# Geometric albedo

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In astronomy, the geometric albedo of a celestial body is the ratio of its actual brightness as seen from the light source (i.e. at zero phase angle) to that of an idealized flat, fully reflecting, diffusively scattering (Lambertian) disk with the same cross-section. (This phase angle refers to the direction of the light paths and is not a phase angle in its normal meaning in optics or electronics.)

## Contents

Diffuse scattering implies that radiation is reflected isotropically with no memory of the location of the incident light source. Zero phase angle corresponds to looking along the direction of illumination. For Earth-bound observers, this occurs when the body in question is at opposition and on the ecliptic.

The visual geometric albedo refers to the geometric albedo quantity when accounting for only electromagnetic radiation in the visible spectrum.

## Airless bodies

The surface materials (regoliths) of airless bodies (in fact, the majority of bodies in the Solar System) are strongly non-Lambertian and exhibit the opposition effect, which is a strong tendency to reflect light straight back to its source, rather than scattering light diffusely.

The geometric albedo of these bodies can be difficult to determine because of this, as their reflectance is strongly peaked for a small range of phase angles near zero. [1] The strength of this peak differs markedly between bodies, and can only be found by making measurements at small enough phase angles. Such measurements are usually difficult due to the necessary precise placement of the observer very close to the incident light. For example, the Moon is never seen from the Earth at exactly zero phase angle, because then it is being eclipsed. Other Solar System bodies are not in general seen at exactly zero phase angle even at opposition, unless they are also simultaneously located at the ascending or descending node of their orbit, and hence lie on the ecliptic. In practice, measurements at small nonzero phase angles are used to derive the parameters which characterize the directional reflectance properties for the body (Hapke parameters). The reflectance function described by these can then be extrapolated to zero phase angle to obtain an estimate of the geometric albedo.

For very bright, solid, airless objects such as Saturn's moons Enceladus and Tethys, whose total reflectance (Bond albedo) is close to one, a strong opposition effect combines with the high Bond albedo to give them a geometric albedo above unity (1.4 in the case of Enceladus). Light is preferentially reflected straight back to its source even at low angle of incidence such as on the limb or from a slope, whereas a Lambertian surface would scatter the radiation much more broadly. A geometric albedo above unity means that the intensity of light scattered back per unit solid angle towards the source is higher than is possible for any Lambertian surface.

## Stars

Stars shine intrinsically, but they can also reflect light. In a close binary star system polarimetry can be used to measure the light reflected from one star off another (and vice versa) and therefore also the geometric albedos of the two stars. This task has been accomplished for the two components of the Spica system, with the geometric albedo of Spica A and B being measured as 0.0361 and 0.0136 respectively. [2] The geometric albedos of stars are in general small, for the Sun a value of 0.001 is expected, [3] but for hotter of lower gravity (i.e. giant) stars the amount of reflected light is expected to be several times that of the stars in the Spica system. [2]

## Equivalent definitions

For the hypothetical case of a plane surface, the geometric albedo is the albedo of the surface when the illumination is provided by a beam of radiation that comes in perpendicular to the surface.

## Examples

The geometric albedo may be greater or smaller than the Bond albedo, depending on surface and atmospheric properties of the body in question. Some examples: [4]

Name Bond albedo Visual geometric albedo
Mercury [5] [6] 0.088

0.142

Venus [7] [6] 0.76

0.689

Earth [8] [6] 0.306

0.434

Moon [9] 0.11

0.12

Mars [10] [6] 0.25

0.17

Jupiter [11] [6] 0.503

0.538

Saturn [12] [6] 0.342

0.499

1.4

Uranus [14] [6] 0.300

0.488

Neptune [15] [6] 0.290

0.442

Pluto 0.4

0.440.61

Eris 0.96

## Related Research Articles

Albedo is the measure of the diffuse reflection of solar radiation out of the total solar radiation and measured on a scale from 0, corresponding to a black body that absorbs all incident radiation, to 1, corresponding to a body that reflects all incident radiation.

Absolute magnitude is a measure of the luminosity of a celestial object, on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it were viewed from a distance of exactly 10 parsecs, without extinction of its light due to absorption by interstellar matter and cosmic dust. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared among each other on a magnitude scale.

In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the direction of the incident light and the surface normal; I = I0cos(θ). The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.

The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

5145 Pholus is an eccentric centaur in the outer Solar System, approximately 180 kilometers in diameter, that crosses the orbit of both Saturn and Neptune. It was discovered on 9 January 1992 by American astronomer David Rabinowitz (uncredited) of UA's Spacewatch survey at the Kitt Peak National Observatory in Arizona, United States. The very reddish object has an elongated shape and a rotation period of 9.98 hours. It was named after the creature Pholus from Greek mythology.

Planetshine is the dim illumination, by sunlight reflected from a planet, of all or part of the otherwise dark side of any moon orbiting the body. Planetlight is the diffuse reflection of sunlight from a planet, whose albedo can be measured.

Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Mirrors exhibit specular reflection.

