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[[Image:2012 Transit of Venus from SF.jpg|right|thumb|256px|The limb darkened Sun - An image of the Sun in visible light showing the limb darkening effect as a drop in intensity towards the edge or "limb" of the solar disk. The image was taken during 2012 [[Transit of Venus]].]]
 
'''Limb darkening''' refers to the diminishing of intensity in the image of a star as one moves from the center of the image of the star to the edge or "[[wikt:limb#Etymology 2|limb]]" of the image. Limb darkening occurs as the result of two effects:
 
* The density of the star diminishes as the distance from the center increases
 
* The temperature of the star diminishes as the distance from the center increases.
 
[[Image:Limb darkening layers.svg|left|thumb|256 px|An idealized case of limb darkening. The outer boundary is the radius at which photons emitted from the star are no longer absorbed. ''L''&nbsp; is a distance for which the optical depth is unity. High-temperature photons emitted at A will just barely escape from the star, as will the low temperature photons emitted at B. Note that for a typical star, this drawing is not to scale. E.g., for the [[Sun]], ''L''&nbsp; would be only a few hundred km.]]
 
Crucial to understanding limb darkening is the idea of [[optical depth]]. An optical
depth of unity is that thickness of absorbing gas from which a fraction of 1/[[e (mathematical constant)|e]] [[photons]] can escape. This is what defines the visible edge of a star since it is at an optical depth of unity that the star becomes opaque. The radiation reaching us is closely approximated by the sum of all the emission along the entire line of sight, up to that point where the optical depth is unity. When we look near the edge of a star, we cannot "see" to the same depth as when we look at the center because the line of sight must travel at an oblique angle through the stellar gas when looking near the limb. In other words, the solar radius at which we see the optical depth as being unity increases as we move our line of sight towards the limb.
 
The second effect is the fact that the [[effective temperature]] of the stellar atmosphere is (usually) decreasing for an increasing distance from the  center of the star. The radiation emitted from a gas is a strongly increasing function of temperature.  For a [[black body]], for example, the spectrally integrated intensity is proportional to the fourth power of the temperature ([[Stefan-Boltzmann law]]). Since when we look at a star, at first approximation the radiation comes from the point at which the optical depth is unity, and that point is deeper in when looking at the center, the temperature will be higher, and the intensity will be greater, than when we look at the limb.
 
In fact,  the temperature in the atmosphere of a star does not always decrease with increasing height, and for certain  [[spectral line]]s, the optical depth is unity in a region of increasing temperature. In this case we see the phenomenon of "limb brightening".
At very long (IR to radio) and very short (EUV to X-ray) wavelengths the situation becomes much more complicated.
A coronal emission such as soft X-radiation will be [[optical depth|optically thin]] and thus be characteristically limb-brightened.
Further complication comes from the existence of rough (three-dimensional) structure.
The classical analysis of stellar limb darkening, as described below, assumes the existence of a smooth hydrostatic equilibrium, and at some level of precision this assumption must fail (most obviously in [[sunspots]] and [[faculae]], but generally everywhere).
The analysis of these effects is presently in its infancy because of its computational difficulty [1].
 
== Calculation of limb darkening ==
 
[[Image:Limb darkening geometry.png|right|frame|Limb darkening geometry.
The star is centered at ''O''&nbsp; and has radius ''R''&nbsp;. The
observer is at point ''P''&nbsp; a distance ''r''&nbsp; from
the center of the star, and is looking at point ''S''&nbsp;
on the surface of the star. From the point of view of the observer,
''S''&nbsp; is at an angle θ from a line through the center of the star, and the edge
or ''limb'' of the star is at angle Ω.]]
 
