Difference between revisions of "Light and distances"
From Universe in Problems
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=== Problem 1: proper distance === | === Problem 1: proper distance === | ||
Determine the "physical" distance -- the proper distance measured along the hypersurface of constant cosmological time -- to an object that is observed with redshift $z$ | Determine the "physical" distance -- the proper distance measured along the hypersurface of constant cosmological time -- to an object that is observed with redshift $z$ | ||
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=== Problem 2: comoving distance in a flat Universe=== | === Problem 2: comoving distance in a flat Universe=== | ||
Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe | Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe | ||
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=== Problem 3: comoving distance in Einstein-de Sitter === | === Problem 3: comoving distance in Einstein-de Sitter === | ||
Solve the previous problem for a flat Universe with domination of non-relativistic matter (the Einstein-de Sitter model) | Solve the previous problem for a flat Universe with domination of non-relativistic matter (the Einstein-de Sitter model) | ||
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=== Problem 4: recession velocity === | === Problem 4: recession velocity === | ||
Determine the recession velocity caused by the cosmological expansion for an object with redshift $z$ in a flat Universe | Determine the recession velocity caused by the cosmological expansion for an object with redshift $z$ in a flat Universe | ||
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''In cosmology the are other types of distances used, besides the proper and comoving one. One of the most frequently used is the photometric distance. Let'' $E$, $[E]=J/s$, ''be the internal absolute luminosity of some source. The observer on Earth detects energy flux'' $F$, $[F]=J/s\cdot m^2$. ''The luminosity distance to the source'' $d_{L}$ ''is then defined through'' | ''In cosmology the are other types of distances used, besides the proper and comoving one. One of the most frequently used is the photometric distance. Let'' $E$, $[E]=J/s$, ''be the internal absolute luminosity of some source. The observer on Earth detects energy flux'' $F$, $[F]=J/s\cdot m^2$. ''The luminosity distance to the source'' $d_{L}$ ''is then defined through'' | ||
\[F=\frac{E}{4\pi d_{L}^{2}}.\] | \[F=\frac{E}{4\pi d_{L}^{2}}.\] | ||
''Thus this would be the distance to the observed object, given its absolute and observed luminosities, in a flat and stationary Universe. Non-stationarity and curvature imply that $d_{L}$ in general does not coincide with the proper distance.'' | ''Thus this would be the distance to the observed object, given its absolute and observed luminosities, in a flat and stationary Universe. Non-stationarity and curvature imply that $d_{L}$ in general does not coincide with the proper distance.'' | ||
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=== Problem 5: luminosity distance in a flat Universe === | === Problem 5: luminosity distance in a flat Universe === | ||
Express the luminosity distance in terms of observed redshift for a spatially flat Universe | Express the luminosity distance in terms of observed redshift for a spatially flat Universe | ||
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<p style="text-align: left;"> solution</p> | <p style="text-align: left;"> solution</p> | ||
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=== Problem 6: generalization to arbitrary curvature === | === Problem 6: generalization to arbitrary curvature === | ||
Generalize the result of the previous problem to the case of arbitrary curvature | Generalize the result of the previous problem to the case of arbitrary curvature | ||
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=== Problem 7: multi-component flat Universe === | === Problem 7: multi-component flat Universe === | ||
Find the expression for the luminosity distance for the multi-component flat Universe | Find the expression for the luminosity distance for the multi-component flat Universe | ||
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=== Problem 8: luminocity distance in terms of deceleration parameter === | === Problem 8: luminocity distance in terms of deceleration parameter === | ||
Express the luminosity distance in a flat Universe in terms of the redshift dependence of deceleration parameter $q(z)$ | Express the luminosity distance in a flat Universe in terms of the redshift dependence of deceleration parameter $q(z)$ | ||
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=== Problem 9: Einstein-de Sitter === | === Problem 9: Einstein-de Sitter === | ||
Express the luminosity distance in terms of redshift for the Einstein-de Sitter model | Express the luminosity distance in terms of redshift for the Einstein-de Sitter model | ||
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=== Problem 10: small redshifts === | === Problem 10: small redshifts === | ||
