Relativistic
Relativistic
Universe
Universe
Quantitative
Quantitative
theory
theory
of
of
star
star
structure
structure
and
and
star
star
evolution
evolution
Relativistic
Relativistic
Universe
Universe
Quantitative
Quantitative
theory
theory
of
of
star
star
structure
structure
and
and
star
star
evolution
evolution
The
The
Hertzprung
Hertzprung
-
-
Russel
Russel
diagram
diagram
Three
Three
main
main
stages
stages
There are three main stages of star
There are three main stages of star
evolution:
evolution:
1.
1.
From molecular cloud to main
From molecular cloud to main
sequrence
sequrence
star;
star;
2.
2.
Main sequence;
Main sequence;
3.
3.
From main sequence, through red giant
From main sequence, through red giant
to white dwarf (neutron star, black hole).
to white dwarf (neutron star, black hole).
Both evolution and internal
Both evolution and internal
structure of stars are now well
structure of stars are now well
understood!
understood!
The
The
interior
interior
of
of
a star
a star
…
…
Can
Can
be
be
described
described
by Euler
by Euler
equation
equation
of
of
fluid
fluid
dynamics
dynamics
1
(
)
u
u
u
P
t
ρ
∂
+ ⋅∇ = −∇Φ − ∇
∂
Euler
Euler
equation
equation
1
(
)
u
u
u
P
t
ρ
∂
+ ⋅∇ = −∇Φ − ∇
∂
u is vector of velocity of small element of the fluid;
ρ
is density;
P
is pressure;
Φ
is Newtonian potential.
In
In
equilibrium
equilibrium
Equation of hydrostatic equilibrium
Equation of hydrostatic equilibrium
P
ρ
∇ = − ∇Φ
Stars in equilibrium
Stars in equilibrium
The equilibrium equation implies
The equilibrium equation implies
relation among pressure, mass, and
relation among pressure, mass, and
radius.
radius.
P
ρ
∇ = − ∇Φ ⇒
(
)
3
2
2
/
( )
M M R
dP
M r
dr
r
R
ρ
∼
∼
On
On
the
the
other
other
hand
hand
2
4
dP
P
M
P
dr
R
R
⇒
∼
∼
For
For
white
white
dwarf
dwarf
The star is composed of degenerate
The star is composed of degenerate
gas of electrons, for which
gas of electrons, for which
5 / 3
5 / 3
5
M
P
R
ρ
∼
∼
2
4
dP
P
M
P
dr
R
R
⇒
∼
∼
It
It
follows
follows
that
that
As the mass of configuration
As the mass of configuration
increases the radius decreases. More
increases the radius decreases. More
massive the white dwarf, the smaller
massive the white dwarf, the smaller
its radius. Adding matter to white
its radius. Adding matter to white
dwarf (by accretion, for example)
dwarf (by accretion, for example)
causes its radius to decrease.
causes its radius to decrease.
3
~
M
R
−
Sooner or later the equation of state
Sooner or later the equation of state
must change over to the fully
must change over to the fully
relativistic one. Here
relativistic one. Here
4 / 3
4 / 3
4
~
~
~
M
P
M
const
R
ρ
⇒
Thus for fully relativistic degenerate
Thus for fully relativistic degenerate
gas, there is a unique mass for which
gas, there is a unique mass for which
the configuration is stable. If this
the configuration is stable. If this
mass is exceeded, the star would
mass is exceeded, the star would
collapse. Thus white dwarfs cannot
collapse. Thus white dwarfs cannot
be more massive than this limiting
be more massive than this limiting
mass, called the Chandrasekhar limit
mass, called the Chandrasekhar limit
and equal 1.4 M
and equal 1.4 M
Let
Let
us
us
derive
derive
a
a
time
time
averaged
averaged
form
form
of
of
Euler
Euler
equation
equation
By standard calculation we can convince
By standard calculation we can convince
ourselves that
ourselves that
(
)
u
du
u
u
t
dt
∂
+ ⋅∇ =
∂
So that we can write
So that we can write
0
du
P
dt
ρ
ρ
+ ∇Φ + ∇ =
We
We
multiply
multiply
by
by
and
and
then
then
integrate
integrate
over
over
volume
volume
of
of
the
the
star
star
0
du
P
dt
ρ
ρ
+ ∇Φ + ∇ =
0
V
V
V
du
r
dV
r
dV
r
P dV
dt
ρ
ρ
⋅
+
⋅∇Φ
+
⋅∇
=
∫
∫
∫
r
Consider
Consider
this
this
equation
equation
term by term
term by term
V
du
r
dV
dt
ρ
⋅
=
∫
2
2
V
d r
r
dV
dt
ρ
⋅
=
∫
2
2
2
2
2
2
1
1
2
2
2
V
V
d
d I
r dV
u dV
T
dt
dt
ρ
ρ
−
=
−
∫
∫
I is the total moment of inertia about the origin,
T is total kinetic energy
V
r
P dV
ρ
⋅∇
=
∫
3
3
(
)
s
S
V
r n P dA
P
r
dV
⋅
−
=
∇ ≡
∫
∫
3
2
V
P dV
U
−
= −
∫
The pressure P vanishes on the surface;
We assume that gas inside the star is ideal.
