (25◦F) and lows at −112◦C (−170◦F). (Phobos does not have an atmosphere to hold heat in during the night.)

Phoebe Moon of Saturn, also designated SIX. It was discovered by Pickering in 1898. Its orbit has an eccentricity of 0.163, an inclination of 177◦, and a semimajor axis of 1.30 ×107 km. Its radius is 110 km, its mass is 4.0 × 1018 kg, and its density is 0.72 g cm−3. Its geometric albedo is 0.05, much lower than the other Saturnian satellites (except the dark part of Iapetus). It orbits Saturn once every 550.5 Earth days. Phoebe’s retrograde orbit indicates that it is probably a captured satellite.

photodissociation region A part of the interstellar medium where the molecules have been destroyed into atoms by ultraviolet light, and the chemical composition and temperature are determined by the intensity of the light. See interstellar medium.

photoelectrons Electrons emitted by the action of light, in particular (in magnetospheric research) of short-wave sunlight impinging on either the upper atmosphere or on the surface of a satellite. Photoelectron emission from the upper atmosphere is the major source of the ionosphere. Photoelectron emission from satellite surfaces causes spacecraft charging and can affect the operation of sensitive instruments in space. See also differential charging.

photoinhibition The decrease in photosynthetic rate with increasing irradiance, caused by surpassing the photosynthetic capacity.

photoionization Ionization of atoms and ions by photons. A photon with energy greater than a threshold energy, the ionization potential, is absorbed, and an electron previously bound to an atom is freed. The energy gained by the electron is equal to the difference between the photon energy hν0 and the ionization potential E0, according to the formula hν = hν0 − E0. Spectral lines are emitted following photoionization when the electron recombines. In astronomy, such photoionized nebulae include HI regions, planetary nebulae, extended envelopes surrounding hot stars, and the line-emitting regions of active galactic nuclei.

photometric binary An orbiting pair of stars, neither of which eclipses the other, but whose gravitational ﬁelds distort each others’ shapes enough so that the surface area we see (hence the brightness of the system) changes through the orbit period. Also ellipsoidal variable or ellipsoidal binary.

photometry The accurate measurement of the ﬂux of light from a source.

photon An elementary particle of zero mass and spin one. The quantum of electromagnetic radiation; the smallest “bundle of energy”

photosphere

of light of a particular frequency, with energy E = hν = hc/λ where h is Planck’s constant, c is the speed of light, ν is the frequency, and λ is the wavelength of the light.

photosphere The layer of the sun or another star that we see in visible light. The photosphere is a quite thin layer and at a temperature between about 2,000 and 150,000 K (5,600 K for the sun). A stellar photosphere produces a continuous spectrum crossed by absorption lines due to the atoms and molecules of the elements and compounds in it.

photosynthesis The manufacture of carbohydrates from carbon dioxide and water in the presence of chlorophyll, by utilizing radiant energy and releasing oxygen; the chemical change induced in chlorophyll by the absorption of a quantum of radiant energy.

photosynthetically available radiation The integral over visible wavelengths (350 to 700 nm or sometimes 400 to 700 nm) of the number of photons available for photosynthesis [photons s−1 m−2]; computed by integrating the spectrally dependent scalar irradiance divided by the photon energy at each wavelength.

photosynthetic capacity	The maximum
photosynthetic rate per unit of biomass.
photosynthetic pigment	Molecules whose

structures efﬁciently absorb light within the 400 to 700 nm range.

phytoplankton Plant forms of plankton, generally with sizes from less than 1 to several hundred µm.

piezometer The elevation of the water table or potentiometric surface as measured in a nonpumping well generally constructed from a small-diameter pipe with screened openings through which water can enter.

piezometric head See hydraulic head.

Pileus cloud A smooth cloud forming at the peak of a mountain, or at the top of a thundercloud.

pillow basalt The form of an eruption that has occurred under water, leading to characteristic rock formations.

pitch angle The angle between the velocity vector of a charged particle relative to the guiding magnetic ﬁeld lines along which the particle is moving. It is given by tanθ = vperp/vpar where subscripts perp and par refer to perpendicular and parallel to the direction of the magnetic ﬁeld.

pitch angle diffusion	See diffusion, in pitch
angle.

