Stellar Properties and Masses

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Questions and Answers

What type of stars are considered the most common?

Main sequence stars (correct)

High mass stars

Brown dwarfs

Giant stars

What is the mass range for stars categorically distinguished from brown dwarfs?

Above 0.1 M⊙ (correct)

0.01 M⊙ to 0.1 M⊙

0.5 M⊙ to 100 M⊙

0.1 M⊙ to 0.5 M⊙

How does the mass-radius relationship on the main sequence behave?

It indicates a linear growth in radius with mass.

It shows no consistent relationship.

It only applies to high mass stars.

The exponent varies depending on mass. (correct)

What is the maximum mass that a star can have, as defined in the content?

100 M⊙ Signup and view all the answers

What is the relationship between the mass of a star and its luminosity?

A higher mass generally indicates higher luminosity. Signup and view all the answers

Why are stars born with a similar distribution of masses?

It is an inherent property of stellar formation. Signup and view all the answers

Which statement about the stellar mass function is true?

It indicates that for every solar mass star, roughly 100 low mass stars are formed. Signup and view all the answers

What is the primary energy generation method in the core of the Sun?

Nuclear fusion Signup and view all the answers

What is the lower mass limit for a star to be able to fuse hydrogen?

0.013M⊙ Signup and view all the answers

What process is primarily used by most stars to fuse hydrogen into helium?

pp-chain Signup and view all the answers

What is produced when two helium-3 nuclei collide in the pp-chain?

Helium-4 and two hydrogen nuclei Signup and view all the answers

What is the main challenge in confirming the fusion occurring within the Sun's core?

Neutrinos hardly interact with matter. Signup and view all the answers

How much energy is released when a helium nucleus is formed in the pp-chain?

4x10^-12 J Signup and view all the answers

What defines an object as a 'brown dwarf'?

Mass between 0.013M⊙ and 0.1M⊙ Signup and view all the answers

What role do neutrinos play in fusion reactions in the Sun?

They are produced in equal numbers to reaction chains. Signup and view all the answers

What percentage of mass is lost when a helium nucleus is formed compared to the original hydrogen nuclei?

0.7% Signup and view all the answers

What percentage less mass does a 4He nucleus have compared to the 4 1H nuclei?

0.7% Signup and view all the answers

How much energy is released for each He created?

4 x 10^-12 J Signup and view all the answers

What is the estimated main sequence lifetime of the Sun?

10 billion years Signup and view all the answers

If a star has twice the mass of the Sun, how much more energy does it produce compared to the Sun?

11 times more Signup and view all the answers

What is the core mass of the Sun that allows for hydrogen fusion?

0.1 M⊙ Signup and view all the answers

What is the lifetime of a 10 M⊙ star on the main sequence?

50 million years Signup and view all the answers

What is the main reason that more massive stars have shorter lifetimes?

They burn fuel much more rapidly. Signup and view all the answers

What is true about stars that are less than 0.1 M⊙?

They are classified as brown dwarfs. Signup and view all the answers

What is the relationship between mass and luminosity for stars on the main sequence?

More massive stars are generally more luminous due to higher energy production. Signup and view all the answers

Which stellar type generally has the lowest mass?

M dwarfs Signup and view all the answers

What process is primarily responsible for the energy production in the Sun?

Nuclear fusion Signup and view all the answers

How does the luminosity of more massive stars compare to less massive stars?

It is usually higher due to increased energy production. Signup and view all the answers

What is a critical factor for fusion to occur between hydrogen nuclei?

High temperatures and pressures Signup and view all the answers

Why do larger stars produce more energy than smaller stars?

They have greater gravitational forces, leading to higher central temperatures and pressures. Signup and view all the answers

What can be inferred about the life span of stars based on their mass?

Higher mass stars generally have shorter life spans than lower mass stars. Signup and view all the answers

What does the mass-luminosity relationship indicate about the energy output of a star?

Signup and view all the answers

What primarily causes more massive stars to be more luminous?

They produce more energy per unit time. Signup and view all the answers

What mass range is typically associated with O dwarfs?

20 - over 100 M⊙ Signup and view all the answers

How does the mass-luminosity relationship vary among different star masses?

It can have an exponent of 3 or 4. Signup and view all the answers

What must be achieved for hydrogen nuclei to overcome the 'Coulomb barrier' during fusion?

High temperatures and pressures. Signup and view all the answers

What is approximately the luminosity of a 10 M⊙ star?

3000 L⊙ Signup and view all the answers

What type of stars are primarily low mass and range from 0.1 to 0.5 M⊙?

M dwarfs Signup and view all the answers

What is needed for the fusion of hydrogen atoms into helium?

High temperatures and pressures. Signup and view all the answers

What is the Sun primarily composed of?

75% hydrogen and 25% helium Signup and view all the answers

Study Notes

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)

M type stars (Most Common)
4.5 Gyr
M type stars (Longest Lifetime)

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Description

This quiz covers the key concepts related to star masses, the relationship between a star's mass, radius, and luminosity, and how stars generate energy through nuclear fusion. Special emphasis is placed on main sequence stars and their properties, along with the initial mass function that governs their formation and development.