V
r
dV
ρ
⋅∇Φ
= Ω
∫
For 1/r
2
force this is just the total potential energy.
Non
Non
-
-
averaged
averaged
virial
virial
theorem
theorem
Finally
Finally
we
we
get
get
2
2
1
2(
)
2
d I
T
U
dt
=
+
+ Ω
Virial
Virial
theorem
theorem
If we consider a system in
If we consider a system in
equilibrium, or at least long
equilibrium, or at least long
-
-
term
term
steady state, then the time average
steady state, then the time average
of moment of inertia vanishes and
of moment of inertia vanishes and
we have
we have
2
2
0
T
U
+
+ Ω =
Consequences
Consequences
Let us estimate the parameters of
Let us estimate the parameters of
the cloud from which star can
the cloud from which star can
eventually form. The internal kinetic
eventually form. The internal kinetic
energy of the gas in the cloud must
energy of the gas in the cloud must
be less than one half the
be less than one half the
gravitational energy, in order for
gravitational energy, in order for
moment of inertia to show any
moment of inertia to show any
accelerative contraction.
accelerative contraction.
For uniform gas confined to a sphere
For uniform gas confined to a sphere
with radius
with radius
R
R
c
c
and of temperature T
and of temperature T
2
2
4
3
2
2
3
c
h
c
R
kT
GM
m
R
π
ρ
μ
⎛
⎞⎛
⎞
≤
⎜
⎟⎜
⎟
⎝
⎠
⎝
⎠
μ
m
h
is an effective mass of a particle in the gas.
This distance is called Jeans length, it is
This distance is called Jeans length, it is
the distance below which a gas cloud
the distance below which a gas cloud
becomes gravitationally unstable.
becomes gravitationally unstable.
For a solar mass of material at the typical
For a solar mass of material at the typical
temperature of 50K, the cloud would be
temperature of 50K, the cloud would be
smaller than about 5
smaller than about 5
×
×
10
10
-
-
3
3
pc, with mean
pc, with mean
density greater than about 10
density greater than about 10
8
8
particles
particles
per cubic cm.
per cubic cm.
These are
These are
not
not
typical conditions in the
typical conditions in the
interstellar cloud!
interstellar cloud!
0.25(
/
)
3
h
c
M M
GM m
R
pc
kT
T
μ
≤
≈
The
The
parameter
parameter
μ
μ
It
It
is
is
convenient
convenient
to
to
devide
devide
the
the
composition
composition
of
of
the
the
stellar
stellar
matter
matter
into
into
three
three
categories
categories
:
:
X
X
–
–
mass
mass
fraction
fraction
of
of
gas
gas
which
which
is
is
hydrogen
hydrogen
Y
Y
–
–
mass
mass
fraction
fraction
of
of
gas
gas
which
which
is
is
helium
helium
Z
Z
–
–
mass
mass
fraction
fraction
of
of
gas
gas
which
which
is
is
everything
everything
else
else
(
(
metals
metals
)
)
Now we want to calculate the
Now we want to calculate the
number of particles in the unit
number of particles in the unit
volume of ionized gas.
volume of ionized gas.
