Effective temperature

The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation.[1] Effective temperature is often used as an estimate of a body's surface temperature when the body's emissivity curve (as a function of wavelength) is not known.

When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), 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 greenhouse effect.

Star

EffectiveTemperature 300dpi e
The effective temperature of the Sun (5777 kelvins) is the temperature a black body of the same size must have to yield the same total emissive power.

The effective temperature of a star is the temperature of a black body with the same luminosity per surface area (FBol) as the star and is defined according to the Stefan–Boltzmann law FBol = σTeff4. Notice that the total (bolometric) luminosity of a star is then L = 4πR2σTeff4, where R is the stellar radius.[2] 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 Rosseland optical depth (usually 1) within the stellar atmosphere.[3][4] The effective temperature and the bolometric luminosity are the two fundamental physical parameters needed to place a star on the Hertzsprung–Russell diagram. Both effective temperature and bolometric luminosity depend on the chemical composition of a star.

The effective temperature of our Sun is around 5780 kelvins (K).[5][6] 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 color index of a star indicates its temperature from the very cool—by stellar standards—red M stars that radiate heavily in the infrared to the very hot blue O stars that radiate largely in the ultraviolet. 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 stellar classifications known as O, B, A, F, G, K, M.

A red star could be a tiny red dwarf, a star of feeble energy production and a small surface or a bloated giant or even supergiant star such as Antares or Betelgeuse, 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 Sun or the giant Capella 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 Vega or Rigel.

Planet

Blackbody temperature

To find the effective (blackbody) temperature of a planet, it can be calculated by equating the power received by the planet to the known power emitted by a blackbody of temperature T.

Take the case of a planet at a distance D from the star, of luminosity 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 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 albedo (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:

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 Stefan–Boltzmann law gives an expression for the power radiated by the planet:

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

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

The effective temperature for Jupiter from this calculation is 88 K and 51 Pegasi b (Bellerophon) is 1,258 K. A better estimate of effective temperature for some planets, such as Jupiter, would need to include the internal heating as a power input. The actual temperature depends on albedo and atmosphere effects. The actual temperature from spectroscopic analysis for HD 209458 b (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 albedo. 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:

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 emissivity 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 Stefan–Boltzmann law gives an expression for the power radiated by the planet:

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

Note the ratio of the two areas. Common assumptions for this ratio are 1/4 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 subsolar point, 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.[7]

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 radioactive decay and also be produced from frictions resulting from tidal forces.

Earth Effective Temperature

The Earth has an albedo of about 0.306.[8] The emissivity is dependent on the type of surface and many climate models set the value of the Earth's emissivity to 1. However, a more realistic value is 0.96.[9] 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 greenhouse effect, which increases the average temperature of the Earth's surface.

See also

References

  1. ^ Archie E. Roy, David Clarke (2003). Astronomy. CRC Press. ISBN 978-0-7503-0917-2.
  2. ^ Tayler, Roger John (1994). The Stars: Their Structure and Evolution. Cambridge University Press. p. 16. ISBN 0-521-45885-4.
  3. ^ Böhm-Vitense, Erika. Introduction to Stellar Astrophysics, Volume 3, Stellar structure and evolution. Cambridge University Press. p. 14.
  4. ^ Baschek (June 1991). "The parameters R and Teff in stellar models and observations". Astronomy and Astrophysics. 246 (2): 374–382. Bibcode:1991A&A...246..374B.
  5. ^ Lide, David R., ed. (2004). "Properties of the Solar System". CRC Handbook of Chemistry and Physics (85th ed.). CRC Press. p. 14-2. ISBN 9780849304859.
  6. ^ Jones, Barrie William (2004). Life in the Solar System and Beyond. Springer. p. 7. ISBN 1-85233-101-1.
  7. ^ Swihart, Thomas. "Quantitative Astronomy". Prentice Hall, 1992, Chapter 5, Section 1.
  8. ^ "Earth Fact Sheet". nssdc.gsfc.nasa.gov. Archived 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)

External links

18 Sagittarii

18 Sagittarii is a single star in zodiac constellation of Sagittarius, located around 550 light years away from the Sun. It is visible to the naked eye as a faint, orange-hued star with an apparent visual magnitude of 5.58. This object is moving closer to the Earth with a heliocentric radial velocity of −19 km/s. The is an aging giant star with a stellar classification of K0 III, which indicates it has exhausted the hydrogen at its core and evolved away from the main sequence. It has about 9 times the Sun's radius and is radiating 309 times the Sun's luminosity from its enlarged photosphere at an effective temperature of 4,341.

