Part i) - Eq. 1.15 dL( r)/ dr =4 r 2 ( r) is and Eq. 3.33

Part i) - Eq. 1.15 dL(r)/dr=4πr²ε(r) is and Eq. 3.33 is εr=ε₀ρ²(r)T^v(r)

Part iii) - Eq. 4.2 is ρ∝M/R³ and Eq. 4.5 is Tc∝(M/R)μ

Part iv) - Eq. 1.18 is L∝M^5.5R^-0.5

Part v) - Eq. 1.1 is L=4πR2σTeff⁴

 

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(i) Combine the energy generation equation (Equation 1.15 one of the four equations of stellar structure) with the energy generation rate equation (Equation 3.33) for the proton-proton chain (assuming the temperature exponent v~ 4), to obtain the rate of change of luminosity with radius as a function of radius, density and temperature. (ii) Use a homology argument, similar to that in Section 4.3, to write a scaling relation for this energy generation equation, i.e. luminosity as a function of radius, density and core temperature. (iii) Substitute scaling relations for density and core temperature as a function of mass and radius into your result from (ii) (Equations 4.2 and 4.5, assuming a constant chemical composition), to obtain a scaling relation for luminosity as a function of mass and radius only. (iv) Combine your result from (iii) with the Eddington mass-luminosity-radius relationship (Equation 1.18) to obtain a scaling relationship between luminosity and radius only. (v) Finally, write a scaling relationship for the Stefan-Boltzmann equation (Equation 1.1) and combine that with your result from (iv) to show that the form of the lower main sequence on the Hertzsprung-Russell diagram may be expressed as log10 L~ log10 Teff + constant. 284 69 (vi) What assumptions about the properties of stars have gone into this derivation? (i) Combine the energy generation equation (Equation 1.15 one of the four equations of stellar structure) with the energy generation rate equation (Equation 3.33) for the proton-proton chain (assuming the temperature exponent v~ 4), to obtain the rate of change of luminosity with radius as a function of radius, density and temperature. (ii) Use a homology argument, similar to that in Section 4.3, to write a scaling relation for this energy generation equation, i.e. luminosity as a function of radius, density and core temperature. (iii) Substitute scaling relations for density and core temperature as a function of mass and radius into your result from (ii) (Equations 4.2 and 4.5, assuming a constant chemical composition), to obtain a scaling relation for luminosity as a function of mass and radius only. (iv) Combine your result from (iii) with the Eddington mass-luminosity-radius relationship (Equation 1.18) to obtain a scaling relationship between luminosity and radius only. (v) Finally, write a scaling relationship for the Stefan-Boltzmann equation (Equation 1.1) and combine that with your result from (iv) to show that the form of the lower main sequence on the Hertzsprung-Russell diagram may be expressed as log10 L~ log10 Teff + constant. 284 69 (vi) What assumptions about the properties of stars have gone into this derivation?

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i To obtain the rate of change of luminosity with radius we can combine Equation 115 and Equation 333 for the protonproton chain energy generation rat... View the full answer