Hydrogens
Hydrogens
contributes
contributes
2
2
X
X
: (
: (
electron
electron
+ proton)
+ proton)
Helium
Helium
contributes
contributes
¾
¾
Y
Y
: (2
: (2
electrons
electrons
+
+
α
α
particle
particle
but
but
the
the
mass
mass
is
is
4
4
times
times
that
that
of
of
hydrogen
hydrogen
)
)
Metals
Metals
contribute
contribute
½
½
Z
Z
: (z
: (z
electrons
electrons
+ 1
+ 1
nuclei
nuclei
, but
, but
the
the
mass
mass
is
is
2z
2z
that
that
of
of
hydrogen
hydrogen
)
)
All
All
together
together
we
we
have
have
But
But
X
X
+
+
Y
Y
+
+
Z
Z
=1
=1
(
)
3
1
2
4
2
h
N
m
ρ
=
+
+
X
Y
Z
(
)
1
1
3
1
2
2
h
N
m
ρ
=
+
+
X
Y
For
For
ideal
ideal
gas
gas
where
where
h
kT
P
NkT
m
ρ
μ
=
=
2
1
1 3
2
μ
=
+
+
X
Y
Interior
Interior
of
of
a star
a star
It
It
is
is
assumed
assumed
that
that
the
the
density
density
is
is
a
a
monotonically
monotonically
decreasing
decreasing
function
function
of
of
radius
radius
( )
( )
for
0
r
r
r
ρ
ρ
≤
>
2
3
0
( )
( )
,
( )
4
( )
4
/ 3
r
M r
r
M r
r r dr
r
ρ
π ρ
π
=
=
∫
Poisson
Poisson
’
’
s
s
equation
equation
In
In
spherical
spherical
coordinates
coordinates
2
4 G
π ρ
∇ Ω =
2
2
4
d
d
r
G r
dr
dr
π ρ
Ω
⎛
⎞ =
⎜
⎟
⎝
⎠
Integrating
Integrating
2
2
2
0
( )
4
r
d
G
GM r
r
dr
dr
r
r
π ρ
Ω
=
=
∫
But
2
( ) ( )
dP
GM r
r
P
dr
r
ρ
ρ
∇ = − ∇Ω ⇒
= −
This is the equation of hydrostatic equilibrium
for spherical stars
We
We
introduced
introduced
total
total
mass
mass
in
in
the
the
interior to r by
interior to r by
using
using
mass
mass
conservation
conservation
2
0
2
( )
4
( )
( )
4
( ),
(0)
0
r
M r
r
r dr
dM r
r
r M
dr
π ρ
π ρ
=
⇔
=
=
∫
Chandrasekhar
Chandrasekhar
variables
variables
For
For
example
example
,
,
total
total
gravitational
gravitational
energy
energy
[
]
,
0
( )
( )
( )
4
r
M r
G
I
r
dM r
r
σ
σ ν
ν
π
≡
∫
2
1,1
0
4
( )
( ) 4
R
dr
I
R
G M r
r
r
π
ρ π
=
∫
Chandrasekhar
Chandrasekhar
variables
variables
can
can
be
be
used
used
to
to
express
express
some
some
useful
useful
quantieties
quantieties
:
:
2,4
0
1,1
0
1,2
0
( )
( )
( )
;
( )
4
( )
( )
;
3
( )
( )
( )
M
M
h
M
I
R
dM r
P
P r
M
M
I
R
m
dM r
T
T r
M
k
M
I
R
dM r
g
g r
M
M
πμ
≡
=
≡
=
≡
=
∫
∫
∫
It
It
can
can
be
be
checked
checked
that
that
/ 3
1
/ 3
/ 3
,
/ 3
1
/ 3
/ 3
2
( )
4
3
1
/ 3
( )
2
( )
4
3
1
/ 3
c
G
M
r
I
r
G
M
r
ν
σ
ν
ν
σ ν
ν
σ
ν
ν
π
ρ
π
σ
ν
π
ρ
π
σ
ν
+ −
+ −
⎡
⎤
≥
⎢
⎥
+ −
⎣
⎦
≥
⎡
⎤
⎢
⎥
+ −
⎣
⎦
It
It
follows
follows
that
that
2
4
2
8
4
6
2
2
2
2
3
5.4 10
;
20
4.61 10
;
5
3
2.05 10
/
4
h
R
GM
M
P
atm
R
M
R
R
G m M
M
T
K
kR
M
R
R
GM
M
g
m s
R
M
R
π
μ
μ
⎛
⎞ ⎛
⎞
≥
=
×
⎜
⎟ ⎜
⎟
⎝
⎠
⎝
⎠
⎛
⎞⎛
⎞
≥
=
×
⎜
⎟⎜
⎟
⎝
⎠
⎝
⎠
⎛
⎞⎛
⎞
≡
=
×
⎜
⎟⎜
⎟
⎝
⎠
⎝
⎠
Polytropes
Polytropes
To complete the theory of processes
To complete the theory of processes
taking place in stars we still need
taking place in stars we still need
equation of state. It turned out that
equation of state. It turned out that
significant insight into the structure
significant insight into the structure
and evolution of stars is provided by
and evolution of stars is provided by
assuming the
assuming the
polytropic
polytropic
equation of
equation of
state.
