Thermodynamic process
A thermodynamic process may be defined as the energetic evolution of a
thermodynamic system proceeding from an initial state to a final state. Paths
through the space of thermodynamic variables are often specified by holding
certain thermodynamic variables constant. It is useful to group these processes
into pairs, in which each variable held constant is one member of a conjugate
pair.
Isochoric Process
An isochoric process, also called an isovolumetric process, is a process during
which volume remains constant. The name is derived from the Greek isos,
"equal", and khora, "place."
If an ideal gas is used in an isochoric process, and the quantity of gas stays
constant, then the increase in energy is proportional to an increase in
temperature and pressure. Take for example a gas heated in a rigid container:
the pressure and temperature of the gas will increase, but the volume will
remain the same.
In the ideal Otto cycle we found an example of an isochoric process when we
assume an instantaneous burning of the gasoline-air mixture in an internal
combustion engine car. There is an increase in the temperature and the
pressure of the gas inside the cylinder while the volume remains the same.
Equations
If the volume stays constant: (ΔV = 0), this implies that the process does no
pressure-volume work, since such work is defined by:
ΔW = PΔV
where P is pressure (no minus sign; this is work done by the system).
By applying the first law of thermodynamics, we can deduce that ΔU the change
in the system's internal energy, is:
ΔU = Q
for an isochoric process: all the heat being transferred to the system is added to
the system's internal energy, U. If the quantity of gas stays constant, then this
increase in energy is proportional to an increase in temperature,
Q = mCVΔT
where CV is molar specific heat for constant volume.
On a pressure volume diagram, an isochoric process appears as a straight
vertical line. Its thermodynamic conjugate, an isobaric process would appear as
a straight horizontal line.
Isochoric Process in the Pressure volume diagram. In this diagram, pressure
increases, but volume remains constant.
Isobaric Process
An isobaric process is a thermodynamic process in which the pressure stays
constant: Δp = 0 The term derives from the Greek isos, "equal," and barus,
"heavy." The heat transferred to the system does work but also changes the
internal energy of the system:
The yellow area represents the work done
According to the first law of thermodynamics, where W is work done by the
system, U is internal energy, and Q is heat. Pressure-volume work (by the
system) is defined as: (Δ means change over the whole process, it doesn't
mean differential)
but since pressure is constant, this means that
.
Applying the ideal gas law, this becomes
assuming that the quantity of gas stays constant (e.g. no phase change during
a chemical reaction). Since it is generally true that
then substituting the last two equations into the first equation produces:
.
The quantity in parentheses is equivalent to the molar specific heat for constant
pressure:
cp = cV + R
and if the gas involved in the isobaric process is monatomic then
and
.
An isobaric process is shown on a P-V diagram as a straight horizontal line,
connecting the initial and final thermostatic states. If the process moves towards
the right, then it is an expansion. If the process moves towards the left, then it is
a compression.
If the volume compresses (delta V = final volume - initial volume < 0), then W <
0. That is, during isobaric compression the gas does negative work, or the
environment does positive work. Restated, the environment does positive work
on the gas.
If the volume expands (delta V = final volume - initial volume > 0), then W > 0.
That is, during isobaric expansion the gas does positive work, or equivalently,
the environment does negative work. Restated, the gas does positive work on
the environment.
Enthalpy. An isochoric process is described by the equation Q = ΔU. It would
be convenient to have a similar equation for isobaric processes. Substituting the
second equation into the first yields
The quantity U + p V is a state function so that it can be given a name. It is
called enthalpy, and is denoted as H. Therefore an isobaric process can be
more succinctly described as
.
Variable density viewpoint. A given quantity (mass M) of gas in a changing
volume produces a change in density ρ. In this context the ideal gas law is
written
R(T ρ) = M P
where T is thermodynamic temperature above absolute zero. When R and M
are taken as constant, then pressure P can stay constant as the densitytempertature quadrant (ρ,T ) undergoes a squeeze mapping.
Isothermal Process
Isothermal process or isothermic process to the change of reversible
temperature in a thermodynamic system is denominated, being this change of
constant temperature in all the system. For a substance that carries out a
change of state static and isothermally, the transferred heat can be calculated
of advisable way according to the second law of the thermodynamics.
Q12
=∫T
dS = T(S2- S1) = mT(s2-s1)…..eq. 1
Combining the previous equation with the first law of the thermodynamics, that
is obtained for a closed system carries out a static change of state isothermally.
Wideal = (U1-T1S1) - (U2-T2S2)
On the other hand, of the definition of the function of Helmholtz or function of
work.
A= U-TS
where: A it is work, U it is internal energy, T is temperature and S is entropy.
Reason why eq 1 can be written like:
Wideal = (A1-A2)T = -(A2-A1)T
One concludes then that the diminution in the function of Helmholtz of a system
represents the maximum work that can develop that one during an isothermal
process by means of some appropriate device. In the same way, the diminution
in the function of Helmholtz of a system that is in an suitable device to absorb
work, the minimum work necessary to carry out the process isothermally. This is
the reason for which the function of Helmholtz also considers a potential
