Hayashi track

A Hayashi track is the nearly vertical path of a fully convective pre-main-sequence star on the Hertzsprung-Russell diagram. Photospheric opacity fixes a weak relation between luminosity and effective temperature.

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Hayashi relation with eighth-power temperature opacity ( $L \propto R^{98/47} M^{28/47}$ )

Match a fully convective stellar polytrope of index

3/2

to the grey-atmosphere convection onset with eighth-power temperature opacity. At the fixed matching optical depth,

T_{b} \propto T_{e}

and

P_{b} \propto g^{2/3} T_{e}^{- 16/3}

. The ideal-gas adiabat

P = K ρ^{5/3}

gives

K \propto T_{b}^{5/3} P_{b}^{- 2/3} \propto T_{e}^{47/9} g^{- 4/9}

. The polytropic mass-radius relation gives

K \propto R M^{1/3}

; with

g \propto M / R^{2}

, this yields

T_{e}^{47} \propto R M^{7}

. The Stefan–Boltzmann law then gives the displayed luminosity. Composition and opacity normalization are held fixed between models.

Hayashi relation with thirteenth-power opacity ( $L \propto M^{8/17} R^{38/17}$ )

Match an efficient, monatomic gas, fully convective star to the grey-atmosphere convection onset with thirteenth-power opacity. Its convective relation is

P = K_{T} T^{5/2}

. At the matching temperature

T_{b} \propto T_{e}

, the stellar atmosphere gives

K_{T}^{2} \propto M L^{- 1} T_{e}^{- 13} \propto M R^{13/2} L^{- 17/4}

using the Stefan–Boltzmann law. The ideal-gas relation converts this to

P = K_{ρ} ρ^{5/3}

with

K_{ρ} = (R / μ)^{5/3} K_{T}^{- 2/3}

. The polytropic mass-radius relation at index

3/2

gives

K_{ρ} \propto G M^{1/3} R

, hence

K_{T}^{2} \propto M^{- 1} R^{- 3}

. Equating the two expressions proves

L^{17/4} \propto M^{2} R^{19/2}

, or

L \propto M^{8/17} R^{38/17}

. The entropy coefficient is constant throughout the convective envelope, but can change between models.

Convective contraction with power-law opacity ( $T_{e}^{8} \propto μ^{4} M^{2} R$ )

For a monatomic fully convective ideal gas star, uniform specific entropy gives

P = K T^{5/2}

with

K \propto μ^{- 5/2} M^{- 1/2} R^{- 3/2}

. Combining this with the stellar surface boundary condition

κ P = 2 g /3

and

κ = κ_{0} ρ^{a} T^{b}

gives

T_{e}^{(3 a + 5) /2 + b} \propto μ^{(3 a + 5) /2} M^{(a + 3) /2} R^{(3 a - 1) /2} .

All composition comparisons here hold

κ_{0}

fixed. For

a = 1, b = 4

,

T_{e}^{8} \propto μ^{4} M^{2} R

, and the Stefan–Boltzmann law gives

L \propto μ^{2} M R^{5/2}

. The stellar virial theorem and Kelvin-Helmholtz contraction then give

R^{- 7/2} - R_{0}^{- 7/2} \propto μ^{2} M^{- 1} (t - t_{0})

, with a positive coefficient. Once the initial-radius term is negligible,

R \propto μ^{- 4/7} M^{2/7} (t - t_{0})^{- 2/7}

.

Contraction law with density-linear fourth-power stellar opacity ( $R^{- 7/2} - R_{0}^{- 7/2} = C μ^{2} M^{- 1} (t - t_{0})$ )

For a fixed-mass, fully convective monatomic gas protostar, write

P = K T^{5/2}

and

K = K_{0} μ^{- 5/2} M^{- 1/2} R^{- 3/2}

. Matching the photosphere to

κ P = 2 g /3

with

κ = κ_{0} ρ T^{4}

gives

T_{e}^{8} = 2 G R μ^{4} M^{2} R / (3 κ_{0} K_{0}^{2})

. The Stefan–Boltzmann law gives

L = πa c [2 G R / (3 κ_{0})]^{1/2} K_{0}^{- 1} μ^{2} M R^{5/2}

. The stellar virial theorem gives total energy

- 3 G M^{2} / (7 R)

, so

L = - 3 G M^{2} \dot{R} / (7 R^{2})

. Integrating proves

C = (49 πa c /6) [2 R / (3 κ_{0} G)]^{1/2} K_{0}^{- 1}

. This holds before stellar nuclear fusion, changing opacity or electron degeneracy pressure invalidates the assumed contraction law.

Hayashi contraction with seventeenth-power opacity ( $L \propto M^{8/21} R^{46/21}, L ≃ \frac{9 G M ^{2}}{67 Rt}$ )

Match an

n = 3/2

stellar polytrope to the grey-atmosphere convection onset with seventeenth-power opacity. At each instant the convective entropy coefficient obeys

P = K T^{5/2}

,

K \propto M^{- 1/2} R^{- 3/2}

at fixed composition, while the boundary gives

K \propto g^{1/2} T_{e}^{- 21/2}

. Thus

T_{e}^{21/2} \propto M R^{1/2}

and the first displayed luminosity law follows. The stellar virial theorem for a monatomic gas gives total energy

- 3 G M^{2} / (7 R)

, so Kelvin-Helmholtz contraction has

\dot{R} = - 7 L R^{2} / (3 G M^{2})

. Integrating at fixed mass gives

R^{- 67/21} - R_{0}^{- 67/21} \propto t

. The second displayed relation holds after the initial-radius term is negligible. The entropy coefficient is spatially constant but changes during contraction.

Composition shift of a convective pre-main-sequence track ( $L \propto μ^{- 8} T_{e}^{20}$ )

For convective contraction with power-law opacity

κ \propto ρ T^{4}

at fixed stellar mass and opacity normalization,

T_{e} \propto μ^{1/2} R^{1/8}

and

L \propto μ^{2} R^{5/2}

. Eliminating

R

gives

L \propto μ^{- 8} T_{e}^{20}

. Thus a larger mean molecular weight moves the track to larger effective temperature at a given luminosity, with

T_{e} ∣_{L} \propto μ^{2/5}

. Contraction decreases both temperature and luminosity for this particular opacity exponent.

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Hayashi track by

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The Hayashi track is a concept in astrophysics, particularly in the study of stellar evolution and the evolution of young stars. It refers to a path on the Hertzsprung-Russell (H-R) diagram, which plots stars according to their luminosity and temperature (or color). The Hayashi track describes the evolutionary phase of pre-main-sequence stars, specifically those that are still in the process of contracting and heating up before they initiate hydrogen fusion in their cores.

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