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| Line 1: |
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| Low-mass stars have nearly vertical evolution tracks until they arrive on the main sequence. For more massive stars, the Hayashi track bends lefts into the [[Henyey track]]. Even more massive stars are born directly onto the Henyey track.<br />
| |
| The end (leftmost point) of every track is labeled with the star's mass in solar masses, and represents its position on the main sequence.
| |
| The red curves labeled in years are isochrones at the given ages. In other words, stars <math>10^5</math> years old lie along the curve labeled <math>10^5</math>, and similarly for the other 3 isochrones.]]
| |
|
| |
|
| The '''Hayashi track''' is a luminosity-temperature relationship obeyed by infant
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| stars of less than 3 solar masses in the [[pre-main-sequence]] phase of stellar
| |
| evolution. On the
| |
| [[Hertzsprung-Russell]] 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
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| track on the HR diagram depends on the star's mass and chemical composition.
| |
| For solar-mass 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 pre-main-sequence. Even heavier
| |
| stars are born onto the main sequence, with no PMS evolution.<ref name=Palla2012/>
| |
| | |
| At a end of a low- or intermediate-mass 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<ref name=hayashi1/><ref name=hayashi2/> that led to the
| |
| concept of the [[pre-main-sequence]] 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]].<ref name=Stahler1988/> He therefore replaced the model by including the effects
| |
| of a thick [[convection zone]]s 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
| |
| Sun-like stars did not follow up on this work and continued to assume radiative
| |
| equilibrium.<ref name=Palla2012/>
| |
| | |
| In his 1961 papers, Hayashi showed that the convective envelope of a star is
| |
| determined by:
| |
| | |
| <math>
| |
| E = 4\pi G^{3/2}(\mu H/k)^{5/2}M^{1/2}R^{3/2}P/T^{5/2}</math>
| |
| | |
| where E is unitless, and not the [[energy]]. Modelling stars as
| |
| [[polytrope]]s with index 3/2--in other words, assuming they follow a
| |
| pressure-density relationship of <math>P=K\rho^{5/3}</math>—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]]
| |
| (luminosity-temperature 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 solar-type stars in NGC2264 and predicted that these stars were
| |
| rapidly-contracting [[T Tauri stars]].
| |
| | |
| In 1962, Hayashi published a 183-page 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
| |
| pre-main-sequence 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 [[Stefan-Boltzmann law]]:
| |
| | |
| :<math>j^{\star} = \sigma T^4</math>
| |
| The star's luminosity is therefore given by:
| |
| :<math>L = 4\pi R^2\sigma T^4</math>
| |
| | |
| 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:
| |
| :<math>\frac{d \ln T}{d \ln P} > 0.4</math>
| |
| where <math>\frac{d \ln T}{d \ln P} = 0.4</math> 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 <math>dT = 0.4\frac{T}{P}dP</math>. 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 <math>P = \frac{\rho RT}{\mu}</math>, 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 <math>\frac{d \ln T}{d \ln P}</math> that only exceeds the
| |
| adiabatic gradient by 1 part in 10 million.<ref name=struct_evol/>
| |
| | |
| 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
| |
| [[free-fall timescale]], which is roughly 100,000 years for
| |
| solar-mass 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 pre-main-sequence stars they are still cool
| |
| and highly [[opacity|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 pre-main-sequence 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]].