function him.
An isothermal curve is a line that on a diagram represents the successive
values of the diverse variables of a system in an isothermal process.
The isotherms of an ideal gas in a diagram P-V, call diagram of Clapeyron, are
equilateral hyperbolas, whose equation is P•V = constant.
Example of an isothermal problem.
Two kilograms of gaseous nitrogen confined in a cylinder that lodges a piston,
static carry out a change of state from 300 k and 101,325 kPa to 300 a final
state to 300 k and 20.000 kPa. Heat transference can be between nitrogen and
a heat deposit that is 300k.
A) It determines the work transferred by nitrogen
B) It determines the total heat transferred between nitrogen and the deposit. .
Because it is a static and isothermal work.
A) 1W 2 = m[(u1-u2)T = -T(s1-s2)]….eq 1
Using data of the table of Nitrogen it is obtained:.
u1 =h1 – p1v1
= (311.163-101.325 x .8786) kJ/ Kg
= 222.138 kJ/Kg
S1 = 6.8418 kJ/Kg K
u2 = h2 – p2 v2
= (279.010 – 20,000 x 0.004704) kJ/Kg
= 184.93 kJ/Kg
s2= 5.1630 kJ/Kg
Replacing the numerical values in the equation.
2[ ( 222.138-184.93 ) – 300(6.8418 – 5.1630) ] kJ
= -932,864 kJ. The negative sign means that the work takes place in the gas.
Since the process static and isothermal, it is had of the second law
1Q2 = m T (s2 – s1)
= 2 x 300 (5.1630 – 6.8418) kJ
= -1007.28 kJ.
The negative sign means that the heat is extracted of the gas.
1W 2 =
BIBLIOGRAPHI
Ingeniería termodinámica. Fundamentos y aplicaciones. Francis. F. Huang,
Primera edición, México 1997. Editorial continental
Adiabatic Process
n thermodynamics, an adiabatic process or an isocaloric process is a
thermodynamic process in which no heat is transferred to or from the working
Ifluid. The term "adiabatic" literally means impassable, coming from the Greek
roots ἀ- ("not"), διὰ- ("through"), and βαῖνειν ("to pass"); this etymology
corresponds here to an absence of heat transfer. Conversely, a process that
involves heat transfer (addition or loss of heat to the surroundings) is generally
called adiabatic.
For example, an adiabatic boundary is a boundary that is impermeable to heat
transfer and the system is said to be adiabatically (or thermally) insulated; an
insulated wall approximates an adiabatic boundary. Another example is the
adiabatic flame temperature, which is the temperature that would be achieved
by a flame in the absence of heat loss to the surroundings. An adiabatic
process that is reversible is also called an isentropic process. Additionally, an
adiabatic process that is irreversible and extracts no work is in an isenthalpic
process, such as viscous drag, progressing towards a nonnegative change in
entropy.
One opposite extreme—allowing heat transfer with the surroundings, causing
the temperature to remain constant—is known as an isothermal process. Since
temperature is thermodynamically conjugate to entropy, the isothermal process
is conjugate to the adiabatic process for reversible transformations.
A transformation of a thermodynamic system can be considered adiabatic when
it is quick enough that no significant heat is transferred between the system and
the outside. At the opposite extreme, a transformation of a thermodynamic
system can be considered isothermal if it is slow enough so that the system's
temperature remains constant by heat exchange with the outside.
Adiabatic heating occurs when the pressure of a gas is increased from work
done on it by its surroundings, ie a piston. Diesel engines rely on adiabatic
heating during their compression stroke to elevate the temperature sufficiently
to ignite the fuel. Similarly, jet engines rely upon adiabatic heating to create the
correct compression of the air to enable fuel to be injected and ignition to then
occur
Adiabatic cooling occurs when the pressure of a substance is decreased as it
does work on its surroundings. Adiabatic cooling does not have to involve a
fluid. One technique used to reach very low temperatures (thousandths and
even millionths of a degree above absolute zero) is adiabatic demagnetisation,
where the change in magnetic field on a magnetic material is used to provide
adiabatic cooling. Adiabatic cooling also occurs in the Earth's atmosphere with
orographic lifting and lee waves, and this can form pileus or lenticular clouds if
the air is cooled below the dew point.
Ideal gas (reversible case only)
For a simple substance, during an adiabatic process in which the volume
increases, the internal energy of the working substance must decrease
The mathematical equation for an ideal fluid undergoing a reversible (i.e., no
entropy generation) adiabatic process is
where P is pressure, V is volume, and
CP being the specific heat for constant pressure and CV being the specific heat
for constant volume. α is the number of degrees of freedom divided by 2 (3/2 for
monatomic gas, 5/2 for diatomic gas). For a monatomic ideal gas, γ = 5 / 3, and
for a diatomic gas (such as nitrogen and oxygen, the main components of air) γ
= 7 / 5. Note that the above formula is only applicable to classical ideal gases
and not Bose-Einstein or Fermi gases.
For reversible adiabatic processes, it is also true that
where T is an absolute temperature.
Bibliography:
http://en.wikipedia.org/wiki/Adiabatic_process
Polytropic Process
When a gas undergoes a reversible process in which there is heat transfer, the
process frequently takes place in such a manner that a plot of the Log P
(pressure) vs. Log V (volume) is a straightline. Or stated in equation form PVn =
a constant.
This type of process is called a polytropic process.
An example of a polytropic process is the expansion of the combustion gasses
in the cylinder of a water-cooled reciprocating engine.
2- Example:Compression or Expansion of a Gas in a Real System such as a
Turbine
Many
processes
can
be
approximated
by
the
law:
where,
P= Pressure,
v= Volume,
n= an index depending on the process type.
Polytropic processes are internally reversible. Some examples are vapors and
perfect gases in many non-flow processes, such as:




