Polytrope
Encyclopedia
In astrophysics
, a polytrope refers to a solution of the Lane-Emden equation
in which the pressure
depends upon the density
in the form
, where 
is pressure, 
is density and 
is a constant
. The constant
is known as the polytropic index. This relation need not be interpreted as an equation of state
, although a gas
following such an equation of state does indeed produce a polytropic solution to the Lane-Emden equation. Rather, this is simply a relation that expresses an assumption about the change of
with radius
in terms of the change of
with radius, yielding a solution to the Lane-Emden equation.
Sometimes the word polytrope may be used to refer to an equation of state that looks similar to the thermodynamic
relation above, although this is potentially confusing and is to be avoided. It is preferable to refer to the fluid
itself (as opposed to the solution of the Lane-Emden equation) as a polytropic fluid. The equation of state of a polytropic fluid is general enough that such idealized fluids find wide use outside of the limited problem of polytropes.
Note that the higher the polytropic index, the more condensed at the centre is the density distribution.
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...
, a polytrope refers to a solution of the Lane-Emden equation
Lane-Emden equation
In astrophysics, the Lane–Emden equation is Poisson's equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. It is named after the astrophysicists Jonathan Homer Lane and Robert Emden...
in which the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
depends upon the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
in the form
, where is pressure, is density and is a constantConstant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
. The constant
is known as the polytropic index. This relation need not be interpreted as an equation of stateEquation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
, although a gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
following such an equation of state does indeed produce a polytropic solution to the Lane-Emden equation. Rather, this is simply a relation that expresses an assumption about the change of
with radiusRadius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
in terms of the change of
with radius, yielding a solution to the Lane-Emden equation.Sometimes the word polytrope may be used to refer to an equation of state that looks similar to the thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
relation above, although this is potentially confusing and is to be avoided. It is preferable to refer to the fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
itself (as opposed to the solution of the Lane-Emden equation) as a polytropic fluid. The equation of state of a polytropic fluid is general enough that such idealized fluids find wide use outside of the limited problem of polytropes.
Example models by polytropic index
- Neutron starNeutron starA neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
s are well modeled by polytropes with index about in the range between
and 
.
- A polytrope with index
is a good model for degenerateDegenerate matterDegenerate matter is matter that has such extraordinarily high density that the dominant contribution to its pressure is attributable to the Pauli exclusion principle. The pressure maintained by a body of degenerate matter is called the degeneracy pressure, and arises because the Pauli principle...
star cores (like those of red giantRed giantA red giant is a luminous giant star of low or intermediate mass in a late phase of stellar evolution. The outer atmosphere is inflated and tenuous, making the radius immense and the surface temperature low, somewhere from 5,000 K and lower...
s), for white dwarfWhite dwarfA white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
s, brown dwarfBrown dwarfBrown dwarfs are sub-stellar objects which are too low in mass to sustain hydrogen-1 fusion reactions in their cores, which is characteristic of stars on the main sequence. Brown dwarfs have fully convective surfaces and interiors, with no chemical differentiation by depth...
s, giant gaseous planetsGas giantA gas giant is a large planet that is not primarily composed of rock or other solid matter. There are four gas giants in the Solar System: Jupiter, Saturn, Uranus, and Neptune...
(like JupiterJupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
), or even for rocky planetTerrestrial planetA terrestrial planet, telluric planet or rocky planet is a planet that is composed primarily of silicate rocks or metals. Within the Solar System, the terrestrial planets are the inner planets closest to the Sun...
s.
- Main sequence starStarA star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s like our SunSunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
are usually modeled by a polytrope with index
, corresponding to the Eddington standard model of stellar structureStellar structureStars of different mass and age have varying internal structures. Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star....
.
- A polytrope with index
has an infinite radius. It corresponds to the simplest plausible model of a self-consistent stellar system, first studied by A. SchusterArthur SchusterSir Franz Arthur Friedrich Schuster FRS was a German-born British physicist known for his work in spectroscopy, electrochemistry, optics, X-radiography and the application of harmonic analysis to physics...
in 1883.
- A polytrope with index
corresponds to what is called isothermal sphere, that is an isothermal self-gravitating sphere of gas, whose structure is identical with the structure of a collisionless system of stars like a globular clusterGlobular clusterA globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is...
.
Note that the higher the polytropic index, the more condensed at the centre is the density distribution.