How to read a photometric diagram

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The effective temperature of a body such as a star or planet is how to read a photometric diagram the of a that would emit the same total amount of. Effective temperature is often used as an estimate of a body's surface temperature when the body's curve (as a function of ) is not known.

When the star's or planet's net in the relevant wavelength band is less than unity (less than that of a ), the actual temperature of the body will be higher than the effective temperature. The net emissivity may be low due to surface or atmospheric properties, including.

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The effective temperature of the (5777 ) is the temperature a black body of the same size must have to yield the same total emissive power.

The effective temperature of a is the temperature of a with the same luminosity per surface area (FBol) as the star and is defined according to the FBol = σTeff4. Notice that the total () luminosity of a star is then L = 4πR2σTeff4, where R is the. The definition of the stellar radius is obviously not straightforward. More rigorously the effective temperature corresponds to the temperature at the radius that is defined by a certain value of the (usually 1) within the. The effective temperature and the bolometric luminosity are the two fundamental physical parameters needed to place a star on the. Both effective temperature and bolometric luminosity depend on the chemical composition of a star.

The effective temperature of our Sun is around 5780  (K). Stars have a decreasing temperature gradient, going from their central core up to the atmosphere. The "core temperature" of the Sun—the temperature at the centre of the Sun where nuclear reactions take place—is estimated to be 15,000,000 K.

The of a star indicates its temperature from the very cool—by stellar standards—red M stars that radiate heavily in the to the very hot blue O stars that radiate largely in the. The effective temperature of a star indicates the amount of heat that the star radiates per unit of surface area. From the warmest surfaces to the coolest is the sequence of known as O, B, A, F, G, K, M.

A red star could be a tiny, a star of feeble energy production and a small surface or a bloated giant or even star such as or, either of which generates far greater energy but passes it through a surface so large that the star radiates little per unit of surface area. A star near the middle of the spectrum, such as the modest or the giant radiates more energy per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as or.

Main article:

Blackbody Temperature[]

The effective (blackbody) temperature of a can be calculated by equating the power received by the planet with the power emitted by a blackbody of temperature T.

Take the case of a planet at a distance D from the star, of L.

Assuming the star radiates isotropically and that the planet is a long way from the star, the power absorbed by the planet is given by treating the planet as a disc of radius r, which intercepts some of the power which is spread over read the surface of a sphere of radius D (the distance of the planet from the star). The calculation assumes the planet reflects some of the incoming radiation by incorporating a parameter called the (a). An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed. The expression for absorbed power is then:

P a b s = L r 2 ( 1 − a ) 4 D 2 {\displaystyle P_{\rm {abs}}={\frac {Lr^{2}(1-a)}{4D^{2}}}} {\displaystyle P_{\rm {abs}}={\frac {Lr^{2}(1-a)}{4D^{2}}}}

The next assumption we can make is that the entire planet is at the same temperature T, and that the planet radiates as a blackbody. The gives an expression for the power radiated by the planet:

P r a d = 4 π r 2 σ T 4 {\displaystyle P_{\rm {rad}}=4\pi r^{2}\sigma T^{4}} P_{{{\rm {rad}}}}=4\pi r^{2}\sigma T^{4}

Equating these two expressions and rearranging gives an expression for the effective temperature:

T = L ( 1 − a ) 16 π σ D 2 4 {\displaystyle T={\sqrt[{4}]{\frac {L(1-a)}{16\pi \sigma D^{2}}}}} {\displaystyle T={\sqrt[{4}]{\frac {L(1-a)}{16\pi \sigma D^{2}}}}}

Note that the planet's radius has cancelled out of the final expression.

The effective temperature for from this calculation is 88 K and (Bellerophon) is 1,258 K.[] A better estimate of effective temperature for some planets, such as Jupiter, would need to include the as a power input. The actual temperature depends on and effects. The actual temperature from for (Osiris) is 1,130 K, but the effective temperature is 1,359 K.[] The internal heating within Jupiter raises the effective temperature to about 152 K.[]

Surface temperature of a planet[]

The surface temperature of a planet can be estimated by modifying the effective-temperature calculation to account for emissivity and temperature variation.

The area of the planet that absorbs the power from the star is Aabs which is some fraction of the total surface area Atotal = 4πr2, where r is the radius of the planet. This area intercepts some of the power which is spread over the surface of a sphere of radius D. We also allow the planet to reflect some of the incoming radiation by incorporating a parameter a called the. An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed. The expression for absorbed power is then:

P a b s = L A a b s ( 1 − a ) 4 π D 2 {\displaystyle P_{\rm {abs}}={\frac {LA_{\rm {abs}}(1-a)}{4\pi D^{2}}}} P_{{{\rm {abs}}}}={\frac {LA_{{{\rm {abs}}}}(1-a)}{4\pi D^{2}}}

The next assumption we can make is that although the entire planet is not at the same temperature, it will radiate as if it had a temperature T over an area Arad which is again some fraction of the total area of the planet. There is also a factor ε, which is the and represents atmospheric effects. ε ranges from 1 to 0 with 1 meaning the planet is a perfect blackbody and emits all the incident power. The gives an expression for the power radiated by the planet:

P r a d = A r a d ε σ T 4 {\displaystyle P_{\rm {rad}}=A_{\rm {rad}}\varepsilon \sigma T^{4}} P_{{{\rm {rad}}}}=A_{{{\rm {rad}}}}\varepsilon \sigma T^{4}

Equating these two expressions and rearranging gives an expression for the surface temperature:

T = A a b s A r a d L ( 1 − a ) 4 π σ ε D 2 4 {\displaystyle T={\sqrt[{4}]{{\frac {A_{\rm {abs}}}{A_{\rm {rad}}}}{\frac {L(1-a)}{4\pi \sigma \varepsilon D^{2}}}}}} {\displaystyle T={\sqrt[{4}]{{\frac {A_{\rm {abs}}}{A_{\rm {rad}}}}{\frac {L(1-a)}{4\pi \sigma \varepsilon D^{2}}}}}}

Note the ratio of the two areas. Common assumptions for this ratio are / for a rapidly rotating body and 1/2 for a slowly rotating body, or a tidally locked body on the sunlit side. This ratio would be 1 for the, the point on the planet directly below the sun and gives the maximum temperature of the planet — a factor of √2 (1.414) greater than the effective temperature of a rapidly rotating planet.

Also note here that this equation does not take into account any effects from internal heating of the planet, which can arise directly from sources such as and also be produced from frictions resulting from.

Earth Effective Temperature[]

Main article:

Let's look at the Earth. The Earth has an albedo of about 0.306. The emissivity is dependent on the type of surface and many set the value of the Earth's emissivity to 1. However, a more realistic value is 0.96. The Earth is a fairly fast rotator so the area ratio can be estimated as 1/4. The other variables are constant. This calculation gives us an effective temperature of the Earth of 252 K (−21 °C). The average temperature of the Earth is 288 K (15 °C). One reason for the difference between the two values is due to the, which increases the average temperature of the Earth's surface.

See also[]

References[]

  1. Archie E. Roy, David Clarke (2003)... ISBN .
  2. Tayler, Roger John (1994). The Stars: Their Structure and Evolution.. p. 16.  .
  3. Böhm-Vitense, Erika. Introduction to Stellar Astrophysics, Volume 3, Stellar structure and evolution.. p. 14.
  4. Baschek (June 1991). "The parameters R and Teff in stellar models and observations". Astronomy and Astrophysics. 246 (2): 374–382. :.
  5. Lide, David R., ed. (2004).. CRC Handbook of Chemistry and Physics (85th ed.).. p. 14-2.  .
  6. Jones, Barrie William (2004)... p. 7.  .
  7. Swihart, Thomas. "Quantitative Astronomy". Prentice Hall, 1992, Chapter 5, Section 1.
  8. . nssdc.gsfc.nasa.gov. from the original on 30 October 2010. Retrieved 8 May 2018.
  9. Jin, Menglin and Shunlin Liang, (2006) “An Improved Land Surface Emissivity Parameter for Land Surface Models Using Global Remote Sensing Observations” Journal of Climate, 19 2867-81. (www.glue.umd.edu/sliang/papers/Jin2006.emissivity.pdf)

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