n=0, results in P=constant i.e. isobaric process.
n=infinity, results in v=constant i.e. isometric process.
n=1, results in P v=constant, which is an isothermal process for a perfect
gas.
n= , which is a reversible adiabatic process for a perfect gas.
Some Polytropic processes are shown in figure below:
The initial state of working fluid is shown by point 0 on the P-V diagram. The
polytropic state changes are:








0 to 1= constant pressure heating,
0 to 2= constant volume heating,
0 to 3= reversible adiabatic compression,
0 to 4= isothermal compression,
0 to 5= constant pressure cooling,
0 to 6= constant volume cooling,
0 to 7= reversible adiabatic expansion,
0 to 8= isothermal expansion.
polytropic process is a thermodynamic process that obeys the relation:
PVn = C,
where P is the pressure, V is volume, n is any real number (the polytropic
index), and C is a constant. This equation can be used to accurately
characterize processes of certain systems, notably the compression or
expansion of a gas, but in some cases, possibly liquids and solids.
Under standard conditions, most gases can be accurately characterized by the
ideal gas law. This construct allows for the pressure-volume relationship to be
defined for essentially all ideal thermodynamic cycles, such as the well-known
Carnot cycle. (Note however that there may be instances where a polytropic
process occurs in a non-ideal gas.)
For certain indices n, the process will be synonymous with other processes:



if n = 0, then PV0=P=const and it is an isobaric process (constant
pressure)
if n = 1, then for an ideal gas PV=NkT=const and it is an isothermal
process (constant temperature)
if n = γ = cp/cV, then for an ideal gas it is an adiabatic process (no heat
transferred)
Note that
adiabatic index)

if n =
,
since
(see:
, then it is an isochoric process (constant volume)
When the index n occurs between any two of the former values (0,1,gamma, or
infinity), it means that the polytropic curve will lie between the curves of the two
corresponding indices.
The equation is a valid characterization of a thermodynamic process assuming
that:


