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.
