Darkening to the edge

The image of the Sun in the visible spectrum obtained using the solar filter, showing the effect of darkening to the edge as a decrease in brightness towards the edge of the disk. The image was obtained during the passage of Venus across the disk of the Sun in 2012 .

Darkening of the disk to the edge is an optical effect when observing stars, including the Sun , in which the central part of the star's disk appears brighter than the edge or limb of the disk. Understanding this effect allowed us to create models of stellar atmospheres taking into account a similar brightness gradient, which contributed to the development of radiation transfer theory.

Fundamentals of Theory

A simplified diagram of the effect of darkening the disc to the edge. The outer boundary is the radius at which the emitted photons do not experience absorption. L is the distance whose optical thickness is unity. The high temperature photons emitted at point A will experience the same absorption as the photons emitted at point B. The diagram given here is not shown to scale, because, for example, for the Sun, L is several hundred km with a radius of the Sun of about 700 thousand km.

The key concept in describing this effect is optical thickness . The distance equal to the optical thickness indicates the thickness of the gas layer, from which only a fraction of photons equal to 1 / e can escape. This value determines the visible edge of the star, because at a depth of several units of optical thickness the star becomes opaque to radiation. The observed star radiation can be represented by the sum of the radiation along the line of sight to the point at which the optical thickness becomes equal to unity. When observing the edge of a star, the observer sees the layers of the star at a shallower depth compared to the observation of the center of the disk, since in the first case the line of sight passes through the layers of gas at a large angle to the normal. In other words, the distance from the center of the star to the layer having a unit optical thickness increases with the displacement of the line of sight from the center of the disk to the edge.

Another effect is that the effective temperature in the star’s atmosphere usually decreases with increasing distance from the center of the star. The properties of radiation are functions of a given temperature. For example, in the case of a star approaching with a completely black body , the spectrum-integrated intensity is proportional to the fourth degree of temperature ( Stefan-Boltzmann law ). Since when we observe a star, in a first approximation, the radiation comes from a layer whose optical thickness is unity and the depth of this layer is greater when observing the center of the star, in the central region of the disk, radiation comes from a layer with a higher temperature, the radiation intensity is higher.

In reality, the temperature in stellar atmospheres does not always strictly decrease with increasing distance from the center of the star, and for some spectral lines a single optical thickness is achieved in the region of increasing temperature. In this case, the observer sees the effect of increasing the brightness to the edge of the disk. For the Sun, the presence of a minimum temperature region means that the effect of enhancing the brightness to the edge of the disk will dominate in the region of far infrared radiation and radio emission . Outside the lower layers of the solar atmosphere, above the region of the minimum temperature of the sun is the solar corona , which has a temperature of about 10 ⁶ K. For most wavelengths, this region is optically thin (has a small optical thickness), and therefore, an increase in brightness to the edge should be observed under the assumption of spherical symmetry.

A classical analysis of the effect assumes the existence of hydrostatic equilibrium, but starting from a certain level of accuracy, such an assumption ceases to be fulfilled (for example, in sunspots , torches ). The boundary between the chromosphere and the solar corona represents a complex transition region, well observed in ultraviolet radiation .

Disc Calculation

The geometric aspect of disc darkening. Point O denotes the center of the star, the radius of the star is equal to R. The observer is at point P at a distance r from the center of the star. Point S is observed on the surface of a star. From the observer's point of view, point S is removed at an angle θ from the line drawn from the observer to the center of the star. The angle between the directions from the observer to the center of the star and to the limb is equal to Ω . The angle between the normal to the surface of the star at point S and the direction from point S to the observer is ψ .

In the figure on the right, the observer is at point P outside the atmosphere of the star. The radiation intensity observed in the θ direction is a function of the angle ψ . The intensity can be represented as a polynomial in powers of cos ψ:

{\frac {I(\psi )}{I(0)}}=\sum _{k=0}^{N}a_{k}\cos ^{k}\psi ,

{\ displaystyle {\ frac {I (\ psi)} {I (0)}} = \ sum _ {k = 0} ^ {N} a_ {k} \ cos ^ {k} \ psi,}

where I (ψ) is the intensity observed at point P along the line of sight forming the angle ψ with the radius vector from the center of the star, I (0) is the intensity from the center of the disk. Since the ratio is equal to unity at ψ = 0, then

\sum _{k=0}^{N}a_{k}=1.

{\ displaystyle \ sum _ {k = 0} ^ {N} a_ {k} = 1.}

In the case of solar radiation at a wavelength of 550 nm, the effect of darkening to the edge can be approximated at N = 2:

a_{0}=1-a_{1}-a_{2}=0.3,

{\ displaystyle a_ {0} = 1-a_ {1} -a_ {2} = 0.3,}

a_{1}=0.93,

{\ displaystyle a_ {1} = 0.93,}

a_{2}=-0.23

{\ displaystyle a_ {2} = - 0.23}

(see Cox, 2000). Often the disc darkening equation is written as

{\frac {I(\psi )}{I(0)}}=1+\sum _{k=1}^{N}A_{k}(1-\cos \psi )^{k},

{\ displaystyle {\ frac {I (\ psi)} {I (0)}} = 1+ \ sum _ {k = 1} ^ {N} A_ {k} (1- \ cos \ psi) ^ {k },}

containing N independent variables. You can specify the relationship between the coefficients a _k and A _k . For example, with N = 2:

A_{1}=-(a_{1}+2*a_{2}),

{\ displaystyle A_ {1} = - (a_ {1} + 2 * a_ {2}),}

A_{2}=a_{2}.

