Entropy (astrophysics)
Encyclopedia
In astrophysics
, what is referred to as "entropy
" is actually the adiabatic constant
derived as follows.
Using the first law of thermodynamics
for a quasi-static
, infinitesimal process for a hydrostatic system
For an ideal gas
in this special case, the internal energy
, U, is only a function of the temperature
T; therefore the partial derivative of heat capacity
with respect to T is identically the same as the full derivative, yielding through some manipulation
Further manipulation using the differential version of the ideal gas law
, the previous equation, and assuming constant pressure, one finds
For an adiabatic process
and recalling 
, one finds
One can solve this simple differential equation to find
This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows
where
is Boltzmann's constant.
Substituting this into the above equation along with
and 
for an ideal monatomic gas
one finds
where
is the mean molecular weight of the gas or plasma
; and
is the mass of the Hydrogen atom
, which is extremely close to the mass of the proton
,
, the quantity more often used in astrophysical theory of galaxy cluster
s.
This is what astrophysicists refer to as "entropy" and has units of [keV cm2]. This quantity relates to the thermodynamic entropy as
where
, the density of states in statistical theory, takes on the value of K as defined above.
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...
, what is referred to as "entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
" is actually the adiabatic constant
Adiabatic invariant
An adiabatic invariant is a property of a physical system that stays constant when changes occur slowly.In thermodynamics, an adiabatic process is a change that occurs without heat flow, and slowly compared to the time to reach equilibrium. In an adiabatic process, the system is in equilibrium at...
derived as follows.
Using the first law of thermodynamics
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...
for a quasi-static
Quasistatic process
In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. However, it is very important of note that no real process is quasistatic...
, infinitesimal process for a hydrostatic system
For an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
in this special case, the internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
, U, is only a function of the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
T; therefore the partial derivative of heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
with respect to T is identically the same as the full derivative, yielding through some manipulation
Further manipulation using the differential version of the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
, the previous equation, and assuming constant pressure, one finds
For an adiabatic process
Adiabatic process
In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...
and recalling , one finds
 |
 |
One can solve this simple differential equation to find
This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows
where
is Boltzmann's constant.Substituting this into the above equation along with
and for an ideal monatomic gasMonatomic gas
In physics and chemistry, monatomic is a combination of the words "mono" and "atomic," and means "single atom." It is usually applied to gases: a monatomic gas is one in which atoms are not bound to each other....
one finds
where
is the mean molecular weight of the gas or plasmaPlasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
; and
is the mass of the Hydrogen atomHydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
, which is extremely close to the mass of the proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
,
, the quantity more often used in astrophysical theory of galaxy clusterGalaxy cluster
A galaxy cluster is a compact cluster of galaxies. Basic difference between a galaxy group and a galaxy cluster is that there are many more galaxies in a cluster than in a group. Also, galaxies in a cluster are more compact and have higher velocity dispersion. One of the key features of cluster is...
s.
This is what astrophysicists refer to as "entropy" and has units of [keV cm2]. This quantity relates to the thermodynamic entropy as
where
, the density of states in statistical theory, takes on the value of K as defined above.