1 Boötis

1 Boötis (1 Boo) is a binary star system in the northern constellation of Boötes, located 318 light years away from the Sun. It is visible to the naked eye as a dim, white-hued star with a combined apparent visual magnitude of 5.71. The pair had an angular separation of 4.660″ as of 2008. It is moving closer to the Earth with a heliocentric radial velocity of −26 km/s.The magnitude 5.78 primary component is an A-type main-sequence star with a stellar classification of A1 V. This star has 2.5 times the mass of the Sun and is radiating 56 times the Sun's luminosity from its photosphere at an effective temperature of 9,863 K. It is 323 million years old and is spinning with a projected rotational velocity of 60 km/s.The system is a source for X-ray emission, which is most likely coming from the companion star. This magnitude 9.60 component is a possible pre-main sequence star with a mass similar to the Sun. It is radiating 76% of the Sun's luminosity at an effective temperature of 6,370 K.

59 Andromedae

59 Andromedae, abbreviated 59 And, is a sixth-magnitude binary star system in the northern constellation of Andromeda. 59 Andromedae is the Flamsteed designation. As of 2017, the pair had an angular separation of 16.60″ along a position angle (PA) of 36°. Compare this to a separation of 15.3″ along a PA of 35°, as measured in 1783. The two stars have an estimated physical separation of 1,370 AU.The magnitude 6.09 primary component is a B-type main-sequence star with a stellar classification of B9 V. It has 2.73 times the Sun's radius and is radiating 84 times the Sun's luminosity from its photosphere at an effective temperature of 10,870 K. It is spinning with a projected rotational velocity of 176 km/s.The secondary is a magnitude 6.82 A-type main-sequence star with a class of A1 Vn, where the 'n' suffix indicates "nebulous" lines due to rapid rotation. It is spinning with a high projected rotational velocity of 233 km/s. The star has 2.23 times the Sun's mass and 2.59 times the Sun's radius. It is radiating 30 times the luminosity of the Sun and has an effective temperature of 9,498 K.

Alpha Lyncis

Alpha Lyncis (α Lyn, α Lyncis) is the brightest star in the northern constellation of Lynx with an apparent magnitude of +3.13. Unusually, it is the only star in the constellation that has a Bayer designation. Based upon parallax measurements, this star is located about 203 light-years (62 parsecs) from the Earth.This is a giant star that has exhausted the hydrogen at its core and has evolved away from the main sequence. It has expanded to about 55 times the Sun's radius and it is emitting roughly 673 times the luminosity of the Sun. The estimated effective temperature of the star's outer envelope is 3,882 K, which is lower than the Sun's effective temperature of 5,778 K, and is giving Alpha Lyncis an orange hue that is characteristic of K-type stars.Alpha Lyncis is a suspected small-amplitude red variable star that changes apparent magnitude from +3.17 up to +3.12. This variability pattern typically occurs in stars that have developed an inert carbon core surrounded by a helium-fusing shell, and suggests that Alpha Lyncis is starting to evolve into a Mira-type variable.

BI Cygni

BI Cygni (BI Cyg, IRC +40408, BD+36 4025) is a red supergiant in the constellation Cygnus. It is an irregular variable star with a maximum brightness of magnitude 8.4 and a minimum of magnitude 9.9. It has a current mass of 20 M☉.It is considered a member of the stellar Cygnus OB1 association, its distance is 1,580 parsecs (5,150 light-years) of the Solar System. It is less than a degree south of another variable red supergiant, BC Cygni.

BI Cyg is a slow irregular variable star classified as type Lc, an irregular supergiant. Its brightness changes between extremes of magnitude 8.4 and 9.9. Frequency analysis of its light curve shows no significant periods.BI Cyg is one of largest known stars with a radius around 1,240 R☉ based on the assumption of an effective temperature of 3,575 K and a bolometric luminosity of 226,000 L☉. More recent studies derive lower luminosities below 130,000 L☉, suggesting an initial mass of 20 M☉, and consequently lower values for the radius.