The process is quasistatic
The values of the heat capacities,cp and cV, are almost constant when 'n'
is not zero or infinity. (In reality, cp and cV are a function of temperature,
but are nearly linear within small changes of temperature).
Polytropic fluids
Polytropic fluids are idealized fluid models that are used often in astrophysics. A
polytropic fluid is a type of barotropic fluid for which the equation of state is
written as:
P = Kρ(1 + 1 / n)
where P is the pressure, K is a constant, ρ is the density, and n is a quantity
called the polytropic index.
This is also commonly written in the form:
P = Kργ
where in this case, γ = (1 + 1 / n) (Note that γ need not be the adiabatic index
(the ratio of specific heats), and in fact often it is not. This is sometimes a cause
for confusion.)
Gamma
In the case of an isentropic ideal gas, γ is the ratio of specific heats, known as
the adiabatic index.
An isothermal ideal gas is also a polytropic gas. Here, the polytropic index is
equal to one, and differs from the adiabatic index γ.
In order to discriminate between the two gammas, the polytropic gamma is
sometimes capitalized, Γ.
To confuse matters further, some authors refer to Γ as the polytropic index,
rather than n. Note that
Bibliography:
www.taftan.com/thermodynamics/POLYTROP.HTM
http://www.answers.com/topic/polytropic-process
http://www.engineersedge.com/thermodynamics/polytropic_process.htm
http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Thermal/ImportantThermalP
rocess.html
interesting article:
During the 19th century many advances in many different fields of physics took
place, creating a new atmosphere conducive to the continuing investigations
into the natural world. One such field, that was investigated rigorously was that
of the stars. Through out the history of mankind, and many years before, it was
evident that the sun had not changed its behavior much. This is a fact that led
to the original idea that one could consider a star a gas sphere, that essentially
must remain in hydrostatic equilibrium with its own gravitation. Any situation
contrary to this would lead to accelerations either inward or outward, which
would cause many short term changes, that have not been observed in normal
stars over the years.i
Although this is an amazing achievement in ones consideration of the
stars, a full understanding of the complex processes was still not at hand. This
description, though informative, still does not address the question of why stars
radiate, or what a star’s energy source may be. Half of this question was
answered with the introduction of quantum theory of blackbody radiation by Max
Plank in 1900.ii After the introduction of the blackbody spectrum, it was
determined that a star was essentially radiating according to the rules of a black
body, with the notable exception of the Fraunhofer absorption lines, which were
discovered by Fraunhofer in 1819, and emissions lines, which originate from the
stellar atmosphere and surrounding gas. This, however, still did not explain the
star’s energy source. Once it was suggested that a star may “live” from its
gravitational potential energy, but this is not the case, as such a star would have
only a fraction of the lifetime of a real star. This being the case a new
assumption was made, that the star had some sort of undefined internal energy
source. With this in mind Karl Schwarzschild began his work on radiative
transfer of energy in stellar atmospheres around 1906.iii This established the
addition of a new dynamic to stellar structure, whose importance is equal to that
of the first assumption based on hydrostatic equilibrium. It was essentially
determined that the structure of a star was supported by internal pressure as
well as radiation pressure against its gravitational pull.
With this theoretical basis in place another important milestone was
reached at about the same time. Hertzsprung and Russell recognized that all
stars are not of the same mass, temperature, and luminosity and therefore have
a number of differing properties dependant upon these quantities. In measuring
the luminosity and effective temperature of each star they found that stars of the
same class occupy distinguishable places in relation to these quantities. With
the creation of the Hertzsprung-Russell diagram many of the classes, such as
the main sequence stars and red giants, that are quite familiar today first
became apparent. Soon after this connection between star class and effective
temperature was established, Eddington established a mass-luminosity
relationship for the main sequence stars and further researched the internal
mechanisms of the stars. In the 1930’s the Dirac – Fermi statistics for a
degenerate electron gas were published. It was quickly realized that this too
was applicable to the stellar interiors of certain star classes, specifically white
dwarfs of high mass. S. Chandrasekar was instrumental in this determination,
moreover he was also able to find the critical maximum mass for a white dwarf
star, known today as the Chandrasekar mass.
It is at this point in history where the Polytrope representation of stars
first was developed. It is a method that today still lends valuable methods and
insights to the internal structure of stars. It has also proven to be most versatile
in the examination of a variety of situations, including the analysis of isothermal
cores, convective stellar interiors, and fully degenerate stellar configurations.
Even the case of an ideal gas can be related to a polytrope of index n = 3/2.
The justification for such a theory is that, as the name implies, it is extremely
versatile. As will be shown later such a class of models allows for the derivation
and prediction of many stellar properties, which continue to be of significant
interest to the astrophysics community. The derivation of polytropic star models
according to R. Kippenhahn and W. Weigert, as well as S. Chandrasekar and
William K. Rose is outlined in the following section.
The polytropic theory of stars essentially follows out of thermodynamic
considerations, that deal with the issue of energy transport, through the transfer
of material between different levels of the star. We simply begin with the
Poisson equation and the condition for hydrostatic equilibrium:
MrG
P
 2
r
r
M r
 4r 2
r
Eq. 1 & 2
where P is the pressure, M(r) is the mass of a star at a certain radius r, and ρ is
the density, at a distance r from the center of a spherical star. iv Combination of
these equations yields the following equation, which as should be noted, is an
equivalent form of the Poisson Equation.
1 d  r 2 dP 

  4G
r 2 dr   dr 
Eq. 3
From these equations one can then obtain the Lane-Emden equation through
the simple supposition that the density is simply related to the density, while
remaining independent of the temperature. We already know that in the case of
a degenerate electron gas that the pressure and density are ρ ~ P3/5. v
Assuming that such a relation exists for other states of the star we are led to
consider a relation of the following form:
P  K
1
1
n
Eq.3
where K and n are constants, at this point it is important to note that n is the
polytropic index. Using this as a basis to classify different interior states within
the star we can also conclude that a non-relativistic degenerate electron gas is
a Polytrope of n = 3/2. Based upon these assumptions we can insert this
relation into our first equation for the hydrostatic equilibrium condition and from
this rewrite equation to:
 K (n  1) n1 1  1 d  2 d 
n
 4G   r 2 dr  r dr   