In the figure on the right, as long as the observer at point P is outside the stellar atmosphere,
the intensity seen in the direction θ will be a function only of the angle of incidence
ψ. This is most conveniently approximated as a polynomial in cos(ψ)
 
:<math>
\frac{I(\psi)}{I(0)} = \sum_{k=0}^N a_k \, \textrm{cos}^k(\psi)
</math>
 
where ''I''(ψ) is the intensity seen at P along a line of sight forming angle
ψ with respect to the stellar radius, and ''I''(0) is the central intensity.
In order that the ratio be unity for ψ=0, we must have:
 
:<math>
\sum_{k=0}^N a_k =1
</math>
 
For example, for a [[Lambert's cosine law|Lambertian]] radiator (no limb darkening) we will have all ''a''<sub>k</sub>=0 except ''a''<sub>0</sub>=1. As another example, for the sun at 550&nbsp;nm, the limb darkening is well expressed by ''N''=2 and
 
:<math>a_0=1-a_1-a_2\,</math>
:<math>a_1=0.93\,</math>
:<math>a_2=-0.23\,</math>
 
(See Cox, 2000). Note - the equation for limb darkening is sometimes more conveniently written as:
 
:<math>
\frac{I(\psi)}{I(0)} = 1+\sum_{k=1}^N A_k \, (1-\cos(\psi))^k
</math>
 
which now has ''N''&nbsp; independent coefficients rather than ''N+1''&nbsp; coefficients which must sum to unity.
 
We can convert from ψ to θ using the relationship:
 
:<math>
\cos(\psi) =
\frac{\sqrt{\cos^2(\theta)-\cos^2(\Omega)}}{\sin(\Omega)}
</math>
 
where Ω is the angle from the observer to the limb of the star.
 
The above approximation can be used to derive an [[analytic expression]] for the ratio of
the mean intensity to the central intensity. The mean intensity I<sub>m</sub> is the integral of the intensity over the disk of the star divided by the solid angle subtended by the disk:
 
:<math>I_m = \frac{\int I(\psi)d\omega}{\int d\omega}</math>
 
where dω=sin(θ)dθdφ is a solid angle element and the integrals are
over the disk: 0≤φ≤2π and 0≤θ≤Ω. Although this equation
can be solved analytically, it is rather cumbersome. However, for an observer at infinite distance from the star, the above equation simplifies to:
 
:<math>\frac{I_m}{I(0)} = 2 \sum_{k=0}^N \frac{a_k}{k+2}</math>
 
== References ==
 
* {{cite book | author=Billings, Donald E. | title=A Guide to the Solar Corona | publisher=Academic Press, New York | year=1966 | id =  }}
 
* {{cite book | author=Cox, Arthur N. (ed) | title=Allen's Astrophysical Quantities | edition=14th| publisher=Springer-Verlag, NY | year=2000 | isbn = 0-387-98746-0 }}
 
* {{cite journal | author=Milne, E.A. | title= | journal=MNRAS | year=1921 | volume=81 | pages=361–375|bibcode = 1921MNRAS..81..361M }}
 
* {{cite journal | author=Minnaert, M. | title= | journal=Z.f. Ap. | year=1930 | volume=1 | pages=209}}
 
* {{cite journal | author=Neckel, H. and Labs, D. | title=Solar Limb Darkening 1986-1990 | journal=Solar Physics | year=1994 | volume=153 | issue=1-2 | pages=91–114 | doi=10.1007/BF00712494 | bibcode=1994SoPh..153...91N}}
* {{cite journal | author=van de Hulst, H. C. | title= | journal=Bull. Astron. Inst. Netherlands | year=1950 | volume=11 | issue=410 | pages=135}}
 
* Steiner, O., ''Photospheric processes and magnetic flux tubes,'' (2007) [http://adsabs.harvard.edu/abs/2007arXiv0709.0081S]
 
[[Category:Stellar phenomena]]
[[Category:Solar phenomena]]

Revision as of 06:25, 20 November 2013

The limb darkened Sun - An image of the Sun in visible light showing the limb darkening effect as a drop in intensity towards the edge or "limb" of the solar disk. The image was taken during 2012 Transit of Venus.

Limb darkening refers to the diminishing of intensity in the image of a star as one moves from the center of the image of the star to the edge or "limb" of the image. Limb darkening occurs as the result of two effects:

  • The density of the star diminishes as the distance from the center increases
  • The temperature of the star diminishes as the distance from the center increases.
An idealized case of limb darkening. The outer boundary is the radius at which photons emitted from the star are no longer absorbed. L  is a distance for which the optical depth is unity. High-temperature photons emitted at A will just barely escape from the star, as will the low temperature photons emitted at B. Note that for a typical star, this drawing is not to scale. E.g., for the Sun, L  would be only a few hundred km.

Crucial to understanding limb darkening is the idea of optical depth. An optical depth of unity is that thickness of absorbing gas from which a fraction of 1/e photons can escape. This is what defines the visible edge of a star since it is at an optical depth of unity that the star becomes opaque. The radiation reaching us is closely approximated by the sum of all the emission along the entire line of sight, up to that point where the optical depth is unity. When we look near the edge of a star, we cannot "see" to the same depth as when we look at the center because the line of sight must travel at an oblique angle through the stellar gas when looking near the limb. In other words, the solar radius at which we see the optical depth as being unity increases as we move our line of sight towards the limb.

The second effect is the fact that the effective temperature of the stellar atmosphere is (usually) decreasing for an increasing distance from the center of the star. The radiation emitted from a gas is a strongly increasing function of temperature. For a black body, for example, the spectrally integrated intensity is proportional to the fourth power of the temperature (Stefan-Boltzmann law). Since when we look at a star, at first approximation the radiation comes from the point at which the optical depth is unity, and that point is deeper in when looking at the center, the temperature will be higher, and the intensity will be greater, than when we look at the limb.

In fact, the temperature in the atmosphere of a star does not always decrease with increasing height, and for certain spectral lines, the optical depth is unity in a region of increasing temperature. In this case we see the phenomenon of "limb brightening". At very long (IR to radio) and very short (EUV to X-ray) wavelengths the situation becomes much more complicated. A coronal emission such as soft X-radiation will be optically thin and thus be characteristically limb-brightened. Further complication comes from the existence of rough (three-dimensional) structure. The classical analysis of stellar limb darkening, as described below, assumes the existence of a smooth hydrostatic equilibrium, and at some level of precision this assumption must fail (most obviously in sunspots and faculae, but generally everywhere). The analysis of these effects is presently in its infancy because of its computational difficulty [1].

Calculation of limb darkening

File:Limb darkening geometry.png
Limb darkening geometry. The star is centered at O  and has radius R . The observer is at point P  a distance r  from the center of the star, and is looking at point S  on the surface of the star. From the point of view of the observer, S  is at an angle θ from a line through the center of the star, and the edge or limb of the star is at angle Ω.

In the figure on the right, as long as the observer at point P is outside the stellar atmosphere, the intensity seen in the direction θ will be a function only of the angle of incidence ψ. This is most conveniently approximated as a polynomial in cos(ψ)

where I(ψ) is the intensity seen at P along a line of sight forming angle ψ with respect to the stellar radius, and I(0) is the central intensity. In order that the ratio be unity for ψ=0, we must have:

For example, for a Lambertian radiator (no limb darkening) we will have all ak=0 except a0=1. As another example, for the sun at 550 nm, the limb darkening is well expressed by N=2 and

(See Cox, 2000). Note - the equation for limb darkening is sometimes more conveniently written as:

which now has N  independent coefficients rather than N+1  coefficients which must sum to unity.

We can convert from ψ to θ using the relationship:

where Ω is the angle from the observer to the limb of the star.

The above approximation can be used to derive an analytic expression for the ratio of the mean intensity to the central intensity. The mean intensity Im is the integral of the intensity over the disk of the star divided by the solid angle subtended by the disk:

where dω=sin(θ)dθdφ is a solid angle element and the integrals are over the disk: 0≤φ≤2π and 0≤θ≤Ω. Although this equation can be solved analytically, it is rather cumbersome. However, for an observer at infinite distance from the star, the above equation simplifies to:

References

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Steiner, O., Photospheric processes and magnetic flux tubes, (2007) [1]
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