The Hayashi track is a luminositytemperature relationship obeyed by infant
stars of less than 3 solar masses in the premainsequence phase of stellar
evolution. On the
HertzsprungRussell diagram, which plots luminosity against temperature,
the track is a nearly vertical curve.
After a protostar ends its phase of rapid contraction and becomes a
T Tauri star, it is extremely luminous. The star then follows the Hayashi
track downwards, becoming several times less luminous but staying at roughly
the same
surface temperature, until either a radiative zone develops, at which point
the star starts following the Henyey track, or nuclear fusion begins,
marking the beginning of the main sequence.
The shape and position of the Hayashi
track on the HR diagram depends on the star's mass and chemical composition.
For solarmass stars, the track lies at a temperature of roughly 4000K.
Stars on the track are nearly fully convective and have their opacity
dominated by the H ion. Stars less than 0.5 solar masses are fully convective
even on the main sequence, but their opacity begins to be dominated by
Kramers' opacity law after nuclear fusion begins, thus moving them off the
Hayashi track. Stars between 0.5 and 3 solar masses develop a radiative
zone prior to reaching the main sequence. Stars between 3 and 10 solar masses
are fully radiative at the beginning of the premainsequence. Even heavier
stars are born onto the main sequence, with no PMS evolution.^{[1]}
At a end of a low or intermediatemass star's life, the star follows an
analogue of the Hayashi track, but in reverse—it increases in luminosity,
expands, and stays at roughly the same temperature, eventually becoming a
red giant.
History
In 1961, Professor Chushiro Hayashi published two papers^{[2]}^{[3]} that led to the
concept of the premainsequence and form the basis of the modern
understanding of early stellar evolution. Hayashi realized that the existing
model, in which stars are assumed to be in radiative equilibrium with no substantial convection zone, cannot explain the shape of the
red giant branch.^{[4]} He therefore replaced the model by including the effects
of a thick convection zones on a star's interior.
A few years prior, Osterbrock proposed deep convection zones with efficient
convection, analyzing them
using the opacity of H ions (the dominant opacity source in cool atmosphres)
in temperatures below 5000K. However, the earliest numerical models of
Sunlike stars did not follow up on this work and continued to assume radiative
equilibrium.^{[1]}
In his 1961 papers, Hayashi showed that the convective envelope of a star is
determined by:
$E\; =\; 4\backslash pi\; G^\{3/2\}(\backslash mu\; H/k)^\{5/2\}M^\{1/2\}R^\{3/2\}P/T^\{5/2\}$
where E is unitless, and not the energy. Modelling stars as
polytropes with index 3/2in other words, assuming they follow a
pressuredensity relationship of $P=K\backslash rho^\{5/3\}$—he found that E=45
is the maximum for a quasistatic star. If a star is not contracting
rapidly, E=45 defines a curve on the HR diagram, to the right of which the star
cannot exist. He then computed the evolutionary tracks and isochrones
(luminositytemperature distributions of stars at a given age)
for a variety of stellar masses and noted that NGC2264, a very young star
cluster, fits the isochrones well. In particular, he calculated much lower
ages for solartype stars in NGC2264 and predicted that these stars were
rapidlycontracting T Tauri stars.
In 1962, Hayashi published a 183page review of stellar evolution. Here, he
discussed the evolution of stars born in the forbidden region. These stars
rapidly contract due to gravity before settling to a quasistatic, fully
convective state on the Hayashi tracks.
In 1965, numerical models by Iben and Ezer & Cameron realistically simulated
premainsequence evolution, including the Henyey track that stars follow
after leaving the Hayashi track. These standard PMS tracks can still be found
in textbooks on stellar evolution.
Forbidden zone and Hayashi limit
The forbidden zone is the region on the HR diagram to the right of the Hayashi
track where no star in hydrostatic equilibrium, even those that are
partially or fully radiative, can be. Newborn protostars start out in
this zone, but are not in hydrostatic equilibrium and will rapidly move towards
the Hayashi track.