{\ displaystyle A_ {2} = a_ {2}.}

Then for solar radiation with a wavelength of 550 nm

A_{1}=-0.47,

{\ displaystyle A_ {1} = - 0.47,}

A_{2}=-0.23.

{\ displaystyle A_ {2} = - 0.23.}

In this model, the radiation intensity at the edge of the solar disk is 30% of the intensity at the center of the disk.

The obtained formulas can be rewritten in terms of the angle θ using the replacement

\cos \psi ={\frac {\sqrt {\cos ^{2}\theta -\cos ^{2}\Omega }}{\sin \Omega }}={\sqrt {1-\left({\frac {\sin \theta }{\sin \Omega }}\right)^{2}}},

{\ displaystyle \ cos \ psi = {\ frac {\ sqrt {\ cos ^ {2} \ theta - \ cos ^ {2} \ Omega}} {\ sin \ Omega}} = {\ sqrt {1- \ left ({\ frac {\ sin \ theta} {\ sin \ Omega}} \ right) ^ {2}}},}

where Ω is the angular distance between the center of the disk and the limb. For small angles θ, we have

\cos \psi \approx {\sqrt {1-\left({\frac {\theta }{\sin \Omega }}\right)^{2}}}.

{\ displaystyle \ cos \ psi \ approx {\ sqrt {1- \ left ({\ frac {\ theta} {\ sin \ Omega}} \ right) ^ {2}}}.}

The approximation considered above can be used to derive an analytical expression of the ratio of the average intensity to the central one. The average intensity I _m is the integral of the intensity over the star’s disk divided by the solid angle occupied by the disk:

I_{m}={\frac {\int I(\psi )\,d\omega }{\int d\omega }},

{\ displaystyle I_ {m} = {\ frac {\ int I (\ psi) \, d \ omega} {\ int d \ omega}},}

where dω = sin θ dθ dφ is an element of the solid angle, the integration variables are in the range: 0 ≤ φ ≤ 2π and 0 ≤ θ ≤ Ω. The integral can be rewritten as

I_{m}={\frac {\int _{\cos \Omega }^{1}I(\psi )\,d\cos \theta }{\int _{\cos \Omega }^{1}d\cos \theta }}={\frac {\int _{\cos \Omega }^{1}I(\psi )\,d\cos \theta }{1-\cos \Omega }}.

{\ displaystyle I_ {m} = {\ frac {\ int _ {\ cos \ Omega} ^ {1} I (\ psi) \, d \ cos \ theta} {\ int _ {\ cos \ Omega} ^ { 1} d \ cos \ theta}} = {\ frac {\ int _ {\ cos \ Omega} ^ {1} I (\ psi) \, d \ cos \ theta} {1- \ cos \ Omega}}. }

This equation can be solved analytically, but it is very difficult. However, for an observer remote at infinite distance $d\cos \theta$ ${\ displaystyle d \ cos \ theta}$ can be replaced by $\sin ^{2}\Omega \cos \psi \,d\cos \psi$ ${\ displaystyle \ sin ^ {2} \ Omega \ cos \ psi \, d \ cos \ psi}$ as a result

I_{m}={\frac {\int _{0}^{1}I(\psi )\cos \psi \,d\cos \psi }{\int _{0}^{1}\cos \psi \,d\cos \psi }}=2\int _{0}^{1}I(\psi )\cos \psi \,d\cos \psi ,

{\ displaystyle I_ {m} = {\ frac {\ int _ {0} ^ {1} I (\ psi) \ cos \ psi \, d \ cos \ psi} {\ int _ {0} ^ {1} \ cos \ psi \, d \ cos \ psi}} = 2 \ int _ {0} ^ {1} I (\ psi) \ cos \ psi \, d \ cos \ psi,}

{\frac {I_{m}}{I(0)}}=2\sum _{k=0}^{N}{\frac {a_{k}}{k+2}}.

{\ displaystyle {\ frac {I_ {m}} {I (0)}} = 2 \ sum _ {k = 0} ^ {N} {\ frac {a_ {k}} {k + 2}}.}

For solar radiation at a wavelength of 550 nm, the average intensity is equal to 80.5% of the central intensity.

Literature

Billings, Donald E. A Guide to the Solar Corona. - Academic Press, New York, 1966.
Cox, Arthur N. (ed). Allen's Astrophysical Quantities. - 14th. - Springer-Verlag, NY, 2000. - ISBN 0-387-98746-0 .
Milne, EA Radiative Equilibrium in the Outer Layers of a Star: the Temperature Distribution and the Law of Darkening // MNRAS : journal. - 1921. - Vol. 81 . - P. 361-375 . - DOI : 10.1093 / mnras / 81.5.361 . - .
Minnaert, M. On the Continuous Spectrum of the Corona and its Polarisation (Eng.) // Astronomy and Astrophysics : journal. - 1930. - Vol. 1 . - P. 209 .
Neckel, H .; Labs, D. Solar Limb Darkening 1986-1990 (Eng.) // Solar Physics . - 1994. - Vol. 153 , no. 1-2 . - P. 91-114 . - DOI : 10.1007 / BF00712494 . - .
van de Hulst; HC The Electron Density of the Solar Corona (Neopr.) // Bull. Astron. Inst. Netherlands. - 1950. - T. 11 , No. 410 . - S. 135 .
Mariska, John. The Solar Transition Region. - Cambridge University Press, Cambridge, 1993. - ISBN 0521382610 .
Steiner, O., Photospheric processes and magnetic flux tubes, (2007) [1]

Darkening to the edge

Fundamentals of Theory

Disc Calculation

Literature

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