Eq.5
Where the additional alteration to the expression for density has been inserted
with λ representing the central density of the star and θ that of a related
dimensionless quantity that are both related to ρ through the following relation.
   n
Eq.6
Additionally, if place this result into the Poisson equation, we obtain a
differential equation for the mass, with a dependance upon the polytropic index
n. Though the differential equation is seemingly difficult to solve, this problem
can be partially alleviated by the introduction of an additional dimensionless
variable ξ, given by the following:
r  a
 (n  1) K
a

 4G
1
1
n



1
2
Eq. 7
Inserting these relations into our previous relations we obtain the famous form
of the Lane-Emden equation, given below:
1 d  2 d 

   n
r 2 d  d 
Eq.8
At this point it is also important to introduce the boundary conditions, which are
based upon the following boundary conditions for hydrostatic equilibrium, and
normalization considerations of the newly introduced quantities ξ and θ. What
follows for r = 0 is
r  0  0
Eq.9
     (0)  1
Eq. 10
Taking these simple relations into consideration, it is also evident that one can
produce additional conditions, based upon a modified form of the Lane-Emden
Equation given by:
2 d d 2
 2   n
 d d
Eq. 11
Here it is apparent that as ξ approaches 0 the first term of the equation
approaches . As a result an additional condition must be introduced in order
to maintain the conditions of Eq. 9 and 10 simultaneously:
d
d
0
 0
Eq. 12
Once the boundary conditions have been determined it is an easy matter to
obtain a number of solutions for the Lane-Emden equation. In addition to
various numerical methods, which will be explained later, this equation actually
has 3 known analytical solutions for polytropes of index n = 0, 1, and 5, given
below:
0  
2
 1, 1  6
6
sin( )
1 
, 1  

5 
1
, 1  
1 2
1 
3
Eq. 13, 14 & 15
where the subscripts represent the index number n for a specified solution and
the ξ subscripts represent its value for θ = 0.
These results are useful in a few respects and deal with some actual
state equation for stars, however they are more important for inferring general
forms of the Lane-Emden solutions. Below are the plots of θ vs. ξ for the
aforementioned solutions, generated using Mathmatica.
Polytrope n = 1
Please note that all the following graphs are modeled after the following: x –
axis gives ξ values and the y – axis give θ values. Here we can see that the
function basically follows the same form as that for an index n = 0, with a few
minor differences, however the Polytrope of index n = 0 also terminates at a
finite radius just as is observed in the relation for a Polytrope of index n = 1.
The other main difference that we observe in this case is that the termination
point of the star is markedly larger, at about a value of 3,15 as opposed to the
termination value of about 2,48 for the polytrope n = 0 as seen below.
Polytrope Index n = 0
Here we can see that the values for ξ start at one in accordance with our
boundary conditions and then eventually reduces to θ = 0. This is essentially
an indicator that the star’s material ceases to exist outside of this area as the
density drops to 0 at this point. The next graph shows many similarities to this
one.
Polytrope Index n = 5
Though these two solutions for n = 1 and n = 0 share many characteristics, the
solution for the Polytrope of index n = 5, contains some radically different and
unexpected characteristics. In this case the behavior of the function is markedly
different than that of its predecessors. Here the density of the star initially
decreases rapidly as radius increases but slows rapidly once a ξ value of
around three is reached. At this point the decrease slows continually. Though
it may not be apparent on the graphic provided, the function never reaches 0. It
is therefore evident that a polytropic star of index n = 5 has an infinite radius,
and in reality cannot exist, however the case itself deserves further study.
Despite this fact such a model provides important theoretical perspective
concerning the theory, as one may view this as the border between polytropic
that are physically feasible. It is also of interest to note, as will be shown later,
that such a stellar model has, in spite of the infinite radius, a finite mass.
Additionally, other stellar models, that are created in a “layered” fashion, where
each layer consists of a Polytrope of a different index, may also utilize this
function for a portion of the star, in which case a finite radius would be possible.
In addition to these relations there are also a number of other
conclusions that one can draw from the polytropic model of stars. For relations
of this type, there exists a relation between the polytropic index, mass of a star
and the radius. It is perhaps evident in the discussion of the analytic solutions
of the polytropic index that one could possibly infer a relation between the
polytropic index of the star and the radius that one would calculate from that
star. In the attempt to find a relation the most immediate result is obtained from
the simple equations of stellar state. Integration of dM/dr on the limit from 0 to
the radius of the star, yields the following expression.
R
M tot   4r 2 dr
0
Eq. 16
This equation, is in itself not possible to integrate as we have an r dependence
in the density ρ. However, this problem can be easily alieveated through the
use of the polytropic variables. By replacing ρ and r by λθ, where lambda is the
central density, and aξ the process of integration can not only be carried out
more easily, but a relation dependent upon central density the maximum Radius
and the other polytropic variables obtained. Integration after such a substition
yeilds the follwoing:
 1 d 
M tot  4R 3 