Because stars emit light via blackbody radiation, the power per unit surface area
they emit is given by the StefanBoltzmann law:
 $j^\{\backslash star\}\; =\; \backslash sigma\; T^4$
The star's luminosity is therefore given by:
 $L\; =\; 4\backslash pi\; R^2\backslash sigma\; T^4$
For a given L, a lower temperature implies a larger radius, and vice versa.
Thus, the Hayashi track separates the HR diagram into two regions: the allowed
region to the left, with high temperatures and smaller radii for each
luminosity, and the forbidden region to the right, with lower temperatures and
correspondingly higher radii. The Hayashi limit can refer to either the
lower bound in temperature or the upper bound on radius defined by the Hayashi
track.
The region to the right is forbidden because it can be shown that a star in
the region must have a temperature gradient of:
 $\backslash frac\{d\; \backslash ln\; T\}\{d\; \backslash ln\; P\}\; >\; 0.4$
where $\backslash frac\{d\; \backslash ln\; T\}\{d\; \backslash ln\; P\}\; =\; 0.4$ for a monatomic ideal gas
undergoing adiabatic
expansion or contraction. A temperature gradient greater than 0.4 is therefore
called superadiabatic.
Consider a star with a superadiabatic gradient. Imagine a parcel of gas that
starts at radial position r, but moves upwards to r+dr in a sufficiently short
time that it exchanges negligible heat with its surroundings—in other words,
the process is adiabatic. The pressure of the
surroundings, as well as that of the parcel, decreases by some amount dP.
The parcel's temperature changes by $dT\; =\; 0.4\backslash frac\{T\}\{P\}dP$. The
temperature of the surroundings also decreases, but by some amount dT' that is
greater than dT. The parcel therefore ends up being hotter than its
surroundings. Since
the ideal gas law can be written $P\; =\; \backslash frac\{\backslash rho\; RT\}\{\backslash mu\}$, a
higher temperature implies a lower density at the same pressure. The parcel
is therefore also less dense than its surroundings. This will cause it to rise
even more, and the parcel will become even less dense than its new surroundings.
Clearly, this situation is not stable. In fact, a superadiabatic gradient
causes convection. Convection tends to lower the temperature gradient
because the rising parcel of gas will eventually be dispersed, dumping its
excess thermal and kinetic energy into its surroundings and heating up said
surroundings. In stars, the convection process is known to be highly efficient,
with a typical $\backslash frac\{d\; \backslash ln\; T\}\{d\; \backslash ln\; P\}$ that only exceeds the
adiabatic gradient by 1 part in 10 million.^{[5]}
If a star is placed in the forbidden zone, with a temperature gradient much
greater than 0.4, it will experience rapid convection that brings the gradient
down. Since this convection will drastically the star's pressure and
temperature distribution, the star is not in hydrostatic equilibrium, and
will contract until it is.
A star far to the left of the Hayashi track has a temperature gradient smaller
than adiabatic.
This means that if a parcel of gas rises a tiny bit, it will be more dense than
its surroundings and sink back to where it came from. Convection therefore
does not occur, and almost all energy output is carried radiatively.
Star formation
Stars form when small regions of a giant molecular cloud collapse under
their own gravity, becoming protostars. The collapse releases gravitational
energy, which heats up the protostar. This process occurs on the
freefall timescale, which is roughly 100,000 years for
solarmass protostars, and ends when the protostar reaches approximately
4000 K. This is known as the Hayashi boundary, and at this point, the protostar
is on the Hayashi track. At this point, they are known as T Tauri stars and
continue to contract, but much more slowly. As they contract, they decrease in
luminosity because less surface area becomes available for emitting light. The
Hayashi track gives the resulting change in temperature, which will be minimal
compared to the change in luminosity because the Hayashi track is nearly
vertical. In other words, on the HR diagram, a T Tauri star starts out on the
Hayashi track with a high luminosity and moves downward along the track as time
passes.
