Semester 1 - Definitions

Question 1

What is the focus when considering intensity within a stellar atmosphere?

Answer 1

The focus is on energy transport within the stellar interior, particularly the photosphere, which is the optical surface of a star. This is where the majority of light originates, and it represents the point beyond which one cannot see “inside.”

Question 2

What are the key components of a generalized equation of transfer for photons in a stellar atmosphere?

Answer 2

The equation of transfer describes the flow of photons through space and time in terms of their velocity and rate of change of momentum (force). The equation involves functions representing the density of photons and their creation and destruction at any given point, related to the intensity of the radiation beam.

Question 3

What assumptions are made to simplify the equation of transfer for stellar atmospheres?

Answer 3

Stars are assumed to be spherical.
Their atmospheres are considered isotropic.
Transport processes are assumed to be frequency-independent.
A continuum description can be used.

Question 4

What is Integrated Intensity (I) in the context of a stellar atmosphere?

Answer 4

Integrated Intensity represents the strength of the radiation field at a specific point in space, time, and direction. It is the overall intensity over all frequencies.

Question 5

What is Specific Intensity (Iv) in the context of a stellar atmosphere?

Answer 5

Specific Intensity is the intensity per unit frequency range. It involves considering photons in a specific frequency range (v to v+dv)

Question 6

What is Flux (F) in the context of a stellar atmosphere?

Answer 6

Flux is a component of energy in a particular direction. It is defined by considering vectors such as the unit vector in the direction of dΩ and the unit vector normal to the surface used for measurements.

Question 7

What does the conservation of energy along a ray path imply?

Answer 7

In the absence of sources or sinks, intensity is constant along a ray path. This conservation principle states that sources supply energy, and sinks remove energy.

Question 8

What are the consequences of intensity being constant along a ray path?

Answer 8

1) the intensity at the surface of a star is the same as at the Earth

2) any measured differences in mean intensity/flux must be dependent on solid angle

Question 9

How is energy density defined in the context of a small volume in a stellar atmosphere?

Answer 9

Energy density (u) is defined as the energy in a small volume V at time t. It involves considering a small cone of solid angle dΩ associated with an area element dA.

Question 10

What are the key parameterizations used to simplify calculations in stellar atmospheres?

Answer 10

Spherical polar coordinates, involving polar angle (θ), azimuthal angle (ϕ), and radial distance (r), provide mathematical shortcuts due to spherical symmetry. This allows a simplification of the equations.

Question 11

Why are observations limited to measuring flux in stellar atmospheres, and what is flux at the stellar surface?

Answer 11

Observations are limited to measuring flux instead of intensity, primarily due to the remoteness of stars. At the stellar surface, all radiation is expected to travel outwards, making the flux (F) equal to zero.

Question 12

What are the losses and gains involved in the interaction of radiation with matter in a stellar atmosphere?

Answer 12

Losses involve absorption and scattering processes, described by absorption and scattering coefficients. Gains involve emission processes, described by the emission coefficient.

Question 13

What are the labeling conventions used for absorption and scattering coefficients?

Answer 13

Subscripts are used, with v indicating a range of frequencies. Unsubscripted parameters could be integrated or frequency-independent (grey atmosphere).

Question 14

What is the Absorption Coefficient (κv)

Answer 14

The absorption coefficient (κv) represents the reduction in specific intensity (Iv) due to absorption processes. It is frequency-dependent.

Question 15

What is the Mean Intensity (J) in the context of Stellar Physics?

Answer 15

The Mean Intensity (J) is the intensity averaged over all directions. For an isotropic and integrated radiation field (i.e. a black body) J(isotropic) = J = I.

Question 16

How is the Mean Free Path (L) related to the Scattering Coefficient (σv)?

Answer 16

The mean free path (L) of a photon is inversely proportional to the scattering coefficient (σv).
L is the average distance a photon travels without being scattered.

Question 17

How are Absorption and Scattering Combined?

Answer 17

All intensity loss processes are combined into a single parameter
kv = κv + σv
, simplifying the representation and highlighting the dominant loss process.

Question 18

What is the Emission Coefficient (jv)?

Answer 18

The emission coefficient (jv) describes how radiation is enhanced as it passes through a medium. It represents the amount of energy emitted by volume.

Question 19

What is the general form of the Equation of Transfer?

Answer 19

The equation of transfer is an algebraic sum of intensity loss and gain processes. It’s represented as a first-order differential equation.

Question 20

What is Optical Depth (τv)?

Answer 20

Optical depth (τv) is the number of mean free paths a photon travels from its original position to the stellar surface. It’s a dimensionless parameter describing how difficult it is to see through the atmosphere.

Question 21

What is the Source Function (Sv)?

Answer 21

The source function (Sv) is the ratio of emission to combined absorption coefficients. It indicates the ratio of intensity gained to intensity lost.

Question 22

How is the Equation of Transfer Simplified for Plane-Parallel Atmosphere Approximation?

Answer 22

In the plane-parallel approximation, the equation of transfer is simplified for a semi-infinite atmosphere. The specific intensity is independent of the angle phi.

Question 23

What are the limiting cases discussed in solving the Equation of Transfer?

Answer 23

The optically thick atmosphere (no spectral lines) and the optically thin atmosphere (spectral lines possible) are considered. They help in understanding the physics of spectral line formation.

Question 24

Describe the Optically Thick Atmosphere.

Answer 24

An optically thick atmosphere has a large optical depth (τv»1), and there is no spectral line formation.

Question 25

Describe the Optically Thin Atmosphere.

Answer 25

An optically thin atmosphere has a small optical depth (τv≈1), allowing for spectral line formation. Spectral lines in emission or absorption depend on the relative values of Iν,0 and Sν.

Question 26

What is the Plane-Parallel Atmosphere Approximation?

Answer 26

In the plane-parallel approximation, the atmosphere is modeled as being composed of infinite, plane-parallel layers in the x and y directions and semi-infinite in the -z direction.

Question 27

How is the Equation of Transfer Modified for Plane-Parallel Atmosphere?

Answer 27

The equation of transfer is modified in the plane-parallel atmosphere approximation, introducing optical depth redux (dtau (ν, mu)).

Question 28

What does Optical Depth Redux involve?

Answer 28

Optical depth redux introduces an angle-dependent definition of optical depth (dτ v,μ), assuming radial symmetry with μ=cosθ. It describes the difficulty of seeing through the atmosphere

Question 29

What is Bolometric Intensity (Bv)?

Answer 29

Bolometric intensity (Bv) is the radiation emitted by a black body per unit frequency range. It represents the integrated intensity over all frequencies and is set equal to the energy absorbed.

Question 30

What are Emission Lines in Astrophysics, and what can this indicate?

Answer 30

Emission lines occur in regions where stellar atmosphere temperature increases outwards such that I(v,0) < Sv. It can indicate regions of high-temperature or back-lit regions.

Question 31

What are Absorption Lines in Astrophysics?

Answer 31

Absorption lines occur in regions where stellar temperatures decreases outwards such that I(v,0) > Sv.

Question 32

What is Line Saturation in Astrophysics?

Answer 32

Saturation of lines occurs if in the optically thin region, there is an inner region of material that is optically thick such that Iv ~ Sv. This results in a Planckian spectrum.

Question 33

What are the moments of the radiation field?

Answer 33

Moments of the radiation field include the mean intensity (J), Eddington flux (Hv), and the K integral. These moments are defined by integrating the product of the distance to a point and a physical quantity raised to a positive power.

Question 34

How are moments of the radiation field related to each other?

Answer 34

The moments are related through integration, differentiation, and mathematical relationships. As the moment order (n) increases, the complexity of the physics involved also increases.

Question 35

What is the Zeroth Moment, and what does it represent?

Answer 35

The zeroth moment represents the mean (specific) intensity (J), analogous to the particle density of the atmosphere.

Question 36

What does the First Moment (Eddington Flux) describe and how does it relate to the zeroth moment?

Answer 36

The first moment, Eddington flux (Hv), is related to the net flow of energy in a specific direction. It is the first derivative of the zeroth moment.

Question 37

What is the Second Moment and how is it related to the First Moment?

Answer 37

The second moment, the K integral, is related to the star's radiation pressure. It is obtained by taking the derivative of the first moment, Eddington flux, with respect to optical depth.

Question 38

What relationship exists between the second and first moments?

Answer 38

The second derivative of the second moment is equivalent to the first derivative of the first moment. This relationship is particularly useful in understanding the radiation field.

Question 39

What is the significance of equilibrium in the context of the stellar atmosphere?

Answer 39

Equilibrium in the stellar atmosphere implies a balance of opposing forces, resulting in constancy in macroscopic parameters like mass, radius, and temperature.

Question 40

What is Local Thermodynamic Equilibrium (LTE) and how is Planck’s law used?

Answer 40

Local thermodynamic equilibrium (LTE) assumes the star behaves like a black body. Planck's law (for surface brightness) is used to relate emission and absorption coefficients.

Question 41

How is Radiative Equilibrium achieved in a Planckian Atmosphere?

Answer 41

Radiative equilibrium in a Planckian atmosphere occurs when radiation is the sole means of energy transport, integrated flux is independent of opacity, such that J = B = S.

Question 42

What does Isotropy imply in the Eddington Approximation?

Answer 42

In the Eddington Approximation, isotropy means the emergent radiation is assumed to be isotropic. The K is expressed as one-third of the mean intensity (I).

Question 43

What does Frequency Independence imply in the Grey Atmosphere, and how does this affect the equation of transfer?

Answer 43

Frequency independence in a grey atmosphere implies that the absorption coefficient (kv) is independent of frequency. This results in a simplified equation of transfer and the removal of frequency dependence.

Question 44

What assumptions are combined to construct a source function for a star?

Answer 44

The assumptions include Eddington's approximation, a grey atmosphere, local and radiative equilibrium. These assumptions help construct a source function for the star.

Question 45

What is Eddington's approximation?

Answer 45

Eddington's approximation, assuming isotropic radiation, influences the source function, leading to K = I /3 = J/3.

Question 46

Why is a Grey Atmosphere characterized by Frequency Independence?

Answer 46

Grey Atmosphere is characterized by frequency independence, meaning the absorption coefficient (kv) is constant across all frequencies, resulting in no colour variations.

Question 47

What is the goal of constructing a source function using the given assumptions?

Answer 47

The goal is to relate flux to intensity by constructing a source function using assumptions such as Eddington's approximation, a grey atmosphere, and local and radiative equilibrium.

Question 48

What characterizes a grey atmosphere, and what assumptions are maintained throughout the discussion?

Answer 48

A grey atmosphere has opacity independent of frequency, and the assumptions from the previous chapter, including radiation pressure, LTE, radiative equilibrium, isotropy, and greyness, are still maintained.

Question 49

What does the temperature profile depend on and how do we determine it?

Answer 49

The expression for the temperature profile depends on the physical depth, i.e., optical depth. The goal is to determine this profile from a grey atmosphere.

Question 50

What is the equation of transfer, and how is it solved to obtain useful results?

Answer 50

The equation of transfer is solved for both outward and inward directions by setting limits for integrations. The boundary conditions are considered, leading to an exact solution in the case of a black body spectrum, suitable for a grey atmosphere.

Question 51

What is the significance of the constant C in the solution of the equation of transfer?

Answer 51

The constant C is determined through calculations and turns out to be 2/3 for the case of a black body spectrum, providing an exact solution for the grey atmosphere when looking radially outward.

Question 52

What is the temperature profile related to and what assumptions are made in constructing it?

Answer 52

Assuming a Planckian profile, the temperature profile is related to the integrated source function. The temperature is Te = T at an optical depth of τ=2/3.

Question 53

Is the Planckian profile a good approximation in the context of temperature profiles?

Answer 53

The Planckian profile is a good approximation at reasonable optical depths but less precise near the surface of stars. A better estimate involves rewriting the integrated mean intensity and source function in terms of the Hopf function.

Question 54

What is the concept of Limb Darkening, and what causes it?

Answer 54

Limb darkening is the observation that the intensity at the center of the solar disk is larger than at its edges. This phenomenon occurs because radiation from the center comes from hotter layers, while radiation from the limbs comes from cooler layers.

Question 55

How is the Eddington-Barbier Relation related to observations of limb darkening, and what assumptions is it based on?

Answer 55

The Eddington-Barbier Relation is a linear model for the source function that matches observations of limb darkening. It is based on the assumption of a Planckian atmosphere (Te = T, τ=2/3) and provides a simple relationship for emergent intensity.

Question 56

Under thermal equilibrium, how is a star primarily described, and what is the goal in defining a mean opacity for a non-grey atmosphere?

Answer 56

Under thermal equilibrium, a star is primarily described in terms of its temperature. The goal in defining a mean opacity for a non-grey atmosphere is to describe the atmosphere with an equivalent grey (or nearly-grey) model.

Question 57

How do we define the Rosseland Mean Opacity (kR)?

Answer 57

The Rosseland Mean Opacity (kR) is defined using the Eddington flux (Hv) specified at large optical depths in the stellar interior.

Question 58

What factors does the Rosseland Mean Opacity depend on, and how does it impact temperature gradients and convective regions?

Answer 58

The Rosseland Mean Opacity depends on local physical conditions of plasma and chemical abundances. When it becomes large, the temperature gradient increases to support it. High opacities make radiative energy transport less efficient, leading to larger temperature gradients and possible convective regions.

Question 59

What is the significance of considering other definitions of opacity, such as the flux-weighted mean opacity and the Planck mean opacity?

Answer 59

Other definitions of opacity, like the flux-weighted mean opacity and the Planck mean opacity, provide alternatives for specific scenarios. The flux-weighted mean opacity excludes scattering, while the Planck mean opacity is suitable for regions of thermal emission, assuming a black body.

Question 60

Why are Rosseland mean opacity calculations typically done numerically?

Answer 60

Rosseland mean opacity calculations are done numerically because the opacity has a complex frequency dependence. Numerical methods are more suitable for handling these dependencies and providing accurate results.

Question 61

Why is a source function needed to describe a stellar atmosphere with spectral lines, and how are absorption lines produced?

Answer 61

Spectral lines, being both frequency and optical depth dependent, require a source function. A radially decreasing source function through the photosphere produces an absorption line.

Question 62

What is the significance of atomic physics in understanding the opacity structure of a stellar atmosphere?

Answer 62

Atomic physics, described by the Boltzmann equation and Saha equation, determines the opacity structure. This involves understanding the relative population of ionization states and energy levels for ions in the atmosphere.

Question 63

What does the Boltzmann equation describe, and how are atomic energy levels populated?

Answer 63

The Boltzmann equation describes the ratio of population of two energy levels for a given ion in a gas at temperature T. The levels are populated inversely exponentially as a function of their energies (Ei,Ej)

Question 64

How is the Saha equation modelled and what does it describe?

Answer 64

The Saha equation is described by a Maxwell-Boltzmann factor. When collisions dominate the saha equation describes ionization rates, indicating that ionization increases with temperature and decreases with increasing electron density.

Question 65

When do stellar atmospheres exhibit departure from thermodynamic equilibrium and when do we have strict thermodynamic equilibrium?

Answer 65

If photons depart statistical equilibrium but particles remain in equilibrium, it's termed local thermodynamic equilibrium (LTE). If photons obey Planck's law and particles obey Maxwell-Boltzmann statistics we have strict thermodynamic equilibrium.

Question 66

What temperature range is valid for the ideal gas law in stars, and how is LTE still valid in this case?

Answer 66

The ideal gas law is valid in stars with effective temperatures ranging from 3000K to 50,000K. Even though spectral lines form due to high pressures, low enough densities maintain LTE.

Question 67

In a simplified two-energy level atom what quantifies emission, absorption and scattering.

Answer 67

In a simplified two-energy level atom, processes include stimulated emission, spontaneous emission, and absorption. These processes are quantified using the Einstein coefficients Aji, Bij, and Bji.

Question 68

How are Einstein coefficients related to the conservation of energy in spectral line processes?

Answer 68

Einstein coefficients Aji, Bij, and Bji, which define the probability of transition between levels i and j occurring in a particular atom, ensure that the same number of photons is emitted as absorbed for any given frequency, in line with the conservation of energy.

Question 69

Describe A and B-processes described by the Einstein coefficients.

Answer 69

A-processes (stimulated emission) do not require photons and do not attract specific intensity (Iv), while B-processes (absorption and stimulated emission) do require photons and attract specific intensity.

Question 70

Describe elastic collisions and explain why LTE remains valid for inelastic processes?

Answer 70

Elastic collisions dominate, tending to thermalize the medium and drive thermo-dynamic equilibrium. LTE remains valid for inelastic processes, as they are rarer in stellar atmospheres.

Question 71

How is the opacity of a spectral line related to the gains and losses of energy in the equation of transfer?

Answer 71

The opacity of a spectral line is related to the gains and losses of energy in the equation of transfer, which algebraically sums the radiation gains and losses through the stellar atmosphere.

Question 72

What parameters quantify the shape of an atomic line, and what is the significance of equivalent width?

Answer 72

The shape of an atomic line is quantified by parameters like equivalent width (Wλ) and line depth (Aλ). Equivalent width represents the width of a hypothetical atomic line of rectangular shape that absorbs all the radiation within it. The amount of energy extracted from the continuum by the line.

Question 73

How are the equivalent width and line depth related to the abundance of elements in stellar atmospheres?

Answer 73

Equivalent width (Wλ) and line depth (Aλ) are proportional to the elemental abundances for optically weak lines.

Question 74

What does the curve of growth predict, and how does it relate to the measurement of equivalent width in stellar spectra?

Answer 74

The curve of growth predicts the relationship between equivalent width (Wλ) and the number of atoms available for absorption (N). It provides a way to determine abundances using the measured equivalent width.

Question 75

What is the description of natural (Lorentz) broadening, and what causes it?

Answer 75

Natural broadening results from Heisenberg's uncertainty principle affecting energy levels. Atoms with electrons in level I can absorb photons not just at the line center but at frequencies surrounding it, leading to a Lorentz profile.

Question 76

How is the natural broadening line shape described, and what is the significance of the constant Γ?

Answer 76

The line exhibits an area-normalized Lorentz profile, and Γ is the radiative damping constant describing the damping of transitions in bound-bound transfers. It's a classical effect.

Question 77

What is the significance of the Full Width Half Maximum (FWHM) in describing a line's width?

Answer 77

FWHM is the width of the line profile at half its maximum intensity. It provides a measure of the broadening of the spectral line.

Question 78

How does Doppler broadening occur, and what is the role of thermal motions in this process?

Answer 78

Doppler broadening results from a distribution of particle velocities, mainly due to thermal motions. Radiating particles moving along the observer's line of sight experience Doppler shifts.

Question 79

How are the particles due to Doppler broadening modeled, and what is the most probable speed in the distribution?

Answer 79

Atmospheric particles are modeled with a Maxwell-Boltzmann velocity distribution. The most probable speed in the distribution is the speed at which the probability is maximized.

Question 80

How are Doppler shifts related to line-of-sight velocity, and what is the significance of positive and negative values?

Answer 80

Doppler shifts (Δv) are related to line-of-sight velocity (Vlos), with positive values measured away from the observer (redshift) and negative values towards the observer.

Question 81

What is the Doppler Full Width Half Maximum (FWHM), and how is it determined from the Doppler profile equation?

Answer 81

Doppler FWHM is found by setting the exponential term in the profile equation to 1/2. This gives a measure of the width of the line due to Doppler broadening.

Question 82

What is the Voigt line profile?

Answer 82

The Voigt line profile combines natural (Lorentz) and Doppler broadening. It converts the Doppler component into wavelength units and convolves it with the Lorentz component.

Question 83

In what regions does Doppler broadening dominate in the line profile, and where does Lorentz broadening dominate?

Answer 83

Doppler broadening dominates in the wings of the line, while Lorentz broadening dominates in the core of the line.

Question 84

What is the significance of microscopic turbulence in line broadening, and how is it modeled?

Answer 84

Microscopic turbulence is modeled as a Maxwell-Boltzmann distribution and contributes to line broadening. It is combined with thermal motions, resulting in a most probable velocity.

Question 85

What is macroscopic turbulence and what challenges arise from the independence of atmospheric elements?

Answer 85

Macroscopic turbulence considers large-scale, optically independent atmospheric elements. Challenges arise due to the non-uniqueness of the line profiles, making analysis model-dependent.

Question 86

What is the result of macroturbulence in a star without limb darkening.

Answer 86

Macroturbulence, in a star without limb darkening, results in a Doppler-shifted line profile. The line profile is dish-shaped, and the central depth decreases with increasing macroturbulent velocity.

Question 87

What is pressure (collisional) broadening, and what causes it?

Answer 87

Pressure broadening is caused by the perturbation of an atom's potential by neighboring particles in a stellar plasma. It results from the modification of energy levels due to the presence of many particles.

Question 88

In which regions of a spectral line does pressure broadening dominate, and what does it imply about stellar atmospheres? Additionally where is it most prevalent?

Answer 88

Pressure broadening dominates in the wings of a spectral line, indicating higher surface gravity. It is more prevalent in main sequence stars compared to red giants with the same surface temperature.

Question 89

What other broadening mechanisms are mentioned, and when do they become significant?

Answer 89

Other broadening mechanisms include hyperfine structure, isotope splitting, and Zeeman splitting. They become significant in specific cases, such as Zeeman splitting being more pronounced in the infrared.

Question 90

What is the Stark effect and where is it most prevalent?

Answer 90

The Stark effect is pressure broadening caused by the splitting of degenerate atomic energy levels due to the presence of external electric fields. It is most prevalent in hydrogen Balmer lines and the shape depends strongly on pressure.

Question 91

How is pressure broadening related to electron density in line-forming regions?

Answer 91

The number of pressure broadening collisions is historically used to estimate electron density in line-forming regions.

Question 92

What is the significance of Zeeman splitting, and how is it used in estimating magnetic fields?

Answer 92

Zeeman splitting, caused by the presence of magnetic fields, can lead to anomalously broad spectral lines. The magnitude of the magnetic field can be estimated from the observed line broadening, although strong magnetic fields are rare.

Question 93

How do different isotopes contribute to line broadening, and when is the effect most noticeable?

Answer 93

Different isotopes, having different nuclear masses, contribute to isotope splitting. The effect is most noticeable in light elements, but it is generally small.

Question 94

What is Hyperfine structure and where is it negligble?

Answer 94

Hyperfine structure causes a splitting of electronic energy levels, but it is usually negligible compared to thermal (Doppler) broadening.

Question 95

What is the difference between optically thick and optically thin in terms of the Voigt Profile?

Answer 95

Optically thick lines will become saturated whereas optically thin lines will not be saturated.

Question 96

Why does pressure broadening dominate in the wings of a spectral line?

Answer 96

The density is higher in the wings and the potential for collision is higher than in the core.

Question 97

What are the assumptions in the Bohr model of the atom?

Answer 97

the continuum refers to electronic transitions from bound to free where when n is free, n -> infinity.

Question 98

Describe the line absorption coefficient.

Answer 98

It is proportional to abundance of the element responsible for the atomic line.

Question 99

Describe the absorption coefficient of the continuum.

Answer 99

It is independent of the abundance of a given element.

Question 100

Describe atomic lines in an isothermal atmosphere.

Answer 100

there is no equivalent width in an isothermal atmosphere hence no atomic lines are visible.