 1 d   1
Eq.17
Though this relation is extremely useful in the polytropic model of stars there
are also a number of additional relations, that can augment usefulness of this
relation as well. The first of these is the relationship between the central density
and the average density of the star. It is easily seen, that for the case of a star
the average density is simply given by the division of the total mass through the
total volume. For the assumption of a spherical star, which yeilds a volume of
4/3πr^3, and using the relation for the total mass. We find simply that the ratio
of the average density to the central density is given by:
  
M tot
3 d

V
1 d

  1
Eq.18
Where ρ is the density and <ρ> is the average density.
Another case, though not nearly as important as the case presented
above, is found in the relationship for the radius of a star. Beginning with the
standard polytropic for the radius (Eq.6) relation and inserting the value for “a”
we obtain:
1
1 n
 K (n  1)  2 2 n
R

1
 4G 
Eq. 20
It is now apparent that the case of n = 1 will, as in the case for the total mass of
the star at n = 3 which will be covered shortly, yield an expression that is
independent of the central density of the star. This relation is extremely
interesting, as it reveals that there is the potential for fundamentally different
classes of stars to exist, whose masses differ, but not the radii. Though this
may seem counterintuitive it is important to note that as the mass in a real star
increases, the equations of state may change, which limits the actual range of
the masses for such a star in practice. This, however, does not diminish from
the theoretically interesting aspects of this model.
There is additionally another case, which is also independent of the
central density as well. For the case of a Polytrope of index n = 5, an analytic
solution is obtainable, as was shown above. Insertion of this relation for θ into
the equation for total mass yields a relation that in itself appears rather
mundane, however we may also insert the value for the maximum radius, which
has already been determined to be infinite. By evaluation of the resulting limit:
M tot

d
 4a  1
d

3
2

2 
3 K 
  4 3a    5
 G 
 1 

1
Eq. 21
It is readily apparent that the total mass of this star with an infinite radius yields
the surprising result of a finite mass. Though the significance of this case is in
practice not important, as there are not any stars that exist with an infinite
radius, it is a case of theoretical interest, as it essentially represents a border
case in polytropic star models, as models with n > 5 diverge. This case,
however, is not nearly groundbreaking or interesting as that of the first relation
or the one to be discussed next, it is merely a theoretical model that provides an
interesting point of consideration, though composite models may contain such
polytropes and make use of such a relation.
In addition to the relations that we discussed here there remains an
additional relation between the mass and the radius that remains to be
discussed. In this case we start with the equations for the maximum radius of
the star as well as the total mass for a star of polytropic index n. Between
these two equations, we can eliminate the central density and obtain a
relationship that is independent of this quantity. Through this we obtain the
N n GM
n 1
n
R
3 n
n
K
1
1  4  n
Nn 


n  1  0 wn n 1 
n 1
n 1
0 wn  1
result given below:
d
d
 1
Eq. 21, 22, & 22
This is one of the most important results of the polytropic theory of stars, as it
not only provides, for a given radius a mass or vice versa, but also provides, in
its simplified form, some of the framework that is used in order to obtain
numerical results for the other intrinsically important values for a star. These
variables and notation are usually used rather universally in the literature
regarding polytropic star models. An additional point that is of paramount
importance is the fact that a polytrope of index n = 3 will yield a mass
expression independent of the radius. As earlier discussed the case where the
n = 3 corresponds to the polytropic star representation of a relativistic
degenerate electron gas. This is essentially the model for the Chandrasekar
mass for stars (given as =1.54 x Sun’s mass), and this can indeed be obtained
by inserting value obtained from the polytropic models in combination with the
state equation of a relativistic degenerate gas. Though this may not seem
important, as the Chandrasekar border mass is a well known value, it is
important to point out that this derivation for the mass is a result as it is in
complete agreement with a value that was originally calculated through
conventional methods through the use of statistical mechanics. This argument
not only lends credence and further support to the border mass concept, but it
also lends credibility to the polytropic theory of stars, while demonstrating the
effectiveness of the method. This is further bolstered by the fact that there are
no known white dwarfs that exceed this mass. This is a powerful statement with
regard to the theory of stellar states, that has been widely accepted for a
number of years.
This being said there remains one final relation for the central pressure
that is easily derivable from the expression for the central density. In order to
obtain this relation we begin with the expression for the central density given by,
P  K
n 1
n
 Pc  K
n 1
n
Eq. 25
As this relation is dependant upon K and the central density it is apparent that
we can replace both of these relations by the values determined for the Mass –
Radius relation as that determined for the value of the average density in
relation to that of the central density. The average density can then be replaced
by its dependence on the mass and the volume and these relations finally yield:
Pc  N n GM
wn 
Eq. 26
n 1
n
R
3 n
n