BO Carinae

BO Carinae (BO Car / HD 93420 / SAO 238447) is a variable star in the constellation Carina.

BO Car has a maximum apparent magnitude of +7.18. Its distance and membership is uncertain, but its possible membership to the star cluster Trumpler 15 allows a distance estimate of approximately 2,500 parsecs (8,150 light-years).BO Car is a red supergiant of spectral type M4Ib with an effective temperature of 3,525 K. It is one of largest stars with a radius of 790 solar radii. Its bolometric luminosity is 78,000 L☉. Mass-loss is on the order of 0.3 × 10−9 solar masses per year.Billed as an irregular variable like TZ Cassiopeiae or V528 Carinae; its apparent brightness fluctuates between magnitude +7.18 and +8.50 without periodicity.

CK Carinae

CK Carinae (CK Car / HD 90382 / SAO 238038) is a variable star in the constellation Carina, the keel of the Argo Navis. Apparent average magnitude +7.59, is a member of the star association Carina OB1-D, so the distance to CK Carinae can be estimated to be around 2,200 parsecs or 7,100 light-years.

CK Carinae is a red supergiant of spectral type M3.5Iab with an effective temperature of 3,550 K. It is one of the largest stars, with a radius over 1,000 times larger than the sun, which means that if it were in the place of the Sun, its surface would reach almost to the orbit of Jupiter, Earth being encompassed within the star. However, it is surpassed by size by stars like VY Canis Majoris, VV Cephei A, and Mu Cephei. Consequently, CK Carinae is also a bright star, being its luminosity 170,000 times that of the Sun.

Billed as a semiregular variable star SRC, CK Carinae's brightness varies magnitude between +7.2 - +8.5 with a period of approximately 525 days.

Extrasolar Planets Encyclopaedia

The Extrasolar Planets Encyclopaedia is an astronomy website, founded in Paris, France at the Meudon Observatory by Jean Schneider in February 1995, which maintains a database of all the currently known and candidate extrasolar planets, with individual pages for each planet and a full list interactive catalog spreadsheet. The main catalogue comprises databases of all of the currently confirmed extrasolar planets as well as a database of unconfirmed planet detections. The databases are frequently updated with new data from peer-reviewed publications and conferences.

In their respective pages, the Planets are listed along with their basic properties such as the year of planet’s discovery, mass, radius, orbital period, semi-major axis, eccentricity, inclination, longitude of periastron, time of periastron, maximum time variation, and time of transit, including all error range values.

The individual planet data pages also contain the data on the parent star such as Name, Distance (pc), Spectral Type, Effective Temperature, Apparent Magnitude V, Mass, Radius, Age, Right Asc. Coord., Decl. Coord. Even when they are known, not all of these figures are listed in the interactive spreadsheet catalog. And many missing planet figures that would simply require the application of Kepler's third law of motion are left blank. Most notably absent on all pages is the star's luminosity.

As of June 2011, the catalog aims to include objects up to 25 Jupiter masses, an increase on the previous inclusion criteria of 20 Jupiter masses.

HD 5789/5788

HD 5789 and HD 5788 is a pair of stars comprising a binary star system in the northern constellation of Andromeda. Located approximately 151 parsecs (490 ly) away, the primary is a hot, massive blue star with an apparent magnitude of 6.06 while the secondary is slightly smaller and cooler, with an apparent magnitude of 6.76. Both stars are main-sequence stars, meaning that they are currently fusing hydrogen into helium in their cores. As of 2016, the pair had an angular separation of 7.90″ along a position angle of 195°. While both have a similar proper motion and parallax, there's still no proof that the pair is gravitationally bound.

The primary component is HD 5789, a B-type main-sequence star with a stellar classification of B9.5Vnn (λ Boo), where the 'n' indicates "nebulous" lines due to rapid rotation. Abt and Morrell (1995) listed it as a Lambda Boötis star, although this is disputed. It has 2.7 times the mass of the Sun and is spinning rapidly with a projected rotational velocity of 249 km/s. The star is radiating 86 times the Sun's luminosity from its photosphere at an effective temperature of 9,977 K.The fainter secondary component is an A-type main-sequence star with a class of A2 Vn. It shows a projected rotational velocity of 270 km/s and has 2.7 times the Sun's mass. The star shines with 73 times the Sun's luminosity at an effective temperature of 9,840 K.

