Astronomy 162:
Introduction to Stars, Galaxies, & the Universe
Prof. Richard Pogge, MTWThF 9:30

Lecture 15: The Main Sequence

Readings: Chapter 20, section 20-4

Key Ideas

Main Sequence stars are those that "burn" Hydrogen into Helium in their cores.: Get slowly brighter with age.
The Main Sequence is a Mass Sequence:: Lower M-S: M < 1.1 M_sun; Upper M-S: M > 1.1 M_sun
The Main-Sequence Lifetime depends on the Mass:: Larger Mass = Shorter Lifetime

Main Sequence Membership

For a star to be located on the Main Sequence in the H-R diagram:

It must be in Hydrostatic Equilibrium (Pressure balances Gravity)
It must be in Thermal Equilibrium (Energy Generation balances Luminosity)
It must generate energy by "burning" Hydrogen into Helium in its core.

Relax any of these conditions and the star must leave the Main Sequence.

Brighter with Age...

Main Sequence stars are in Hydrostatic Equilibrium. This requires a high central Pressure.

Since we have an ideal gas, Pressure = Density x Temperature:

Temperature: mean speeds of the nuclei
Density: number of nuclei per cubic centimeter

As Hydrogen is fused into Helium in the core there are fewer nuclei around (4 H become 1 He). As a consequence:

The remaining nuclei must move faster to maintain the same high Pressure as before.
The gas at the center gets slowly hotter.
This causes fusion to run faster as the temperature rises.

The result is that Main Sequence stars get slowly brighter as they age.

It's a small effect: ~0.7% brighter every 100 Myr

We would not notice this on a human timescale, but there is evidence of changes in the Sun's brightness over geologic time (specifically the Sun has gotten about 30% brighter over the last 4.5Gyr since the formation of the Earth).

The Main Sequence is a Mass Sequence

The location of a star along the M-S is determined by its Mass.

Low-Mass Stars: Cooler and Fainter
High-Mass Stars: Hotter and Brighter

As we saw in Lecture 11, M-S stars obey a strong Mass-Luminosity Relation:

(In words: High-mass M-S stars are more luminous than low-mass M-S stars proportional to the 4th power of their Mass.)

Internal Structure

Nuclear reaction rates are very sensitive to core temperature:

P-P Chain: rate ~ T⁴
CNO Cycle: rate ~ T¹⁸ !

The leads to:

Differences in internal structure.
Division into Upper and Lower Main-Sequences by mass.

The dividing line is at about 1.1 M_sun, the mass at which the P-P chain and CNO cycle both contribute equally to the total energy generation in the core.

Upper Main Sequence

Upper Main-Sequence stars have

M > 1.1 M_sun
T_Core > 18 Million K

Hydrogen fusion occurs via the CNO Cycle

Internal Structure:: Convective Core; Radiative Envelope

[Schematic of an Upper M-S Star interior]

Lower Main Sequence

Lower Main-Sequence stars have

M < 1.1 M_sun
T_Core < 18 Million K

Hydrogen fusion occurs via the Proton-Proton Chain

Internal Structure:: Radiative Core; Convective Envelope

[Schematic of a Lower M-S Star interior]

The Lowest Mass Stars

Mass Range: 0.25 > M_* > 0.08 M_sun:

Generate energy by the P-P Chain.

These stars have Fully Convective Interiors:

Convective Core and
Convective Envelope

Genericaly called Red Dwarf Stars

[Schematic of a red dwarf interior]

The Nuclear Timescale

How long a star can continue to generate energy by fusing H into He in its core depends upon how much fuel it has (total mass of the star), and how fast it is burning it (luminosity).

We call this the Nuclear Timescale:

where

f = fraction of nuclear fuel available for fusion

e = efficiency of matter-energy conversion

M = mass of the star

L = luminosity of the star

For the Sun:

t_nuc = 10 Gyr given that f=10% of the Sun's H is available for fusion into He with a matter-energy conversion efficiency of e=0.7%

Main Sequence Lifetime

The Nuclear Timescale above depends on the Mass (M) and Luminosity (L). But, we know from the Mass-Luminosity Relation for Main Sequence Stars that

L = M⁴

If we combine this with the formula for the Nuclear Timescale, we get the Main Sequence Lifetime:

t_MS ~ 1 / M³

The consequence is that the M-S lifetime is strongly dependent on the Mass of the star, in the sense that:

High-Mass M-S Stars have short M-S lifetimes

Low-Mass M-S Stars have long M-S lifetimes

Examples:

Sun:: M = 1 M_sun, and t_MS = 10 Gyr
Massive Star (10 M_sun):: t_MS = 10 Gyr / (10 M_sun)³ = 10 Million Years
Low-Mass Star (0.1 M_sun):: t_MS = 10 Gyr / (0.1 M_sun)³ = 10 Trillion Years

Consequences:

Some observational consequences of the Main-Sequence lifetime depending so strongly on the Mass of the star:

If you see an O or B dwarf star, it must be relatively young as O and B stars evolve very rapidly and live for only a few Million years.
You can't tell how old an M dwarf is because their M-S lifetimes are extremely long and they evolve very slowly.
The Sun is ~5 Billion years old, so it will only last for about another ~5 Billion years.

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