Phase curve (astronomy)
Encyclopedia
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 one 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.
The phase curve is useful for characterizing an object’s regolith
(soil) and atmosphere. It is also the basis for computing the geometrical albedo and the Bond albedo
of the body. In ephemeris
generation, the phase curve is used in conjunction with the distances from the object to the Sun and the Earth to calculate the apparent magnitude.
Mercury
The phase curve of Mercury
is very steep which is characteristic of a body on which bare regolith
(soil) is exposed to view. At phase angles exceeding 90° (crescent
phase) the brightness falls off especially sharply. The shape of the phase curve indicates a mean slope on the surface of Mercury of about 16° which is slightly smoother than that of the Moon
. