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Stellar Structure and Evolution > Semester 1 - Definitions > Flashcards

Semester 1 - Definitions Flashcards

1
Q

What is flux, and how is it related to luminosity and apparent magnitude?

A

Flux describes energy passing through a unit area per unit time and is measured in Joules per second per square meter. It is related to luminosity, the total energy radiated per second, and apparent magnitude (m) through the equation
m=−2.5log10 (F/FVega).

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2
Q

How is stellar distance measured?

A

Stellar distance is measured using parallax, where a is the Earth-Sun distance, θ is the angle of the star with respect to background stars, and p is the parallax measured in parsec (pc).

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3
Q

What is effective temperature?

A

Effective temperature is the temperature of a black body providing the same luminosity as the star.

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4
Q

What is color temperature, and how is it determined from the observed spectrum of a star?

A

Color temperature is determined by fitting the observed spectrum with a Planck function. If a clear peak is observed, Wien’s Displacement Law can be used.

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5
Q

What is absolute magnitude, and how is it related to apparent magnitude and stellar distance?

A

Absolute magnitude (M) is the apparent magnitude a star would have if it were at a distance of 10 parsecs (10pc). It is related to apparent magnitude (m) and stellar distance (d) through the equation
m−M=5log 10 (d/10pc).

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6
Q

What is stellar color, and how is it used as an alternative measure to temperature?

A

Stellar color is the difference between pairs of magnitudes (e.g., B−V). It is used as an alternative measure to temperature since stellar temperature is hard to define directly.

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7
Q

What are color-magnitude diagrams (HR diagrams), and how are they used in comparing stellar populations?

A

HR diagrams are plots of absolute magnitude against color. They compare results of stellar physics calculations to observations. Different age clusters have different features in their HR diagrams.

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8
Q

What is the main sequence in HR diagrams, and what does it represent about stable stars?

A

The main sequence represents the part of the HR diagram where most stars are found. Stable stars spend a large fraction of their life on the main sequence, indicating a stable condition.

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9
Q

What is the main-sequence mass-luminosity relationship, and why is it significant for understanding stellar properties?

A

The main-sequence mass-luminosity relationship shows a tight correlation between stellar mass and luminosity for main-sequence stars. It is significant for understanding how luminosity depends on mass.

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10
Q

What is the internal energy source in main-sequence stars?

A

The internal energy source is fusion of H (hydrogen) to He (Helium).

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11
Q

How is stellar mass measured, and what is the significance of binary star systems in determining mass?

A

Stellar mass is measured in binary star systems where stars orbit a common center-of-mass. Observations of binary systems provide information on masses, especially in visual binaries where the stars are resolved.

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12
Q

What is a star, and what are the three types of pressure that support a star’s internal structure?

A

A star is a self-gravitating sphere of gas that radiates due to internal energy sources. The three types of pressure supporting a star’s internal structure are gas pressure, radiation pressure, and degeneracy pressure.

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13
Q

What are the Kelvin-Helmholtz, Einstein, and dynamical timescales, and how do they relate to stellar energy sources?

A

The Kelvin-Helmholtz timescale is the timescale on which a star loses energy through radiation. The Einstein timescale is related to the conversion of rest-mass energy to energy available for nuclear fusion. The dynamical timescale is the timescale on which a star responds to changes in its dynamical equilibrium.

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14
Q

What is the Virial Theorem?

A

The Virial Theorem states that the total internal energy of a star in hydrostatic equilibrium equals half the gravitational potential energy. It relates the energy contributions to a star’s stability.

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15
Q

How are pressure and temperature estimated in a star supported by gas pressure, and what assumptions are made in the estimation?

A

Pressure and temperature are estimated based on the assumption of hydrostatic equilibrium. Gas pressure depends on temperature and density, and the gas heats as it contracts. The estimation assumes slow contraction, approximating the star to be in quasi-hydrostatic equilibrium.

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16
Q

Where does star formation take place and what is the main seed for star formation?

A

Star formation takes place in interstellar gas clouds. The main seed is thought to be supernova shocks.

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17
Q

What is the Eddington luminosity, and what role does it play in determining the mass of a star?

A

The Eddington luminosity is the maximum luminosity above which a star begins to drive off its outer layers due to radiation pressure. It determines the Eddington mass, a limit beyond which stars become unstable.

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18
Q

What is the concept of the Jeans mass, and how does it relate to the collapse of interstellar gas clouds?

A

The Jeans mass is the maximum mass for a given density and temperature that can be in hydrostatic equilibrium. It is crucial in the collapse of interstellar gas clouds, where overdense regions collapse under self-gravity.

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19
Q

What is the significance of the ignition temperature, and how does it influence star formation?

A

The ignition temperature is the temperature at which nuclear burning starts, leading to the formation of a star. It sets a minimum mass for a protostar to undergo nuclear fusion and enter the main sequence.

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20
Q

How is the maximum temperature in a contracting protostar determined?

A

The maximum temperature in a contracting protostar is determined by considering gas and degeneracy pressure. Gas pressure heats the star as it contracts, but degeneracy pressure can halt contraction before reaching the required temperature for fusion.

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21
Q

What is the minimum mass for a star to enter the main sequence?

A

The minimum mass occurs when the maximum core temperature of the proto-star exceeds kbTign = 1keV for the p-p chain.

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22
Q

What is the maximum stellar mass, and what role does the Eddington luminosity play in determining this limit?

A

The maximum stellar mass is limited by the Eddington luminosity, where radiation pressure drives off the outer layers. So at some value of M, we will have Frad > Fg.

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23
Q

How do observations support the idea that stellar luminosity strongly depends on mass?

A

Observationally, stellar luminosity depends strongly on mass. The gravitational force on a particle increases with mass, while radiation force increases linearly with luminosity.

think eddington luminosity graph

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24
Q

Why is stellar mass and radius used as independent variables in constructing models for the interior of a star?

A

Stellar mass is used as the independent variable instead of volume, making it directly measurable. Stellar radius is then inferred from other observations.

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25
Q

What is the Lane-Emden equation?

A

The Lane-Emden equation describes the variation of a dimensionless density variable θ with a dimensionless radial coordinate η. It is a second-order differential equation that arises from the polytropic models for stellar interiors.

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26
Q

What is the polytropic equation of state, and how is it used in deriving solutions for the Lane-Emden equation?

A

The polytropic equation of state is
P=Kρ^γ, where P is pressure, ρ is density, K is a constant, and γ is the adiabatic index. An ideal gas under adiabatic conditions is polytropic. This equation allows the derivation of restricted solutions for stellar structure equations.

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27
Q

What are the boundary conditions for the Lane-Emden equation, and how are they determined?

A

The boundary conditions for the Lane-Emden equation involve setting θ to be finite at η=0 and ensuring that dθ/dη is zero at the center of the star (η=0). The latter arises from dP/dη=0 at the center of the star.

28
Q

What is the significance of the polytropic index (n), and how does it affect the solutions of the Lane-Emden equation?

A

The polytropic index (n) is used in the polytropic equation of state and the Lane-Emden equation. The edge of the star is at θ = 0. For n ≥ 5 there is no edge the star is unbounded - unphysical.

29
Q

Describe the Lane-Emden equation for a perfect adiabatic gas.

A

In the case of a perfect, adiabatic gas, the Lane-Emden equation describes the linear variation of dimensionless temperature with dimensionless radius.

30
Q

What are the characteristics of the Lane-Emden solutions for specific values of n?

A

The n = 3 solution corresponds to the ‘Eddington Standard Model,’ describing a star in which the ratio of radiation pressure to total pressure is constant throughout. For n = 1, stellar radius does not depend on total mass.

31
Q

How does the Eddington Standard Model compare with a full stellar model, and under what conditions does it perform well?

A

The Eddington Standard Mode provides a fairly good description below the convection zone. It compares well with the full stellar model, especially in regions where radiation pressure dominates.

32
Q

Describe the nuclear reaction rate and the two contributors.

A

The nuclear reaction rate strongly depends on temperature. The energy release rate depends on both the reaction rate and the density of reacting particles. The proton-proton chain and CNO cycle are primary contributors to energy release.

33
Q

What is the significance of the proton-proton chain in energy generation, and when does it become active in a star’s life cycle?

A

The proton-proton chain is the first source of nuclear reactions activated as a star contracts. It involves the fusion of protons with light nuclei like deuterium, lithium, beryllium, and boron. This phase starts during contraction when the core temperature reaches 10^6 K.

34
Q

Describe hydrogen burning and it’s role in main-sequence stars.

A

Hydrogen burning involves converting four protons into helium. While the probability of simultaneous fusion is small, the reaction proceeds in steps. In main-sequence stars, hydrogen burning is the primary energy source.

35
Q

What is the CNO cycle and what is the reaction rate give by?

A

The CNO cycle operates in the cores of massive stars with higher temperatures. It transforms protons into helium using carbon, nitrogen, and oxygen as ‘catalysts.’ The reaction rate is determined by the slowest step in the cycle.

36
Q

What determines the form of the temperature gradient and what considerations are given for different modes?

A

The form of the equation for the temperature gradient depends on the mode of energy transport (radiation, convection, or conduction). Different considerations, such as radiation properties, opacity, and the Schwartzschild criterion, are introduced for each mode.

37
Q

Describe how energy is transported through radiation and the how this affects the temperature gradient.

A

Photons carry energy in a stellar interior, interacting with electrons and ions. On average each interaction causes the photon to lose energy to the electrons/ions. The radiative temperature gradient is influenced by the luminosity and the opacity.

38
Q

What provides the outward radiation pressure and how does this relate to opacity?

A

The interaction of radiation with ions and electrons provides the outward radiation pressure. Where the effectiveness of radiation and matter interactions is given by the opacity.

39
Q

How is opacity defined, and what is the significance of the Rosseland mean opacity (kR)?

A

Opacity (Kv) is defined as the frequency-dependent interaction of photons with stellar plasma. The Rosseland mean opacity (kR) is a frequency-averaged opacity that helps describe the temperature gradient in stellar interiors.

40
Q

What factors contribute to opacity in stars, and how does Kramers opacity play a role?

A

Opacity in stars is influenced by processes such as electron scattering, bound-bound absorption, bound-free absorption, and free-free absorption. Kramers opacity is the opacity due to free-free absorption, is essential for high-temperature conditions.

41
Q

Describe electron scattering.

A

Electron scattering: photon scatters from a free electron such that the photon energy is unchanged. It is independent of density and temperature and dominant in hot, massive stars.

42
Q

Where is electron scattering opacity and Kramers opacity relevant?

A

Electron scattering and Kramers opacity are crucial at high temperatures. Electron scattering opacity is independent of temperature, while Kramers opacity decreases with increasing temperature.

43
Q

What is the Eddington Standard Model and the Eddington quartic equation?

A

The Eddington Standard Model assumes a polytropic equation of state with n=3. It describes a star where the ratio of radiation pressure to total pressure remains constant throughout. The Eddington quartic equation relates properties of the star.

44
Q

Under what conditions does radiative transport break down, and how does convection take over?

A

Radiative transport breaks down if luminosity or energy generation rate is too high. In such cases, either the star is not in hydrostatic equilibrium, or radiation pressure cannot be maintained. Convection takes over when radiative transport is no longer effective.

45
Q

How does the mass effect whether the star is convective or radiative?

A

Massive stars have high luminosities hence have convective interiors and radiative exteriors while less massive stars have radiative interiors and convective exteriors.

46
Q

What determines the onset of convection?

A

Convection occurs if a temperature gradient larger than the critical value is reached. The critical temperature gradient is determined by the adiabatic index (γ). If γ is close to 1, convection can start.

47
Q

Describe the breakdown of radiating transport for high opacity.

A

Radiative transport is not possible for high Kramers opacity. The Kramers opacity increases with decreasing T. But at lower temperatures there are fewer free electrons so other forms of opacity dominate. Convection thus takes over from radiation

48
Q

When does convection occur, and what assumptions are made? DELETE

A

Convection occurs if an element of gas in a stellar interior, when displaced upwards, continues to rise. This happens if the gas is buoyant. We assume the element rises quickly enough to be considered adiabatic. But it also rises slowly enough that pressure inside and outside the element are equal, Pe=Pi.

49
Q

What assumptions are made in the Eddington standard model and what is the result of this?

A

The Eddington Standard Model assumes that κη ~ constant. This implies that the ratio of radiation pressure to total pressure is constant throughout the star, providing a polytropic equation of state with n=3.

50
Q

Where is convection likely to occur?

A

Convection is likely to occur in zones of partial ionisation in stars as the energy input goes into increasing its ionisation state.

51
Q

What is the assumption for homologous stars, and what are the implications of this?

A

Equations for homologous stars are derived assuming elemental uniform composition so that they are independent of the star’s total mass. Hence L ∝ M^3.

52
Q

What is the temperature-luminosity relationship in the H-R diagram, and how does it vary with stellar mass?

A

The temperature-luminosity relationship in the H-R diagram shows a slope that varies with stellar mass. Above 1M⊙, the gradient is steeper for more massive stars. The change in slope is attributed to the transition from proton-proton to CNO burning in more massive stars.

53
Q

What is the standard luminosity-temperature diagram?

A

If we assume stars emit like black bodies and have the properties of a black body. The temperature can be used to determine the shape of the spectrum and the flux in one band is uniquely related to the total flux of a black body.

54
Q

Explain which opacity is dominant in high mass stars and why the other types are not dominant.

A

Electron scattering because it is fully ionised so no bound bound or bound free opacity occurs. Free free opacity is proportional to temperature hence it is not as dominant.

55
Q

Which types of opacity are more dominant in the outer parts of a solar mass star?

A

Bound bound and bound free dominate because lower temperatures allow for electrons to recombine.

56
Q

Why is stellar temperature not as useful?

A

Stars do not radiate exactly as black bodies. The outer layers are not in local thermodynamic equilibrium hence stellar temperature is intrinsically ill-defined.

57
Q

Describe stellar clusters.

A

Stellar clusters are believed to be formed at the same time and have the same chemical composition. Open clusters contain young stars while globular clusters contain old stars.

58
Q

What pressure dominates in the interior of most main sequence stars?

A

Gas pressure dominates.

59
Q

Where is nuclear burning limited to and why?

A

The stellar core where density and temperature are high enough to allow Coulomb barrier penetration.

60
Q

Describe how gas and degenerate pressure vary with ignition temperature.

A

When gas pressure dominates ignition temperature increases with density. When degenerate pressure dominates temperature no longer increases with density.

61
Q

When do electrons have the biggest cross section?

A

For interactions with radiation - Thomson cross section.

62
Q

What does the temperature gradient depend on?

A

The stars luminosity and the opacity where opacity is the strength of the interaction between the photons and the stellar material.

63
Q

Describe bound-bound absorption.

A

A photon absorbed by an atom, excites a bound electron to a higher energy bound state.

64
Q

Describe bound-free absorption.

A

A photon absorbed by a bound electron, giving it energy above the ionisation potential.

65
Q

Describe free-free absorption and it’s temperature dependence.

A

A photon absorbed by a free electron, in the presence of an ion, increasing the electrons energy. The strength of free-free absorption decreases with increasing temperature.

66
Q

What is the mean free path?

A

The characteristic distance between interactions for a photon. The average distance that a photon can travel before changing direction.

67
Q

What does the Eddington quartic equation describe?

A

The dependence of the ratio of radiation/gas pressure in the states properties.

Stellar Structure and Evolution (4 decks)

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