Semester 1 - Definitions Flashcards
What is the focus when considering intensity within a stellar atmosphere?
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.”
What are the key components of a generalized equation of transfer for photons in a stellar atmosphere?
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.
What assumptions are made to simplify the equation of transfer for stellar atmospheres?
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.
What is Integrated Intensity (I) in the context of a stellar atmosphere?
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.
What is Specific Intensity (Iv) in the context of a stellar atmosphere?
Specific Intensity is the intensity per unit frequency range. It involves considering photons in a specific frequency range (v to v+dv)
What is Flux (F) in the context of a stellar atmosphere?
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.
What does the conservation of energy along a ray path imply?
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.
What are the consequences of intensity being constant along a ray path?
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
How is energy density defined in the context of a small volume in a stellar atmosphere?
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.
What are the key parameterizations used to simplify calculations in stellar atmospheres?
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.
Why are observations limited to measuring flux in stellar atmospheres, and what is flux at the stellar surface?
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.
What are the losses and gains involved in the interaction of radiation with matter in a stellar atmosphere?
Losses involve absorption and scattering processes, described by absorption and scattering coefficients. Gains involve emission processes, described by the emission coefficient.
What are the labeling conventions used for absorption and scattering coefficients?
Subscripts are used, with v indicating a range of frequencies. Unsubscripted parameters could be integrated or frequency-independent (grey atmosphere).
What is the Absorption Coefficient (κv)
The absorption coefficient (κv) represents the reduction in specific intensity (Iv) due to absorption processes. It is frequency-dependent.
What is the Mean Intensity (J) in the context of Stellar Physics?
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.
How is the Mean Free Path (L) related to the Scattering Coefficient (σv)?
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.
How are Absorption and Scattering Combined?
All intensity loss processes are combined into a single parameter
kv = κv + σv
, simplifying the representation and highlighting the dominant loss process.
What is the Emission Coefficient (jv)?
The emission coefficient (jv) describes how radiation is enhanced as it passes through a medium. It represents the amount of energy emitted by volume.
What is the general form of the Equation of Transfer?
The equation of transfer is an algebraic sum of intensity loss and gain processes. It’s represented as a first-order differential equation.
What is Optical Depth (τv)?
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.
What is the Source Function (Sv)?
The source function (Sv) is the ratio of emission to combined absorption coefficients. It indicates the ratio of intensity gained to intensity lost.
How is the Equation of Transfer Simplified for Plane-Parallel Atmosphere Approximation?
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.
What are the limiting cases discussed in solving the Equation of Transfer?
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.
Describe the Optically Thick Atmosphere.
An optically thick atmosphere has a large optical depth (τv»1), and there is no spectral line formation.
Describe the Optically Thin Atmosphere.
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.
What is the Plane-Parallel Atmosphere Approximation?
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.
How is the Equation of Transfer Modified for Plane-Parallel Atmosphere?
The equation of transfer is modified in the plane-parallel atmosphere approximation, introducing optical depth redux (dtau (ν, mu)).
What does Optical Depth Redux involve?
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
What is Bolometric Intensity (Bv)?
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.
What are Emission Lines in Astrophysics, and what can this indicate?
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.
What are Absorption Lines in Astrophysics?
Absorption lines occur in regions where stellar temperatures decreases outwards such that I(v,0) > Sv.
What is Line Saturation in Astrophysics?
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.
What are the moments of the radiation field?
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.
How are moments of the radiation field related to each other?
The moments are related through integration, differentiation, and mathematical relationships. As the moment order (n) increases, the complexity of the physics involved also increases.
What is the Zeroth Moment, and what does it represent?
The zeroth moment represents the mean (specific) intensity (J), analogous to the particle density of the atmosphere.
What does the First Moment (Eddington Flux) describe and how does it relate to the zeroth moment?
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.
What is the Second Moment and how is it related to the First Moment?
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.
What relationship exists between the second and first moments?
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.
What is the significance of equilibrium in the context of the stellar atmosphere?
Equilibrium in the stellar atmosphere implies a balance of opposing forces, resulting in constancy in macroscopic parameters like mass, radius, and temperature.
What is Local Thermodynamic Equilibrium (LTE) and how is Planck’s law used?
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.
How is Radiative Equilibrium achieved in a Planckian Atmosphere?
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.
What does Isotropy imply in the Eddington Approximation?
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).
What does Frequency Independence imply in the Grey Atmosphere, and how does this affect the equation of transfer?
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.
What assumptions are combined to construct a source function for a star?
The assumptions include Eddington’s approximation, a grey atmosphere, local and radiative equilibrium. These assumptions help construct a source function for the star.
What is Eddington’s approximation?
Eddington’s approximation, assuming isotropic radiation, influences the source function, leading to K = I /3 = J/3.
Why is a Grey Atmosphere characterized by Frequency Independence?
Grey Atmosphere is characterized by frequency independence, meaning the absorption coefficient (kv) is constant across all frequencies, resulting in no colour variations.
What is the goal of constructing a source function using the given assumptions?
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.
What characterizes a grey atmosphere, and what assumptions are maintained throughout the discussion?
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.
What does the temperature profile depend on and how do we determine it?
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.
What is the equation of transfer, and how is it solved to obtain useful results?
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.
What is the significance of the constant C in the solution of the equation of transfer?
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.
What is the temperature profile related to and what assumptions are made in constructing it?
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.
Is the Planckian profile a good approximation in the context of temperature profiles?
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.
What is the concept of Limb Darkening, and what causes it?
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.
How is the Eddington-Barbier Relation related to observations of limb darkening, and what assumptions is it based on?
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.
Under thermal equilibrium, how is a star primarily described, and what is the goal in defining a mean opacity for a non-grey atmosphere?
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.
How do we define the Rosseland Mean Opacity (kR)?
The Rosseland Mean Opacity (kR) is defined using the Eddington flux (Hv) specified at large optical depths in the stellar interior.
What factors does the Rosseland Mean Opacity depend on, and how does it impact temperature gradients and convective regions?
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.
What is the significance of considering other definitions of opacity, such as the flux-weighted mean opacity and the Planck mean opacity?
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.
Why are Rosseland mean opacity calculations typically done numerically?
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.
Why is a source function needed to describe a stellar atmosphere with spectral lines, and how are absorption lines produced?
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.
What is the significance of atomic physics in understanding the opacity structure of a stellar atmosphere?
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.
What does the Boltzmann equation describe, and how are atomic energy levels populated?
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)
How is the Saha equation modelled and what does it describe?
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.
When do stellar atmospheres exhibit departure from thermodynamic equilibrium and when do we have strict thermodynamic equilibrium?
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.
What temperature range is valid for the ideal gas law in stars, and how is LTE still valid in this case?
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.
In a simplified two-energy level atom what quantifies emission, absorption and scattering.
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.
How are Einstein coefficients related to the conservation of energy in spectral line processes?
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.
Describe A and B-processes described by the Einstein coefficients.
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.
Describe elastic collisions and explain why LTE remains valid for inelastic processes?
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.
How is the opacity of a spectral line related to the gains and losses of energy in the equation of transfer?
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.
What parameters quantify the shape of an atomic line, and what is the significance of equivalent width?
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.
How are the equivalent width and line depth related to the abundance of elements in stellar atmospheres?
Equivalent width (Wλ) and line depth (Aλ) are proportional to the elemental abundances for optically weak lines.
What does the curve of growth predict, and how does it relate to the measurement of equivalent width in stellar spectra?
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.
What is the description of natural (Lorentz) broadening, and what causes it?
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.
How is the natural broadening line shape described, and what is the significance of the constant Γ?
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.
What is the significance of the Full Width Half Maximum (FWHM) in describing a line’s width?
FWHM is the width of the line profile at half its maximum intensity. It provides a measure of the broadening of the spectral line.
How does Doppler broadening occur, and what is the role of thermal motions in this process?
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.
How are the particles due to Doppler broadening modeled, and what is the most probable speed in the distribution?
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.
How are Doppler shifts related to line-of-sight velocity, and what is the significance of positive and negative values?
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.
What is the Doppler Full Width Half Maximum (FWHM), and how is it determined from the Doppler profile equation?
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.
What is the Voigt line profile?
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.
In what regions does Doppler broadening dominate in the line profile, and where does Lorentz broadening dominate?
Doppler broadening dominates in the wings of the line, while Lorentz broadening dominates in the core of the line.
What is the significance of microscopic turbulence in line broadening, and how is it modeled?
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.
What is macroscopic turbulence and what challenges arise from the independence of atmospheric elements?
Macroscopic turbulence considers large-scale, optically independent atmospheric elements. Challenges arise due to the non-uniqueness of the line profiles, making analysis model-dependent.
What is the result of macroturbulence in a star without limb darkening.
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.
What is pressure (collisional) broadening, and what causes it?
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.
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?
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.
What other broadening mechanisms are mentioned, and when do they become significant?
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.
What is the Stark effect and where is it most prevalent?
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.
How is pressure broadening related to electron density in line-forming regions?
The number of pressure broadening collisions is historically used to estimate electron density in line-forming regions.
What is the significance of Zeeman splitting, and how is it used in estimating magnetic fields?
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.
How do different isotopes contribute to line broadening, and when is the effect most noticeable?
Different isotopes, having different nuclear masses, contribute to isotope splitting. The effect is most noticeable in light elements, but it is generally small.
What is Hyperfine structure and where is it negligble?
Hyperfine structure causes a splitting of electronic energy levels, but it is usually negligible compared to thermal (Doppler) broadening.
What is the difference between optically thick and optically thin in terms of the Voigt Profile?
Optically thick lines will become saturated whereas optically thin lines will not be saturated.
Why does pressure broadening dominate in the wings of a spectral line?
The density is higher in the wings and the potential for collision is higher than in the core.
What are the assumptions in the Bohr model of the atom?
the continuum refers to electronic transitions from bound to free where when n is free, n -> infinity.
Describe the line absorption coefficient.
It is proportional to abundance of the element responsible for the atomic line.
Describe the absorption coefficient of the continuum.
It is independent of the abundance of a given element.
Describe atomic lines in an isothermal atmosphere.
there is no equivalent width in an isothermal atmosphere hence no atomic lines are visible.
