Fundamentals of Astrophysics Stan Owocki

I Stellar Properties

Chapter 8: Stellar Ages and Lifetimes

8.1 Shortness of chemical burning timescale for the Sun and 8.2 Kelvin-Helmholtz timescale for gravitational contraction 8.3 Nuclear burning timescale 8.4 Age of stellar clusters from main-sequence turnoff point 8.5 Questions and exercises What powers stars on the ?

Most stars fall somewhere on the main sequence of the H-R diagram, with

3 L ⎛ M ⎞ ≈ L ⎝⎜ M ⎠⎟ ⊙ ⊙ Shortness of chemical burning timescale

Efficiency of reaction yielding ΔE energy ΔE from mass m ϵ ≡ mc2

For chemical reactions, ΔE~ eV, while mc2~Gev, so ε~10-9

Thus chemical reaction can’t power Sun for long:

3 Kelvin-Helmholz timescale for gravitational contraction

Gravitational binding energy (for constant density Sun)

Suppose luminosity L is powered by gravitational contraction:

The K-H time is how long this can work:

For Sun, we find:

This is longer than tchem, but still << age of Earth (Sun/solar system) 4 Nuclear burning timescale

Hydrogen fusion (requires T~15 MK)

Eγ ~ 0.007 mHe c2 => ε~0.007

Moreover, only central 10% (fraction f=0.1) of Sun is hot enough, so

Such H-fusion time sets the main-sequence lifetime of Sun!

5 Age of a stellar cluster is marked by main-sequence turnoff point On main sequence, luminosity is proportional to cube of mass : which implies that the MS lifetime scales as:

Thus for a cluster with stars born at ~ same time, the MS turnoff indicates cluster age:

old => age ~ 8 Gyr

6 Age of a stellar cluster is marked by main-sequence turnoff point

−2/3 L age ≈ 10 Gyr to ( L⊙ ) 7 Age of a stellar cluster is marked by main-sequence turnoff point

−2/3 L age ≈ 10 Gyr to ( L⊙ )

8 Age of a stellar cluster is marked by main-sequence turnoff point

age ≲ 10 Gyr

9 Quick Questions

10 Question What is the main-sequence lifetime of a with mass M = 16 Msun?

A. 2.5 Myr

B. 25 Myr

C. 250 Myr

D. 2.5 Gyr

E. 25 Gyr Question What is the main-sequence lifetime of a star with mass M = 2 Msun?

A. 2.5 Myr

B. 25 Myr

C. 250 Myr

D. 2.5 Gyr = 10 Gyr/22

E. 25 Gyr Question What is the age of a cluster with main-sequence turnoff at a luminosity Lto = 1000 Lsun?

A. 10 Myr

B. 100 Myr

C. 1 Gyr

D. 10 Gyr

E. It depends on the mass of the turn-off star. Question What is the age of a cluster with main-sequence turnoff at a luminosity Lto = 1000 Lsun?

A. 10 Myr

B. 100 Myr = 10 Gyr/10002/3

C. 1 Gyr

D. 10 Gyr

E. It depends on the mass of the turn-off star.