HD 77370

HD 77370, also called b² Carinae (b² Car), is a star in the constellation Carina.

b² Carinae is a yellow-white F-type main sequence dwarf with an apparent magnitude of +5.17. It is approximately 85.4 light years from Earth.

It has a projected rotational velocity of 60.4±3.0 km/s and an effective temperature of 6,609 K.

Photosphere

The photosphere is a star's outer shell from which light is radiated. The term itself is derived from Ancient Greek roots, φῶς, φωτός/phos, photos meaning "light" and σφαῖρα/sphaira meaning "sphere", in reference to it being a spherical surface that is perceived to emit light. It extends into a star's surface until the plasma becomes opaque, equivalent to an optical depth of approximately 2/3, or equivalently, a depth from which 50% of light will escape without being scattered.

In other words, a photosphere is the deepest region of a luminous object, usually a star, that is transparent to photons of certain wavelengths.

Psi1 Aurigae

Psi1 Aurigae (ψ1 Aur, ψ1 Aurigae) is a star in the northern constellation of Auriga. It is faintly visible to the naked eye with an apparent visual magnitude of 4.91. Based upon a measured annual parallax shift of 0.82 mas, it is approximately 4,000 light-years (1,200 parsecs) distant from the Earth.

This is a massive supergiant star with a stellar classification of M0 I. It is a slow irregular variable of the LC type, with its brightness varying in magnitude by 0.44. The star is more than 14 times as massive as the Sun and is blazing with 63,579 times the Sun's luminosity. This energy is being radiated into outer space from its outer atmosphere at an effective temperature of 3,750 K, giving it the orange-red hue of a cool M-type star.

Stefan–Boltzmann law

The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. The value of the constant is

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance (watts per square metre per steradian) is given by

A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, :

The radiant emittance has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. is the emissivity of the grey body; if it is a perfect blackbody, . In the still more general (and realistic) case, the emissivity depends on the wavelength, .

To find the total power radiated from an object, multiply by its surface area, :

Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

UBV photometric system

The UBV photometric system (Ultraviolet, Blue, Visual), also called the Johnson system (or Johnson-Morgan system), is a wide band photometric system for classifying stars according to their colors. It is the first known standardized photoelectric photometric system. The letters U, B, and V stand for ultraviolet, blue, and visual magnitudes, which are measured for a star then two subtractions are performed in a specific order to classify it in the system.The choice of colors on the blue end of the spectrum is because of the bias that photographic film has for those colors. It was introduced in the 1950s by American astronomers Harold Lester Johnson and William Wilson Morgan. A 13 in (330 mm) telescope and the 82 in (2,100 mm) telescope at McDonald Observatory were used to define the system.The filters are selected so that the mean wavelengths of response functions (at which magnitudes are measured to mean precision) are 364 nm for U, 442 nm for B, 540 nm for V. Zero points were calibrated in the B−V (B minus V) and U−B (U minus B) color indices selecting such A0 main sequence stars which are not affected by interstellar reddening. These stars correspond with a mean effective temperature (Teff (K)) of between 9727 and 9790 Kelvin, the latter being stars with class A0V.

The UBV system has some disadvantages. The short wavelength cutoff that is the U filter is defined mainly by the terrestrial atmosphere rather than the filter itself; thus, it (and observed magnitudes) can vary with altitude and atmospheric conditions. However, a large number of measurements have been made in this system, including many of the bright stars.

ULAS J003402.77−005206.7

ULAS J003402.77-005206.7 (also ULAS J0034-00) is a T-type brown dwarf in the constellation of Cetus.ULAS J0034-00 is one of the coolest brown dwarfs known. It was first identified in data from the UK Infrared Telescope (UKIRT) Infrared

Deep Sky Survey (UKIDSS). Infrared spectra subsequently taken with the IRS instrument on the Spitzer Space Telescope give an estimated effective temperature of between 550 and 600 K and does not emit any visible light. Its mass is estimated at between 5 and 20 Jupiter masses and its age at between 0.1 and 2.0 billion years.