n 1
n
 GM 2 
 wn  4 
 R 
1
 d  
4(n  1)   
 d   1 
2
Though it is often not mentioned it is interesting to note that this relation can
also be used as a good check on the accuracy of ones stellar model. It is
apparent that as the central pressure changes that the equation of state may
change with it as well. For example if it is known that the electron gas is
degenerate we have two cases for the degeneracy. The first of these is the
relativistic case and the second of these is the non – relativistic case. Each of
these possesses its own equation of state that is different the first. For the
relativistic case we have a polytrope of n = 3 and for the non - relativistic case
we have a polytrope of n = 3/2. In any case if we have a star of a known mass,
radius, and general chemical composition one could easily determine the
polytropic index. For the case of n = 3 this is of extreme importance as we are
able to determine the border mass. This gives a maximum value of the central
density before the collapse of a star into a neutron star, and the value for the
non-relativistic case give values that correspond to approaching values of
largest central pressure. At the point where the model exceeds the pressure of
the Chandrasekar mass we know that the model most certainly is that of a
neutron star or just invalid. In any case it provides a nice guide to the models of
the stars that one would be concerned with and also provides a good basis for
the inclusion of models or exclusion of models, based on the validity of the
equations of state at different values for the pressure.
Having demonstrated the importance of the polytropic star relations, it is
now important to consider cases, in which one is not able to use the exact
solutions of the polytropic relations. Examples of such systems would be that of
a non-relativistic electron gas, where, as mentioned earlier, the polytropic index
is n = 3/2. Even the all important case of the fully degenerate electron gas (n =
3) is not covered by the exact solutions. There are a number of ways that this
problem could be attacked. The first of these is to use a program such as
“Mathmatica” in order to obtain solutions to the Lane – Emden equations.
However under the assumption that one does not have access to such a
program there are also a number of other ways that the problem at hand can be
attacked. The first of these methods is to attempt a power series solution for
the Lane-Emden equation:
2 d d 2

  n
 d d 2
Eq. 11
One can then proceed to formulate a power series solution to the Lane –
Emden equation. Which is of the general form:
 ( )   cn n
Eq. 31
where c subscript n represent arbitrary constants.
By matching the coefficients and inserting the appropriate border conditions, the
resulting solution is given by
 ( )  1 
1
6
2