The Hayashi track describes a fully convective star. This is
a good approximation for very young premainsequence stars they are still cool
and highly opaque, so that radiative transport is insufficient
to carry away the generated energy and convection must occur. Stars lighter
than 0.5 solar masses remain fully convective, and therefore remain on the Hayashi track, throughout their premainsequence stage,
joining the main sequence at the bottom of the Hayashi track. Stars heavier
than 0.5 solar masses have higher interior temperatures, which decreases their
central opacity and allows radiation to away large amounts of energy. This
allows a radiative zone to develop around the star's core. The star is then
no longer on the Hayashi track, and experiences a period of rapidly increasing
temperature at nearly constant luminosity. This is called the
Henyey track, and ends when temperatures are high enough to ignite hydrogen
fusion in the core. The star is then on the main sequence.
Lowermass stars follow the Hayashi track until the track intersects with the
main sequence, at which point hydrogen fusion begins and the star follows the
main sequence. Even lowermass 'stars' never achieve the conditions necessary
to fuse hydrogen and become brown dwarfs.
Derivation
The exact shape and position of the Hayashi track can only be computed
numerically using computer models. Nevertheless, we can make an extremely
crude analytical argument that captures most of the track's properties. The
following derivation loosely follows that of Kippenhahn, Weigert, and Weiss in
Stellar Structure and Evolution.^{[5]}
In our
simple model, a star is assumed to consist of a fully convective interior
inside of a fully radiative atmosphere.
The convective interior is assumed to be an ideal monatomic gas with a
perfectly adiabatic temperature gradient:
 $\backslash frac\{d\backslash ln\{T\}\}\{d\backslash ln\{P\}\}\; =\; 0.4$
This quantity is sometimes labelled $\backslash nabla$. The following
adiabatic equation therefore holds true for the entire interior:
 $P^\{1\backslash gamma\}T^\{\backslash gamma\}\; =\; C$
where $\backslash gamma$ is the adiabatic gamma, which is 5/3 for an ideal
monatomic gas. The ideal gas law says:
 $P\; =\; NkT/V$
 $=\; \backslash frac\{\backslash rho\; kT\}\{\backslash mu\; H\}$
 $=\; (\backslash frac\{k\backslash rho\; C\}\{\backslash mu\; H\})^\backslash gamma$
where $\backslash mu$ is the molecular weight per particle and H is (to a very good
approximation) the mass of a hydrogen atom. This equation represents a
polytrope of index 1.5, since a polytrope is defined by
$P\; =\; K\backslash rho^\{1\; +\; 1/n\}$, where n=1.5 is the polytropic index. Applying
the equation to the center of the star gives:
$P\_c\; =\; (\backslash frac\{k\backslash rho\_c\; C\}\{\backslash mu\; H\})^\backslash gamma$
We can solve for C:
 $C\; =\; \backslash frac\{\backslash mu\; HP\_c^\{1/\backslash gamma\}\}\{\backslash rho\_c\; k\}$
But for any polytrope, $P\_c\; =\; W\_n\backslash frac\{GM^2\}\{R^4\}$,
$\backslash rho\_c\; =\; K\_n\backslash rho\_\{avg\}$, and
$R^\{\backslash frac\{3n\}\{n\}\}\; M^\{\backslash frac\{n1\}\{n\}\}\; =\; \backslash frac\{K\}\{GN\_n\}$. $W\_n,\; K\_n,\; N\_n,$ and K are all constants independent of pressure and density,
and the average density is defined as
$\backslash rho\_\{avg\}\; \backslash equiv\; \backslash frac\{M\}\{4/3\; \backslash pi\; R^3\}$. Plugging all 3 equations
into the equation for C, we have:
 $C\; \backslash sim\; M^\{2\backslash gamma\}\; R^\{3\backslash gamma\; \; 4\}$
where all multiplicative constants have been ignored. Recall that our original
definition of C was:
 $P^\{1\backslash gamma\}T^\backslash gamma\; =\; C$
We therefore have, for any star of mass M and radius R:
 $P^\{1\backslash gamma\}T^\backslash gamma\; \backslash sim\; M^\{2\; \; \backslash gamma\}\; R^\{3\backslash gamma\; \; 4\}$

We need another relationship between P, T, M, and R, in order to eliminate P.