Lambertian reflectance is the property that defines an ideal "matte" or diffusely reflecting surface. The apparent brightness of a Lambertian surface to an observer is the same regardless of the observer's angle of view. More technically, the surface's luminance is isotropic, and the luminous intensity obeys Lambert's cosine law. Lambertian reflectance is named after Johann Heinrich Lambert, who introduced the concept of perfect diffusion in his 1760 book Photometria.

The planet Neptune has 14 known moons, which are named for minor water deities in Greek mythology. By far the largest of them is Triton, discovered by William Lassell on October 10, 1846, 17 days after the discovery of Neptune itself; over a century passed before the discovery of the second natural satellite, Nereid. Neptune's outermost moon Neso, which has an orbital period of about 26 Julian years, orbits farther from its planet than any other moon in the Solar System.

The Yarkovsky–O'Keefe–Radzievskii–Paddack effect, or YORP effect for short, changes the rotation state of a small astronomical body – that is, the body's spin rate and the obliquity of its pole(s) – due to the scattering of solar radiation off its surface and the emission of its own thermal radiation.

The rings of Uranus are intermediate in complexity between the more extensive set around Saturn and the simpler systems around Jupiter and Neptune. The rings of Uranus were discovered on March 10, 1977, by James L. Elliot, Edward W. Dunham, and Jessica Mink. William Herschel had also reported observing rings in 1789; modern astronomers are divided on whether he could have seen them, as they are very dark and faint.

The bidirectional reflectance distribution function is a function of four real variables that defines how light is reflected at an opaque surface. It is employed in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, , and outgoing direction, , and returns the ratio of reflected radiance exiting along to the irradiance incident on the surface from direction . Each direction is itself parameterized by azimuth angle and zenith angle , therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr−1, with steradians (sr) being a unit of solid angle.

For most numbered asteroids, almost nothing is known apart from a few physical parameters and orbital elements and some physical characteristics are often only estimated. The physical data is determined by making certain standard assumptions.

The definition of the BSDF is not well standardized. The term was probably introduced in 1980 by Bartell, Dereniak, and Wolfe. Most often it is used to name the general mathematical function which describes the way in which the light is scattered by a surface. However, in practice, this phenomenon is usually split into the reflected and transmitted components, which are then treated separately as BRDF and BTDF.

The Bond albedo, named after the American astronomer George Phillips Bond (1825–1865), who originally proposed it, is the fraction of power in the total electromagnetic radiation incident on an astronomical body that is scattered back out into space.

The Hapke parameters are a set of parameters for an empirical model that is commonly used to describe the directional reflectance properties of the airless regolith surfaces of bodies in the Solar System. The model has been developed by astronomer Bruce Hapke at the University of Pittsburgh.

The opposition surge is the brightening of a rough surface, or an object with many particles, when illuminated from directly behind the observer. The term is most widely used in astronomy, where generally it refers to the sudden noticeable increase in the brightness of a celestial body such as a planet, moon, or comet as its phase angle of observation approaches zero. It is so named because the reflected light from the Moon and Mars appear significantly brighter than predicted by simple Lambertian reflectance when at astronomical opposition. Two physical mechanisms have been proposed for this observational phenomenon: shadow hiding and coherent backscatter.

In astronomy a phase curve describes the brightness of a reflecting body as a function of its phase angle. The brightness usually refers the object's absolute magnitude, which, in turn, is its apparent magnitude at a distance of astronomical unit from the Earth and Sun. The phase angle equals the arc subtended by the observer and the sun as measured at the body.

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## References

1. See for example this discussion of Lunar albedo Archived April 13, 2009, at the Wayback Machine by Jeff Medkeff.
2. Bailey, Jeremy; Cotton, Daniel V; Kedziora-Chudczer, Lucyna; De Horta, Ain; Maybour, Darren (2019-04-01). "Polarized reflected light from the Spica binary system". Nature Astronomy. 3 (7): 636–641. arXiv:. Bibcode:2019NatAs...3..636B. doi:10.1038/s41550-019-0738-7.
3. Gilbert, Lachlan (2019-04-02). "Scientists prove that binary stars reflect light from one another". UNSW Newsroom. UNSW. Retrieved 2019-04-02.
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8. Williams, David R. (2014-04-25). "Moon Fact Sheet". NASA. Retrieved 2015-03-02.
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10. Hanel, R.A.; et al. (1983). "Albedo, internal heat flux, and energy balance of Saturn". Icarus. 53: 262. Bibcode:1983Icar...53..262H. doi:10.1016/0019-1035(83)90147-1.
11. See the discussion here for explanation of this unusual value above one.
12. Pearl, J.C.; et al. (1990). "The albedo, effective temperature, and energy balance of Uranus, as determined from Voyager IRIS data". Icarus. 84: 12–28. Bibcode:1990Icar...84...12P. doi:10.1016/0019-1035(90)90155-3.
13. Pearl, J.C.; et al. (1991). "The albedo, effective temperature, and energy balance of Neptune, as determined from Voyager data". J. Geophys. Res. 96: 18, 921–18, 930. Bibcode:1991JGR....9618921P. doi:10.1029/91JA01087.
• NASA JPL glossary
• K.P. Seidelmann, Ed. (1992) Explanatory Supplement to the Astronomical Almanac, University Science Books, Mill Valley, California.