n
...
120 4
Eq. 32
One can see immediately that this result is consistent with the analytical
solution of n = 0, as the additional terms for the series solution are n dependant.
These solutions, though not the only available for use in determination of
physical quantities, are generally satisfactory for representing different
polytropic indices.
A method that is readily available is a variation of the Euler method, for
second order differential equations. In considering this method on creates an
arbitrary step, suitable to the task at hand, and from initial values of the
equation extrapolates a solution. In this case one considers the initial value of
dξ/dθ as well as θ. At different itterations one can now extrapolate values from
the function’s slope. This yields a recursive relation for this method of:
 'i 1   'i h ''i
i 1   'i h 'i
Eq. 33
where h is the step size and the primes represent derivatives of the function θ.
Using the initial values and the Lane-Emden equation, one can then find
approximate values of the function θ and θ’ for the next step. Reinserting these
values yields a new value of θ’’ allowing one to extrapolate another value.
Repetition of this process allows one to crate an approximate data curve,
thereby describing a polytrope of a certain type.
Although this method may be of use it is quite evident that there remain a
number of other techniques that may be superior, which result in numerical
results that one can use in order to determine the necessary factors that may be
placed in the various relations, which were introduced in the preceding section,
in order to make predictions for different stellar properties. The first, and also
the most used throughout the literature (See Schwarzschild, Chandrasekar,
Kippenhahn and Weigert) is the Runge-Kutta Method. In order to determine the
numerical solutions of these equations, one must first begin with initial values as
in the preceding section. The first of these is given by our normalization
condition, and the second of these is acquired through observation of the LaneEmden equation, in the expanded form, given by equation 13. As the
normalization conditions requires that θ(0)=1 it is apparent that the second term
of the equation will diverge if the value of dθ(0)/dξ is not equal to zero. It is
therefore apparent that a second condition exists, which makes the
determination of solutions through numerical methods possible. The general
procedure for the Runge-Kutta method is as follows. As the Lane-Emden
equation is a second order differential, one must use an expanded form.
According to the methods outlined in a number of mathematical texts the
various values of θ, dθ/dξ, for certain values of ξ can be determined through a
step mehtod. The size of the steps h are taken to be constant and provide an
iterated view of the function θ(ξ).vi
h
 k  2 k 2 n  2 k 3 n  k 4 n   n '
6 1n
 h ' n n
 n 1 ' 
 n 1
Eq. 31
Where the separate coefficients k are given by,
 ' n ' '  f (n ,n ,n ')
k1n  n ' '
k 2 n  f (n .5h,n .5hn ',n '.5k1n )
k 3n  f (n .5h,n .5hn ',n '.5k 2 n )
k 2 n  f (n  h,n  hn ',n '.5k 3n )
Eq. 32
Calculation of these parameters is in this case not limited to the exact polytropic
solutions, but can be tabulated for any polytropic index n, note that n of the
Runge-Kutta method is not n the polytropic index. As an example a list of a
number of solutions is included to below.
n
0
1
1.5
3
4
5
?
2.4494
3.14159
1.65375
6.89485
14.97755
Infinite
?(d?/d?)
4.8988
3.14159
5.99071
2.01824
1.79723
1.73205
?(central)/?(average)
1
3.28987
5.99071
54.1825
622.408
Infinite
o_w_n
0.333
132.3843
2.01824
0.729202
0
N_n
0.63662
0.42422
0.36394
0.4772
Infinite
w_n
0.11936
0.26227
0.7714
11.05666
247.558
Infinite
- [(n+1) ? [ d?/d? ] ]^-1
0.5
0.5
0.53849
0.85432
1.66606
Infinite
Where each of these variables represent the variables defined in the primary
relations for polytropic relations.vii
Now in order to convince the reader of the validity of such relations, in
terms of actual theory – data agreement, the model of a fully degenerate white
dwarf is considered. As indicated earlier such a model is that of a polytrope
with index n = 3. It is also of use to remind the reader that the mass relation
introduced earlier produces the Chandrasekar mass, which defines the
maximum mass of an exhausted star, before which it would simply collapse
further until it would reach the state of a neutron star. Inserting the appropriate
values into the mass – radius relations, given by Equations 21, 22, and 23, we
find the radius of the star to be on the order of 28km - 40km.4 This radius is
necessarily much smaller than that of a typical star, but larger than that of a
neutron star, on the order of ~10km. This being a crude, however satisfying,
example of a polytropic star model’s practical use, is only one of many
conceivable uses for such a theory.
Throughout this survey on the Polytropic theory of stars, it has been
demonstrated that in many respects these models provide a versatile and
accurate method for the determination of many stellar parameters. Though the
Lane-Emden equation itself only has three known solutions, this difficulty can
easily be overcome with the introduction of a number of numerical solutions,
that can be found through a numerous variety of methods, for which
approximate solutions can be found to varying degrees of accuracy. This being
the case the polytropic theory of Stars has proven to be a versatile and able
theory worthy of its name.
Martin Schwarzschild. Structure and Evolution of Stars.
Princeton University Press, Princeton 1958
H.H. Voigt. Abriß der Astronomie (5. Überarbeitete Auflage).
Bibliografisches Institut und F.A. Brockhaus A.G., Mannheim 1991.
William K. Rose. Advanced Stellar Astrophysics.
Cambridge University Press, Cambridge (UK) 1998.
R. Kippenhahn, A. Weigert. Stellar Structure and Evolution.
Springer-Verlag, Heidelberg und Berlin 1990.
S. Chandrasekar. An Introduction the Study of Stellar Structure.
Chicago University Press, Chicago 1931.
William E. Boyce, Richard C. DiPrima. Elementary Differential Equations and
Boundary Value Problems. John Wiley and Sons, New York 1997
Branden T. Allen. Various Lectures given at Göttingen, Germany on the
polytropic theory of Stars.
PROBLEMS OF TERMODINAMICS PROCESS
A mass of 0.6 [g] of air are in a cylinder with a piston 3 pressure [bar], a
temperature of 176 [oC] and an initial volume of 260 [cm3]. For each point
calculate the final state, the work, the heat involved in the process and the type
of process that question.
a. Adiabatic expands until the initial pressure is 1 [bar] according to the PVK
where k = constant = 1.4.
b. The pressure is reduced to one tenth of the initial pressure unchanged in
volume.
c. Is compressed to one quarter of its original volume assuming that change is
in accordance with the relationship PV1.3 = constant.
d. The pressure is increased to twice the initial value according to the PV =
constant relationship.
e. Is compressed at constant pressure and temperature is tripled.
f. The volume expands 5 times the initial value according to the PV = constant
relationship where P +3 = 5 V.