This relationship will come from the atmosphere model.
The atmosphere is assumed to be thin, with average opacity k. Opacity is
defined to be optical depth divided by density. Thus, by definition, the
optical depth of the stellar surface, also called the photosphere, is:
 $\backslash frac\{d\backslash tau\}\{dr\}\; =\; k\backslash rho$
 $\backslash tau\; =\; \backslash int\_R^\backslash infty\; k\backslash rho\; dr$
 $=\; k\backslash int\_R^\backslash infty\; \backslash rho\; dr$
where R is the stellar radius, also known as the position of the photosphere.
The pressure at the surface is:
 $P\_0\; =\; \backslash int\_R^\backslash infty\; g\backslash rho\; dr$
 $=\; \backslash frac\{GM\}\{R^2\}\backslash int\_R^\backslash infty\; \backslash rho\; dr$
 $=\; \backslash frac\{GM\backslash tau\}\{kR^2\}$
The optical depth at the photosphere turns out to be $\backslash tau\; =\; 2/3$. By
definition, the temperature of the photosphere is $T\; =\; T\_\{eff\}$ where effective
temperature is given by $L\; =\; 4\backslash pi\; R^2T\_\{eff\}^4$. Therefore,
the pressure is:
 $P\_0\; =\; \backslash frac\{GM\}\{R^2\}\backslash frac\{2\backslash tau\}\{3k\}$
We can approximate the opacity to be:
 $k\; =\; k\_0P^aT^b$
where a=1, b=3. Plugging this into the pressure equation, we get:
 $P\_0\; =\; const(\backslash frac\{M\}\{R^2T\_\{eff\}^b\})^\{\backslash frac\{1\}\{a+1\}\}$

) 
2}}
Finally, we need to eliminate R and introduce L, the luminosity. This can be
done with the equation:
 $L\; =\; 4\backslash pi\; R^2T\_\{eff\}^4$

+ const
3}}
Equation 1 and 2 can now be combined by
setting $T=T\_\{eff\}$ and $P=P\_0$ in Equation 1, then eliminating $P\_0$.
R can be eliminated using Equation 3. After some algebra,
and after setting $\backslash gamma\; =\; 5/3$, we get:
 $\backslash ln\{T\_\{eff\}\}\; =\; A\backslash ln\{L\}\; +\; B\backslash ln\{M\}\; +\; const$
where
 $A\; =\; \backslash frac\{0.75a0.25\}\{5.5a+b+1.5\}$
 $B\; =\; \backslash frac\{0.5a\; +\; 1.5\}\{5.5a+b+1.5\}$
In cool stellar atmospheres (T < 5000 K) like those of newborn stars,
the dominant source of opacity is the H ion, for which
$a\; \backslash approx\; 1$ and $b\; \backslash approx\; 3$, we get
$A\; =\; 0.05$ and $B\; =\; 0.2$.
Since A is much smaller than
1, the Hayashi track is extremely steep: if the luminosity changes by a factor
of 2, the temperature ony changes by 4 percent. The fact that B is positive
indicates that the Hayashi track shifts left on the HR diagram, towards higher
temperatures, as mass increases. Although this model is extremely crude, these
qualitative observations are fully supported by numerical simulations.
At high temperatures, the atmosphere's opacity begins to be dominated by
Kramers' opacity law instead of the H ion, with a=1 and b=4.5 In that
case, A=0.2 in our crude model, far higher than 0.05, and the star is no longer
on the Hayashi track.