Ultra-cool dwarf

An ultra-cool dwarf is a stellar or sub-stellar object of spectral class M that has an effective temperature under 2,700 K (2,430 °C; 4,400 °F). TRAPPIST-1 is a widely known example of an ultra-cool dwarf star.

Upsilon Carinae

Upsilon Carinae, Latinized from υ Carinae, is a double star in the southern constellation of Carina. It is part of the Diamond Cross asterism in southern Carina. The Upsilon Carinae system has a combined apparent magnitude of +2.97 and is approximately 1,400 light years (440 parsecs) from Earth.In Chinese, 海石 (Hǎi Dàn), meaning Sea Rock, refers to an asterism consisting of υ Carinae, ε Carinae, ι Carinae, HD 83183 and HD 84810. Consequently, υ Carinae itself is known as 海石五 (Hǎi Dàn wǔ, English: the Fifth Star of Sea Rock.)The primary component, υ Carinae A, has a stellar classification of A8 Ib, making it a supergiant star that has exhausted the hydrogen at its core and evolved away from its brief main sequence lifetime as an O9 V star. With an apparent magnitude of +3.08, it has an effective temperature of about 7,600 K, giving it a white hue. The companion, υ Carinae B, is a giant star with a classification of B7 III, although Mandrini and Niemela (1986) suggested it may be a subgiant star with a classification of B4–5 IV. The outer envelope of this star has an effective temperature of around 23,000 K, resulting in the blue-white hue of a B-type star.

The two stars have an angular separation of 5.030 arcseconds. As a binary star system, they would have an estimated orbital period of at least 19,500 years and a present-day separation of around 2,000 Astronomical Units. This system is roughly 12 million years old.In the next 7500 years, the south Celestial pole will pass close to this stars and Iota Carinae (8100 CE).

V528 Carinae

V528 Carinae (V528 Car, HD 95950, HIP 54021) is a variable star in the constellation Carina.

V528 Carinae has an apparent visual magnitude of +6.75. It is a distant star but the exact distance is uncertain. The Hipparcos satellite gives a negative annual parallax and is not helpful. Its Carina OB2 membership allows the distance to be estimated at 3,850 light-years.V528 Carinae is a red supergiant of spectral type M2 Ib with an effective temperature of 3,700 K. It has a radius of 700 solar radii, making it one of the largest stars. In the visible spectrum luminosity is 11,900 times higher than the sun, but the bolometric luminosity considering all wavelengths reaches 81,000 L☉. It loses mass at 0.5×10−9 M☉ per year.It is classified as a slow irregular variable whose prototype is TZ Cassiopeiae.

Xi2 Centauri

Xi2 Centauri, Latinized from ξ2 Centauri, is a triple star system in the southern constellation of Centaurus. It is visible to the naked eye with an apparent visual magnitude of 4.30, and forms a wide double star with the slightly dimmer ξ1 Centauri. Based upon an annual parallax shift of 6.98 mas, Xi2 Centauri lies roughly 470 light years from the Sun. At that distance, the visual magnitude is diminished by an interstellar extinction factor of 0.32 due to intervening dust.This system was discovered to be a single-lined spectroscopic binary in 1910 by American astronomer Joseph Haines Moore. The pair, component A, orbit each other with a period of 7.6497 days and an eccentricity of 0.35. The primary is a B-type star with a stellar classification of B1.5 V or B2 IV, depending on the source. This indicates it may be a main sequence star or a more evolved subgiant star. It has about 8.1 times the mass of the Sun and radiates 1,702 times the solar luminosity from its outer atmosphere at an effective temperature of 20,790 K.A third star, component B, is a magnitude 9.38 F-type main sequence star with a classification of F7 V. It has 1.25 times the mass of the Sun and radiates 2.4 times the solar luminosity at an effective temperature of 6,194 K. It lies at an angular separation of 25.1 arc seconds from the inner pair. They share a common proper motion, indicating they may be gravitationally bound with an orbital period of around 41,000 years.The system has a peculiar velocity of 16.2±4.2 km/s. It belongs to the Scorpius–Centaurus Association and appears to be a member of the Gould's Belt.

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