Show that in the first order by $z\ll 1$ luminosity distance is $d_{L}\approx z /H_{0}$ and find the second order correction | Show that in the first order by $z\ll 1$ luminosity distance is $d_{L}\approx z /H_{0}$ and find the second order correction | ||
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=== Problem 11: the closed and open dusty Universes === | === Problem 11: the closed and open dusty Universes === | ||
Derive the luminosity distance as function of redshift for the closed and open models of the Universe, dominated by non-relativistic matter (dust) | Derive the luminosity distance as function of redshift for the closed and open models of the Universe, dominated by non-relativistic matter (dust) | ||
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\[d_{A}=\frac{\delta l}{\delta \theta}.\] | \[d_{A}=\frac{\delta l}{\delta \theta}.\] | ||
''Again, in a stationary and flat Universe this is reduced to the ordinary distance, while in general they differ.'' | ''Again, in a stationary and flat Universe this is reduced to the ordinary distance, while in general they differ.'' | ||
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=== Problem 12: angular diameter distance in terms of redshift === | === Problem 12: angular diameter distance in terms of redshift === | ||
Express the angular diameter distance in terms of the observed redshift | Express the angular diameter distance in terms of the observed redshift | ||
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Revision as of 10:51, 27 September 2012
Contents
- 1 Problem 1: proper distance
- 2 Problem 2: comoving distance in a flat Universe
- 3 Problem 3: comoving distance in Einstein-de Sitter
- 4 Problem 4: recession velocity
- 5 Problem 5: luminosity distance in a flat Universe
- 6 Problem 6: generalization to arbitrary curvature
- 7 Problem 7: multi-component flat Universe
- 8 Problem 8: luminocity distance in terms of deceleration parameter
- 9 Problem 9: Einstein-de Sitter
- 10 Problem 10: small redshifts
- 11 Problem 11: the closed and open dusty Universes
- 12 Problem 12: angular diameter distance in terms of redshift
Problem 1: proper distance
Determine the "physical" distance -- the proper distance measured along the hypersurface of constant cosmological time -- to an object that is observed with redshift $z$
solution
solution
Problem 2: comoving distance in a flat Universe
Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe
solution
solution
Problem 3: comoving distance in Einstein-de Sitter
Solve the previous problem for a flat Universe with domination of non-relativistic matter (the Einstein-de Sitter model)
solution
solution
Problem 4: recession velocity
Determine the recession velocity caused by the cosmological expansion for an object with redshift $z$ in a flat Universe
solution
solution
In cosmology the are other types of distances used, besides the proper and comoving one. One of the most frequently used is the photometric distance. Let $E$, $[E]=J/s$, be the internal absolute luminosity of some source. The observer on Earth detects energy flux $F$, $[F]=J/s\cdot m^2$. The luminosity distance to the source $d_{L}$ is then defined through
\[F=\frac{E}{4\pi d_{L}^{2}}.\]
Thus this would be the distance to the observed object, given its absolute and observed luminosities, in a flat and stationary Universe. Non-stationarity and curvature imply that $d_{L}$ in general does not coincide with the proper distance.
Problem 5: luminosity distance in a flat Universe
Express the luminosity distance in terms of observed redshift for a spatially flat Universe
solution
solution
Problem 6: generalization to arbitrary curvature
Generalize the result of the previous problem to the case of arbitrary curvature
solution
solution
Problem 7: multi-component flat Universe
Find the expression for the luminosity distance for the multi-component flat Universe
solution
solution
Problem 8: luminocity distance in terms of deceleration parameter
Express the luminosity distance in a flat Universe in terms of the redshift dependence of deceleration parameter $q(z)$
solution
solution
Problem 9: Einstein-de Sitter
Express the luminosity distance in terms of redshift for the Einstein-de Sitter model
solution
solution
Problem 10: small redshifts
Show that in the first order by $z\ll 1$ luminosity distance is $d_{L}\approx z /H_{0}$ and find the second order correction
solution
solution
Problem 11: the closed and open dusty Universes
Derive the luminosity distance as function of redshift for the closed and open models of the Universe, dominated by non-relativistic matter (dust)
solution
solution
Another distance used is the angular diameter distance. It is defined through the angular dimension of the object $\delta \theta$ and its proper transverse size $\delta l$ as
\[d_{A}=\frac{\delta l}{\delta \theta}.\]
Again, in a stationary and flat Universe this is reduced to the ordinary distance, while in general they differ.
Problem 12: angular diameter distance in terms of redshift
Express the angular diameter distance in terms of the observed redshift
solution
solution