Pitot tube (After Henri Pitot, 1695–1771) An open-ended tube used to measure the stagnation pressure of the ﬂuid for subsonic ﬂow; or the stagnation pressure behind the tube’s normal shock wave for supersonic ﬂow. In application, it is immersed in a moving ﬂuid with its mouth pointed upstream. When combined with a “static” measurement, the pressure difference can be used to determine the speed of the tube through the ﬂuid. The Pitot tube is in wide use to determine airspeed in aircraft, where the static pressure is obtained through an opening in the instrument case within the aircraft.

plage An extended chromospheric emission feature of an active region overlying photospheric faculae.

plagioclase (triclinic feldspar) Common mineral, essential constituents of most igneous rocks; composed of mixture of NaAlSi3O8, Ca Al2 Si2O8, and occasionally barium silicates. Also called oligoclase.

Planck constant (h) The elementary quantum of action, which relates energy to frequency

through the equation E = hν. In metric units h = 6.6261971 × 10−27 gm cm2/sec. Also,

Planck’s reduced constant, h¯ = 1.05459 × 10−27 gm cm2/sec.

Planck length The unique combination of the Planck constant h¯, the gravitational constant G, and the speed of light c of dimension length:

GP = Gh/c¯ 3 1/2 1.616 × 10−33cm .

planetary boundary layer

The Planck length is the approximate scale at which quantum effects become strong in gravitating systems.

For a black hole, the typical length is set by the mass (half the gravitational radius), RH /2 = Gm/c2. The typical associated quantum length is the Compton wavelength rc = h/mc¯ . The Planck length can be seen to coincide with RH and rc when the two are equal. Smaller masses (smaller gravitational radii) are strongly quantum because the quantum effect (their Compton wavelength) extends outside the classical black hole. Thus, the Planck length is the scale at which a quantum gravity description (not yet perfected) becomes necessary.

Planck mass The mass of an object whose Compton wavelength equals the Planck length:

	=	hc	1/2		×
	=	G			×
m		¯		2.177		10−5 grams .

Planck’s black body radiation Radiation at all frequencies, such as would be emitted by a “black body”. A black body is one that absorbs all radiation incident upon it. The emissivity of a black body is unity so the radiation that it emits is a function of temperature only. The peak of a black body spectrum is given by λpeak = (2.9×106)/T nm, where T is measured in degrees Kelvin.

Planck’s Law That the energy of light is directly proportional to its frequency is known as Planck’s Law: E = hν, where E is the

energy of a photon, ν the photon’s frequency, and h = 6.626076 × 10−34 J s, known as

Planck’s constant. This is a result of Planck’s attempts to explain the observed black body radiation curves and is regarded as the beginning of the quantum theory of radiation and ultimately led to the foundation of quantum mechanics. Planck’s law for the energy density of black body or cavity radiation is

H(λ) dλ = 8πchλ−5[exp(ch/λkT ) − 1] dλ

where c is the speed of light, h is Planck’s constant, k is the Boltzmann constant, T is the temperature, and λ is the wavelength of the radiation. This law contains the Wien displacement law and the Stefan–Boltzmann law and

avoids the “ultraviolet catastrophe” inherent to Rayleigh’s derivation of black body radiation.

Planck’s radiation law Electromagnetic radiation propagates in discrete quanta, each with an energy equal to the product of Planck’s constant times the frequency of the radiation.

Planck time The shortest length of time for which the classical theory of gravitation (general relativity) is useful, 10−43sec. Below this scale general relativity must be improved to include quantum theory. It can be expressed as

tP = GP /c = Gh/c¯ 5 1/2 5.4 × 10−44 sec

where G is Newton’s gravitational constant, h¯ is Planck’s constant, and c is the velocity of light in vacuum.

plane-fronted waves Gravitational waves in which normal of the wave fronts is a covariantly constant null vector. See pp-waves.

planet A large body (generally larger than 2000 km in diameter) which orbits around a star. In our solar system, nine planets orbit around the sun.

planetary boundary layer The turbulent boundary layer of the atmosphere, about 1 to 1.5 km altitude from ground in which the ﬂow ﬁeld is strongly inﬂuenced directly by interaction with the Earth’s surface. In the planetary boundary layer, the frictional effects of the underlying surface generate turbulence; the air movement is completely turbulent, and the turbulent friction force is equal or larger than the orders of geostrophic force and pressure gradient force. In theoretical analysis one usually assumes that the planetary boundary layer can be represented by stationary homogeneous turbulence. The planetary boundary layer is the heat and moist source and the momentum sink of whole atmosphere. The free atmosphere interacts with the surface through the planetary boundary layer. Based on the different features of turbulence, the planetary boundary layer can be divided into three layers: the laminar sublayer, which is just above the ground less than 2 m thick; the surface layer above the laminar