state.
(
1) /
( )
( )
n
n
P r
K
r
ρ
+
=
Using equation of state we can
Using equation of state we can
eliminate pressure from the equation
eliminate pressure from the equation
of hydrostatic equilibrium for
spherical stars
2
( ) ( )
dP
GM r
r
dr
r
ρ
= −
We
We
differentiate
differentiate
this
this
equation
equation
and
and
use
use
2
( )
4
( )
dM r
r
r
dr
π ρ
=
2
2
( )
4
( )
( )
d
r
dP
d
GM r
Gr
r
dr
r dr
dr
π
ρ
ρ
⎛
⎞
= −
= −
⎜
⎟
⎝
⎠
1/
1
n
dP
n
d
K
dr
n
dr
ρ
ρ
+
=
Polytropic
Polytropic
star (
star (
Lame
Lame
-
-
Emden
Emden
)
)
equation
equation
2
2
(
1) /
(
1)
4
;
n
n
d
Kr n
d
Gr
dr
n
dr
ρ
π
ρ
ρ
−
⎛
⎞
+
= −
⎜
⎟
⎝
⎠
Initial
Initial
conditions
conditions
We want to use natural initial
We want to use natural initial
conditions
conditions
ρ(0) =
ρ(0) =
ρ
ρ
c
c
;
;
ρ(
ρ(
R
R
) = 0.
) = 0.
But this
But this
naive choice does not work because it
naive choice does not work because it
follows from the equilibrium condition that
follows from the equilibrium condition that
0 = P (0)
0 = P (0)
∼
∼
ρ
ρ
’
’
(0).
(0).
Thus the initial conditions
Thus the initial conditions
are
are
ρ(0) =
ρ(0) =
ρ
ρ
c
c
,
,
ρ
ρ
’
’
(0) = 0
(0) = 0
, while radius of the
, while radius of the
star R is to be computed from
star R is to be computed from
ρ(
ρ(
R
R
) = 0.
) = 0.
Solutions
Solutions
Polytropic
Polytropic
equation
equation
can
can
be
be
solved
solved
analytically
analytically
only
only
for
for
few
few
(not
(not
particularly
particularly
interesting
interesting
)
)
values
values
of
of
n.
n.
It
It
can
can
be,
be,
however
however
,
,
easily
easily
solved
solved
numerically
numerically
for
for
any
any
n.
n.
For
For
all
all
polytropes
polytropes
(
1) /
(3
) /
const( )
n
n
n
n
M
R
n
−
−
=
For
For
example
example
, for
, for
isothermal
isothermal
star
star
(n=4) we
(n=4) we
have
have
1/ 3
M
R
≈
Realistic
Realistic
stars
stars
Real stars are composed of several
Real stars are composed of several
polytropic
polytropic
layers. For example red
layers. For example red
giant has isothermal (n=4) helium
giant has isothermal (n=4) helium
core surrounded by convective
core surrounded by convective
(n=1.5) hydrogen envelope. One can
(n=1.5) hydrogen envelope. One can
model such a star by appropriate
model such a star by appropriate
matching these two phases at some
matching these two phases at some
radius
radius
.
.
Nuclear
Nuclear
reactions
reactions
in
in
stars
stars
There
There
are
are
two
two
major
major
processes
processes
that
that
are
are
sources
sources
of
of
energy
energy
in
in
stars
stars
:
:
p
p
-
-
p
p
cycle
cycle
and
and
CNO
CNO
cycle
cycle
.
.