In Stellar Interiors, Hansen, Kawaler, and Trimble go through a similar
derivation without neglecting multiplicative constants,^{[6]}
and arrived at:
 $T\_\{eff\}\; =\; (2600\; K)\backslash mu^\{13/51\}(\backslash frac\{M\}\{M\_\{\backslash odot\}\})^\{7/51\}(\backslash frac\{L\}\{L\_\{\backslash odot\}\})^\{1/102\}$
where $\backslash mu$ is the molecular weight per particle. The authors note that the coefficient of 2600K is too
low—it should be around 4000K—but this equation nevertheless shows that
temperature is nearly independent of luminosity.
Numerical results
The diagram at the top of this article shows numerically computed stellar
evolution
tracks for various masses. The vertical portions of each track is the Hayashi
track. The endpoints of each track lie on the main sequence.
The horizontal segments for highermass stars show the Henyey track.
It is approximately true that:
 $\backslash frac\{\backslash partial\; \backslash ln\{T\_\{eff\}\}\}\{\backslash partial\; \backslash ln\{M\}\}\; \backslash approx\; 0.1$.
The diagram to the right shows how Hayashi tracks change with changes in
chemical composition. Z is the star's metallicity, the mass fraction not
accounted for by hydrogen or helium. For any given hydrogen mass fraction,
increasing Z leads to increasing molecular weight. The dependence of
temperature on molecular weight is extremely steep—it is approximately
 $\backslash frac\{\backslash partial\; \backslash ln\{T\_\{eff\}\}\}\{\backslash partial\; \backslash ln\{\backslash mu\}\}\; \backslash approx\; 26$.
Decreasing Z by a factor of 10 shifts the track right, changing
$\backslash ln\{T\_\{eff\}\}$ by about 0.05.
Chemical composition affects the Hayashi track in a few ways. The
track depends strongly on the atmosphere's opacity, and this opacity is
dominated by the H ion. The abundance of the H ion is proportional to the
density of free electrons, which, in turn, is higher if there are more metals
because metals are easier to ionize than hydrogen or helium.
Observational status
Observational evidence of the Hayashi track comes from colormagnitude plots—the observational equivalent of HR diagrams—of young star clusters.^{[1]} For
Hayashi, NGC 2264 provided the first evidence of a population of contracting
stars. In 2012, data from NGC 2264 was reanalyzed to account for dust
reddening and extinction. The resulting colormagnitude plot is shown at
right.
In the upper diagram, the isochrones are curves along which stars of a certain
age
are expected to lie, assuming that all stars evolve along the Hayashi track.
An isochrone is created by taking stars of every conceivable mass, evolving
them forwards to the same age, and plotting all of them on the colormagnitude
diagram.
Most of the stars in NGC 2264 are already on the main sequence (black line),
but a substantial population lies between the isochrones for 3.2 million and 5
million years, indicating that the cluster is 3.25 million years old and a
large population of T Tauri stars is still on their respective Hayashi tracks.
Similar results have been obtained for NGC 6530, IC 5146, and NGC 6611.^{[1]}
The lower diagram shows Hayashi tracks for various masses, along with T Tauri
observations collected from a variety of sources. Note the bold curve to
the right, representing a stellar birthline. Even though some Hayashi tracks
theoretically extend above the birthline, few stars are above it. In effect,
stars are 'born' onto the birthline before evolving downwards along their
respective Hayashi tracks.
The birthline exists because stars form from overdense cores of giant molecular
clouds in an insideout manner.^{[4]} That is, a small central region first
collapses in on itself while the outer shell is still nearly static. The outer
envelope then accretes onto the central protostar. Before the accretion is
over, the protostar is hidden from view, and therefore not plotted on the
colormagnitude diagram. When the envelope finishes accreting, the star is
revealed and appears on the birthline.
References
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