Stellar Properties and Masses

Lecture Aims

• Students will gain an understanding of the distribution of star masses.
• Students will learn about the relationship between a star's mass, radius, and luminosity, mainly on the main sequence.
• Students will understand how energy is generated within the Sun's core through nuclear fusion.
• Students will learn how the mass-luminosity relationship aids in estimating the lifetime of stars.

Physical Properties of Stars

• Combining HR diagrams and eclipsing binaries allows the study of how stellar properties change with mass, leading to an understanding of stellar evolution.
• Focus is placed on main sequence stars as they are the most common and well-understood type of star.

Stellar Masses

• Star masses vary significantly, ranging from approximately 10⁻² to 10² solar masses (M☉).
• Objects with less than ~0.1 M☉ are classified as brown dwarfs, not stars, as they do not undergo hydrogen (H) fusion.

The Initial Mass Function

• Stars are born with a relatively consistent mass distribution, as demonstrated by observations of star clusters like the Pleiades.
• Low-mass stars (0.1-0.5 M☉) are the most prevalent type of star.
• The mass distribution follows a log-normal pattern. For every solar-mass star, roughly 100 low-mass stars are formed.

The Mass-Radius Relationship

• A relationship exists between a star's mass and radius during the main sequence.
• The exponent in the relationship varies between ~0.6 for lower-mass stars and ~0.8 for higher-mass stars.
• This relationship is helpful for deriving stellar mass from measured radius, or vice versa, and relating these to surface gravity.

The Mass-Luminosity Relationship

• A clear relationship exists between a star's mass and luminosity on the main sequence.
• For typical solar-mass stars, luminosity is approximately proportional to the 3.5 power of the mass.
• The relationship differs slightly depending on the mass range considered.
• The steepness of the relationship strongly correlates with differences in stellar life spans.

Mass-Luminosity Relation (Graph)

• A graph demonstrates the correlation between mass (in solar masses) and luminosity (in solar luminosities).
• High-mass stars occupy the upper region of the graph, and low-mass stars occupy the lower region.
• The Sun sits roughly around the middle ground of the plot.

The Mass-Luminosity Relationship (HR Diagram)

• The HR diagram reveals a relationship between spectral type and luminosity, crucial for connecting these with stellar masses during a star's main sequence stage.
• M-dwarfs are of low mass (0.1 – 0.5 M☉)
• G-dwarfs are mid-mass (~0.8-1.2 M☉)
• O-dwarfs have high mass (20 M☉ to > 100 M☉)

The Mass-Luminosity Relationship (Why are some stars more luminous?)

• More luminous stars produce more energy per unit time (higher luminosity).
• This increased energy production is a direct consequence of their greater mass.
• Stars with 10 solar masses produce approximately 3000 times the energy per unit time compared to the Sun.
• Stars with 0.1 solar masses produce approximately 0.0003 times the energy per unit time compared to the Sun.

Energy Generation

• Radiometric dating of Earth indicates an age of about 4.5 Gyr, which sets a minimum age for the Sun.
• The Sun's energy production (luminosity) is maintained by nuclear fusion, converting mass into energy (E=mc²).
• The Sun's composition (75% Hydrogen, 25% Helium) makes H to He fusion the dominant energy-producing process.

Fusion

• High temperatures (T) and pressures (P) are necessary for nuclear fusion processes (e.g., H to He).
• High temperature leads to fast-moving nuclei, increasing the chance of collisions.
• High pressure increases the likelihood of nuclei coming into contact.
• More massive stars have higher central T and P due to stronger gravitational forces. This results in increased energy production and higher luminosity.

Coulomb Barrier

• The electromagnetic force (Coulomb barrier) initially repels positively charged nuclei.
• Overcoming this barrier requires nuclei to get close enough (~10⁻¹⁵ m) for the strong nuclear force to dominate and cause fusion.
• Graph depicts the repulsive force (Coulomb repulsion) decreasing with distance to become weaker than the strong nuclear attraction at short distances.

Fusion Limit

• Stars with less than ~0.1 solar masses don't achieve high enough central T and P for hydrogen fusion.
• These objects are classified as brown dwarfs if they can fuse deuterium (2H).
• Objects with insufficient mass (0.013 M☉ - 13 Jupiter masses) cannot even fuse deuterium and are categorized as planets.

The pp-chain

• The pp-chain is the primary method stars fuse H to He.
• The initial step involves making deuterium (a proton and a neutron) within the reaction chains.
• Positrons (and neutrinos) are produced and annihilate, releasing energy in the form of photons.
• Multiple steps (reactions) lead to the conversion of hydrogen to helium.

The pp-chain (Variations)

• Stars with mass exceeding ~1.3 solar masses utilize the CNO cycle.
• The CNO cycle relies on Carbon, Nitrogen, and Oxygen.
• The pp-chain, however, is the dominant route in the Sun.
• A final 4He nucleus has 0.7% less mass than the 4 Hydrogen nuclei — this difference is converted into energy at a rate of ~4x10⁻¹² J per He nucleus. An enormous number of reactions is required to sustain energy production.

Solar Neutrinos

• Neutrinos are produced as a by-product of fusion reactions within the Sun's core.
• Neutrinos interact weakly with matter and can therefore easily pass through the Sun and the Earth.
• Early detection of neutrinos did not match the expected numbers (solar neutrino problem), but this has been resolved by the recognition that neutrinos change flavor.

The Main Sequence Lifetime of the Sun

• Main sequence stars generate energy by fusing H to He.
• The Sun's core, which fuels this process, constitutes about 10% of the Sun's total mass.
• A 0.1 solar mass core = roughly 2x10²⁹ kilograms.
• This mass of available fuel, converted into energy at a rate of ~3.8 x 10²⁶ W, dictates the Sun's lifetime on the main sequence.
• The Sun's main sequence lifetime is estimated to be around 10 billion years.

The Main Sequence Lifetime of Stars

• Main sequence lifetime is highly dependent on a star's mass.
• More massive stars burn through their fuel much more quickly.
• Less massive stars burn fuel slowly, resulting in much longer lifetimes. A 10 solar mass star exists for much less time than a 0.1 solar mass star

Key Points to Remember

• Stars span a wide range of masses, with most (over 90%) being 0.1-0.5M.
• Stars less than 0.1M are brown dwarfs.
• A clear relationship exists between stellar mass and luminosity on the main sequence (higher masses result in higher luminosities).
• Stellar spectral class provides a useful way to gauge mass on the main sequence.
• Stars generate energy by fusing H to He.
• Stars with M > 1.3 M☉ utilize the CNO cycle.
• The mass difference generated between 4 Hydrogen nuclei and a Helium nucleus is converted into energy at ~4x10⁻¹² J/He .
• A Star remains within the Main Sequence until it exhausts its Hydrogen core.
• Energy generating rate is strongly dependent on mass meaning high mass stars produce energy at extremely high rates however this results in exceedingly short lifetimes whereas low mass stars have extremely slow energy production rates but extremely long lifetimes.

Example Questions (Answers)

1. M type stars (Most Common)
2. 4.5 Gyr
3. M type stars (Longest Lifetime)