The effectiveness of a process can be
The effectiveness of a process can be
measured by amount of energy
measured by amount of energy
produced by one gram of stellar
produced by one gram of stellar
material in unit time
material in unit time
0
T
ν
ε ε ρ
=
p
p
-
-
p
p
cycle
cycle
1
H
P
2
H
β
+
γ
P
3
He
3
He
4
He
ν
2p
Other
Other
p
p
-
-
p
p
cycles
cycles
4
He
7
Be
γ
3
He
β
−
ν
3
H
4
He
Triple
Triple
α
α
process
process
0
T
ν
ε ε ρ
=
It is a general property of these types of reaction
rates that the temperature dependence
"weakens" as the temperature increases. At the
same time the efficiency ε
0
increases. In general,
the efficiency of the nuclear cycles rate is
governed by the slowest process taking place. In
the case of p-p cycles, this is always the
production of deuterium given in step 1. For the
CNO cycle, the limiting reaction rate depends on
the temperature. At moderate temperatures, the
production of
15
O (step 4) limits the rate at which
the cycle can proceed. However, as the
temperature increases, the reaction rates of all
the capture processes increase, but the steps
involving inverse β decay (particularly step 5),
which do not depend on the state variables, do
not and therefore limit the reaction rate. So there
is an upper limit to the rate at which the CNO
cycle can produce energy independent of the
conditions which prevail in the star.
Collapse
Collapse
of
of
protostar
protostar
If the cloud is gravitationally confined
within a sphere of the Jeans' length, the
cloud will experience rapid core collapse
until it becomes optically thick. If the
outer regions contain dust, they will
absorb the radiation produced by the core
contraction and reradiate it in the infrared
part of the spectrum. After the initial free-
fall collapse of a 1M
⊙
cloud, the inner core
will be about 5 AU surrounded by an outer
envelope about 20000 AU.
Jeans length
Jeans length
2
2
4
3
2
2
3
c
h
c
R
kT
GM
m
R
π
ρ
μ
⎛
⎞⎛
⎞
≤
⎜
⎟⎜
⎟
⎝
⎠
⎝
⎠
3
h
J
GM m
R
kT
μ
=
When the core temperature reaches
about 2000 K, the
2
H molecules
dissociate, thereby absorbing a
significant amount of the internal
energy. The loss of this energy
initiates a second core collapse of
about 10 percent of the mass with
the remainder following as a "heavy
rain".
After a time, sufficient matter has
rained out of the cloud, and the
cloud becomes relatively transparent
to radiation and falls freely to the
surface, producing a fully convective
star. While this scenario seems
relatively secure for low mass stars
(i.e., around 1M
⊙
), difficulties are
encountered with the more massive
stars.
Contraction onto Main Sequence
Once the protostar has become opaque to
radiation, the energy liberated by the
gravitational collapse of the cloud cannot
escape to interstellar space. The collapse
slows down dramatically and the future
contraction is limited by the star's ability
to transport and radiate the energy away
into space. Hayashi showed that there
would be a period after the central regions
became opaque to radiation during which
the star would be in convective
equilibrium.
Convection
Convection
by
by
buoyance
buoyance
δ
T>0
Hayashi Evolutionary Tracks
Once convection is established, it is
incredibly efficient at transporting energy.
Thus, as long as there are no sources of
energy other than gravitation, the future
contraction is limited by the star's ability
to radiate energy into space rather than
by its ability to transport energy to the
surface. The structure of a fully convective
star is that of a polytrope of index n =
1.5. We may combine these two
properties of the star to approximately
trace the path it must take on the
Hertzsprung-Russell diagram.
Some of the energy generated by
contraction will be released from the
stellar surface in the form of
photons. As long as the process is
slow, the virial theorem will hold and
<T> ≈ 0. Thus
½ <Ω> = - <U>
This implies that one-half of the
change in the gravitational energy
will go into raising the internal
kinetic energy of the gas. The other
half is available to be radiated away.
Therefore
Therefore
Since the
luminosity is
related to the
surface
parameters by
(def. of effective
temperature)
the change in the
luminosity with
respect to the
radius will be
2
2
2
1
1
2
2
d
GM
GM dR
L
dt
R
R
dt
⎛
⎞
=
= −
⎜
⎟
⎝
⎠
2
4
4
e
L
R
T
π σ
∗
=
4
2
e
e
dT
dL
L
L
dR
T dR
R
∗
∗
∗
=
+
As long as there are no sources of
energy other than gravitation, the
contraction is limited by the star's
ability to radiate energy into space
rather than by its ability to transport
energy to the surface.
