Stellar Scaling Relations and basic observations (roughly sections 3.2-3.5 in Choudhuri s book) Astrophysics-I, HS2017 week 3, Oct. 17 & 18, 2017 Benny Trakhtenbrot
Stellar structure - recap density mass Hydrostatic equilibrium Radiation generation Radiative transfer (3.1) (3.2) (3.15) (3.16) Convection (3.22)
Stellar structure - recap Trivial boundaries: (3.1) (3.2) Rough boundaries: (3.15) (3.16) (3.22) 4 equations with 4 boundary conditions: Is it solvable? Are the solutions unique?
Stellar structure - recap Trivial boundaries: (3.1) (3.2) Rough boundaries: (3.15) (3.16) (3.22) 4 equations with 4 boundary conditions: Is it solvable? yes! numerically Are the solutions unique? yes! generally
Order-of-Magnitude Estimates Goals: To make drastic simplifying assumptions, in order to derive rough stellar properties of interest Gain insights on relevant physics / improvements
The Sun s Central Pressure and Temperature (in Section 3.2.1) We want a (very) rough estimate of these quantities, to understand the physics in play Let s start with the basic (observable) quantities: Equation 3.2 says: Let s plug in some rough/ global values:
P P c R r
The Sun s Central Pressure and Temperature (in Section 3.2.1) We want a (very) rough estimate of these quantities, to understand the physics in play Let s start with the basic (observable) quantities: Equation 3.2 says: Let s plug in some rough/ global values: So we get: - 6 billion atmospheres! Detailed models give a higher value
The Sun s Central Pressure and Temperature (in Section 3.2.1) We already mentioned our assumption of ideal gas: In the center of the sun, the hydrogen-dominated gas is fully ionized, so every atom gives 2 particles: So we get: average density : So we get, which is not too far from detailed models (1.57x )
The life-time and fuel source of the Sun (in Section 3.2.2) What drives the luminosity of the sun? Helmholtz (1854) and Kelvin (1861) had a theory: Assume the sun shrinks - R decreases The gravitational energy will decrease: But note that increases When deriving the virial theorem we saw: So the thermal energy increases the star becomes hotter BUT: the thermal energy takes only half of the change in The other half is radiated away and is what we see as L?
The life-time and fuel source of the Sun (in Section 3.2.2) The other half is radiated away and is what we see as L? If this is true, then the sun had already radiated away It did so at an energy-loss rate of L So a rough estimate for the age of the sun is: which we call the Kelvin-Helmholtz timescale. Plugging in some numbers: Which results in: - only 10 million years?!
Stellar Scaling Relations Goals: To make drastic simplifying assumptions, in order to derive observable relations Test these scaling relations against observations Hopefully, move on
Temperature Mass-Size Relation As we already saw, we can linearize the pressure gradient: We can also write a scaling relation for the density: So starting from Eq. 3.2: We obviously get: In an ideal gas the E.o.S. is: Which can be combined to produce:
Luminosity Mass Relation Let s (ab)use Eq. 3.16 in a similar way: Using a linear Temperature gradient, and scaling relations: But we already saw above in Eq. 3.35 that: So we can finally write: This is very important, because we observe L from (groups of) stars Imagine 1000 sun-like stars with M and L, and then 10 stars with 10 M. The sun-like stars are ~90% of te mass, but only ~9% of the light! If you have only one 10 M star, it will be 50% of the light and 1% of the mass!
Luminosity Mass Relation from Choundhuri s book Notice the units are log10(x), log10(y)
Luminosity Temperature Relation The surface of the star is observed to have an effective temperature, T eff and we can assume: We don t specify which temp. Just a generalized property It looks like a blackbody, so we can write: We ve already seen that So we can derive: The massive stars are also hotter and dominate the blue light!
Luminosity Temperature Relation Notice the units are log10(x), log10(y)
Luminosity Temperature Relation 100 in L
Luminosity Temperature Relation
Lifetime Mass Relation We can assume that a fraction of the stellar mass is involved in fueling the radiation So the star will exhaust its fuel after a time that scales like: Plugging in the mass-luminosity relation we get: Massive stars shine bright, are hot, and die young!
Lifetime Mass Relation
Lifetime Mass Relation missing stars from H-R?
Lifetime Mass Relation missing stars from H-R?
Lifetime Mass Relation