| |
| | |
| Lower-mass 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 lower-mass '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''.<ref name=struct_evol/>
| |
| | |
| 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:
| |
| | |
| :<math>\frac{d\ln{T}}{d\ln{P}} = 0.4</math>
| |
| | |
| This quantity is sometimes labelled <math>\nabla</math>. The following
| |
| adiabatic equation therefore holds true for the entire interior:
| |
| | |
| :<math>P^{1-\gamma}T^{\gamma} = C</math>
| |
| | |
| where <math>\gamma</math> is the [[adiabatic gamma]], which is 5/3 for an ideal
| |
| monatomic gas. The ideal gas law says:
| |
| | |
| :<math>P = NkT/V</math>
| |
| :<math>= \frac{\rho kT}{\mu H}</math>
| |
| :<math>= (\frac{k\rho C}{\mu H})^\gamma</math>
| |
| | |
| where <math>\mu</math> 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
| |
| <math>P = K\rho^{1 + 1/n}</math>, where n=1.5 is the polytropic index. Applying
| |
| the equation to the center of the star gives:
| |
| <math>P_c = (\frac{k\rho_c C}{\mu H})^\gamma</math>
| |
| We can solve for C:
| |
| :<math>C = \frac{\mu HP_c^{1/\gamma}}{\rho_c k}</math>
| |
| | |
| But for any polytrope, <math>P_c = W_n\frac{GM^2}{R^4}</math>,
| |
| <math>\rho_c = K_n\rho_{avg}</math>, and
| |
| <math>R^{\frac{3-n}{n}} M^{\frac{n-1}{n}} = \frac{K}{GN_n}</math>. <math>W_n, K_n,
| |
| N_n,</math> and K are all constants independent of pressure and density,
| |
| and the average density is defined as
| |
| <math>\rho_{avg} \equiv \frac{M}{4/3 \pi R^3}</math>. Plugging all 3 equations
| |
| into the equation for C, we have:
| |
| | |
| :<math> C \sim M^{2-\gamma} R^{3\gamma - 4}</math>
| |
| | |
| where all multiplicative constants have been ignored. Recall that our original
| |
| definition of C was:
| |
| | |
| :<math>P^{1-\gamma}T^\gamma = C</math>
| |
| | |
| We therefore have, for any star of mass M and radius R:
| |
| | |
| :<math>P^{1-\gamma}T^\gamma \sim M^{2 - \gamma} R^{3\gamma - 4}</math>
| |
| {{NumBlk|:|<math>\ln P = \frac{2-\gamma}{1-\gamma}\ln M + \frac{3\gamma - 4}
| |
| {1 - \gamma}\ln R - \gamma \ln T</math>|{{EquationRef|1}}}}
| |
| | |
| 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:
| |
| | |
| :<math>\frac{d\tau}{dr} = k\rho</math>
| |
| :<math>\tau = \int_R^\infty k\rho dr</math>
| |
| :<math> = k\int_R^\infty \rho dr</math>
| |
| | |
| where R is the stellar radius, also known as the position of the photosphere.
| |
| The pressure at the surface is:
| |
| | |
| :<math>P_0 = \int_R^\infty g\rho dr</math>
| |
| :<math>= \frac{GM}{R^2}\int_R^\infty \rho dr</math>
| |
| :<math> = \frac{GM\tau}{kR^2}</math>
| |
| | |
| The optical depth at the photosphere turns out to be <math>\tau = 2/3</math>. By
| |
| definition, the temperature of the photosphere is <math>T = T_{eff}</math> where effective
| |
| temperature is given by <math>L = 4\pi R^2T_{eff}^4</math>. Therefore,
| |
| the pressure is:
| |
| | |
| :<math>P_0 = \frac{GM}{R^2}\frac{2\tau}{3k}</math>
| |
| | |
| We can approximate the opacity to be:
| |
| | |
| :<math>k = k_0P^aT^b</math>
| |
| | |
| where a=1, b=3. Plugging this into the pressure equation, we get:
| |
| | |
| :<math>P_0 = const(\frac{M}{R^2T_{eff}^b})^{\frac{1}{a+1}}</math>
| |
| :{{NumBlk|:|<math>\ln P_0 = \ln{const} + \frac{1}{a+1}(\ln{M} - 2\ln{R} -
| |
| b\ln{T_{eff}}) </math>|{{EquationRef|2}}}}
| |
| | |
| Finally, we need to eliminate R and introduce L, the luminosity. This can be
| |
| done with the equation:
| |
| | |
| :<math>L = 4\pi R^2T_{eff}^4</math>
| |
| :{{NumBlk|:|<math>\ln{R} = 0.5\ln{L} - 2\ln{T_{eff}} + const
| |
| </math>|{{EquationRef|3}}}}
| |
| | |
| Equation {{EquationNote|1}} and {{EquationNote|2}} can now be combined by
| |
| setting <math>T=T_{eff}</math> and <math>P=P_0</math> in Equation 1, then eliminating <math>P_0</math>.
| |
| R can be eliminated using Equation {{EquationNote|3}}. After some algebra,
| |
| and after setting <math>\gamma = 5/3</math>, we get:
| |
| | |
| :<math>\ln{T_{eff}} = A\ln{L} + B\ln{M} + const</math>
| |
| | |
| where
| |
| :<math>A = \frac{0.75a-0.25}{5.5a+b+1.5}</math>
| |
| :<math>B = \frac{0.5a + 1.5}{5.5a+b+1.5}</math>
| |
| | |
| In cool stellar atmospheres (T < 5000 K) like those of newborn stars,
| |
| the dominant source of opacity is the H- ion, for which
| |
| <math>a \approx 1</math> and <math>b \approx 3</math>, we get
| |
| <math>A = 0.05</math> and <math>B = 0.2</math>.