So as long as the stellar luminosity is
determined solely by the change in
gravity, and the energy loss is
dictated by the atmosphere, we
might expect that T
e
remains
unchanged.
Thus dT
e
/dR
*
is approximately zero,
and we expect the star to move
vertically down the Hertzsprung-
Russell (H-R) diagram with the
luminosity changing roughly as R
*
2
until the internal conditions within
the star change. For the Hayashi
tracks
ln
0,
2
ln
e
e
dT
dT
d
L
dR
dL
d
R
∗
∗
=
=
=
log L
log T
e
While the location of a specific track
depends on the atomic physics of the
photosphere, the relative location of
these tracks for stars of differing
mass is determined by the fact that
the star is a polytrope of index n =
3/2. From the polytropic mass-radius
relation
1/ 3
log
1
log
3
d
R
M
R
const
d
M
∗
∗
=
⇒
= −
If we inquire as to the spacing of the vertical
Hayashi tracks in the H-R diagram, then we
can look for the effective temperatures for
stars of different mass but at the same
luminosity.
2
4
4
e
L
R
T
π σ
∗
=
4
2
e
e
dT
dL
L
L
dR
T dR
R
∗
∗
∗
=
+
2
4
4
e
L
R
T
π σ
∗
=
4
2
e
e
dT
dL
L
L
dR
T dR
R
∗
∗
∗
=
+
2
4
e
e
dT
dR
dL
L
L
dM
R dM
T dM
∗
∗
+
⇒
=
log
1
log
6
e
d
T
d
M
=
This extremely weak dependence of the
effective temperature on mass means that
we should expect all the Hayashi tracks
for the majority of main sequence stars to
be bunched on the right side of the H-R
diagram. Since the star is assumed to be
radiating as a blackbody of a given T
e
and
is in convective equilibrium, no other
stellar configuration could lose its energy
more efficiently. Thus no stars should lie
to the right of the Hayashi track of the
appropriate mass on the H-R diagram;
this is known as the Hayashi zone of
avoidance.
We may use arguments like these to
describe the path of the star on the
H-R diagram followed by a
gravitationally contracting fully
convective star. This contraction will
continue until conditions in the
interior change as a result of
continued contraction.
As the star moves down the Hayashi
track, the internal temperature
increases so that T = µM/R.
At some point, depending on the
dominant source of opacity, and
convection will cease.
At that point the mode of collapse
will change because the primary
barrier to energy loss will move from
the photosphere to the interior and
the diffusion of radiant energy.
As the star continues to shine, the gravitational
energy continues to become more negative,
and to balance it, in accord with the virial
theorem, the internal energy continues to rise.
This results in a slow but steady increase in the
temperature gradient which results in a steady
increase in the luminosity as the radiative flux
increases. This increased luminosity combined
with the ever-declining radius produces a
sharply rising surface temperature as the
photosphere attempts to accommodate the
increased luminosity. This will yield tracks on
the H-R diagram which move sharply to the left
while rising slightly.
We may quantify this by asking how
the luminosity changes in time.
If we further invoke the virial
theorem and require that the
contraction proceed so as to keep
the second derivative of the moment
of inertia equal to zero, then
2
2
2
2
2
2
2
1
1
2
2
2
dR
d R
dL
d
GM
dt
dt
R
R
dt
dt
∗
∗
∗
∗
⎛
⎞
Ω
⎛
⎞
= −
= −
−
+
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
(
)
2
2
2
2
2
2
2
2
0
0
dR
d R
d I
d
MR
R
dt
dt
dt
dt
α
∗
∗
∗
∗
⎛
⎞
=
=
⇒
+
=
⎜
⎟
⎝
⎠
Thus
Thus
(
(
Henyey
Henyey
evolution
evolution
)
)
Moreover
Moreover
log
3
log
d
L
d
R
∗
= −
2
4
e
e
dT
dR
dL
L
L
dM
R dM
T d
R dM
M dR
M
∗
∗
∗
∗
⎛
⎞
⎜
⎟
⎝
=
⎠
+
⇒
log
5
log
12
,
log
4
log
5
e
e
d
T
d
L
d
R
d
T
∗
= −
=