| |
| | |
| 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,<ref name=interiors/>
| |
| and arrived at:
| |
| | |
| :<math>T_{eff} = (2600 K)\mu^{13/51}(\frac{M}{M_{\odot}})^{7/51}(\frac{L}{L_{\odot}})^{1/102}</math> | |
| | |
| where <math>\mu</math> 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==
| |
| [[File:Hayashi composition.png|thumb|Hayashi tracks of a 0.8 solar mass star with helium mass fraction 0.245, for 3 different metallicities]]
| |
| | |
| 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 higher-mass stars show the [[Henyey track]].
| |
| | |
| It is approximately true that:
| |
| :<math>\frac{\partial \ln{T_{eff}}}{\partial \ln{M}} \approx 0.1</math>.
| |
| | |
| 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
| |
| :<math>\frac{\partial \ln{T_{eff}}}{\partial \ln{\mu}} \approx -26</math>.
| |
| Decreasing Z by a factor of 10 shifts the track right, changing
| |
| <math>\ln{T_{eff}}</math> 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==
| |
| [[File:Ngc 2264.png|thumb|The young star cluster NGC 2264, with a large number of T Tauri stars contracting towards the main sequence. The solid line represents the main sequence, while the two lines above that are the <math>10^{6.5}</math> yr (upper) and <math>10^{6.7}</math> yr (lower) isochrones.]]
| |
| | |
| Observational evidence of the Hayashi track comes from color-magnitude plots—the observational equivalent of HR diagrams—of young star clusters.<ref name=Palla2012/> For
| |
| Hayashi, [[NGC 2264]] provided the first evidence of a population of contracting
| |
| stars. In 2012, data from NGC 2264 was re-analyzed to account for dust
| |
| reddening and extinction. The resulting color-magnitude 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 color-magnitude
| |
| 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 3.2-5 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.<ref name=Palla2012/>
| |
| | |
| [[File:T Tauri tracks.png|thumb|The numbered curves show the Hayashi tracks of stars of that mass (in solar masses). The small circles represent observational data of T Tauri stars. The bold curve to the right is the birthline, above which few stars exist.]]
| |
| | |
| 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.
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| | |
| The birthline exists because stars form from overdense cores of giant molecular
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| clouds in an inside-out manner.<ref name=Stahler1988/> That is, a small central region first
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| collapses in on itself while the outer shell is still nearly static. The outer
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| envelope then accretes onto the central protostar. Before the accretion is
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| over, the protostar is hidden from view, and therefore not plotted on the
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| color-magnitude diagram. When the envelope finishes accreting, the star is
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| revealed and appears on the birthline.
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| | |
| ==References==
| |
| {{Reflist|colwidth=30em|refs=
| |
| <ref name=struct_evol>{{Cite book | title = Stellar structure and evolution | year = 2012 | publisher = Springer | location = New York | isbn = 978-3-642-30255-8 | pages = 271–282}}</ref>
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| | |
| <ref name=interiors>{{Cite book | last1 = Hansen | first1 = Carl J. | last2 = Kawaler | first2 = Steven D. | last3 = Trimble | first3 = Virginia. | title = Stellar interiors : physical principles, structure, and evolution | year = 2004 | publisher = Springer | location = New York | isbn = 978-0-387-20089-7 | pages = 367–374}}</ref>
| |
| | |
| <ref name="Palla2012">{{cite journal|last1=Palla|first1=Francesco|title=1961–2011: Fifty years of Hayashi tracks|year=2012|pages=22–29|issn=0094243X|doi=10.1063/1.4754323}}</ref>
| |
| | |
| <ref name="Stahler1988">{{cite journal|last1=Stahler|first1=Steven W.|title=Understanding young stars - A history|journal=Publications of the Astronomical Society of the Pacific|volume=100|year=1988|pages=1474|issn=0004-6280|doi=10.1086/132352|bibcode = 1988PASP..100.1474S }}</ref>
| |
| | |
| <ref name="hayashi1">{{Cite journal
| |
| | last = Hayashi
| |
| | first = C.
| |
| | title = Stellar evolution in early phases of gravitational contraction
| |
| | journal = Publ. Astron. Soc. Jap.
| |
| | volume=13
| |
| | pages=450–452
| |
| | year = 1961
| |
| |bibcode = 1961PASJ...13..450H }}</ref>
| |
| | |
| <ref name="hayashi2">{{Cite journal
| |
| | last = Hayashi
| |
| | first = C.
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| | title = The Outer Envelope of Giant Stars with Surface Convection Zone
| |
| | journal = Publ. Astron. Soc. Jap.
| |
| | volume=13
| |
| | pages=442–449
| |
| | year = 1961
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| |bibcode = 1961PASJ...13..450H }}</ref>
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| | |
| }}
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| | |
| {{Star}}
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| [[Category:Star formation]]
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| [[Category:Hertzsprung–Russell classifications]]
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| [[Category:Astrophysics]]
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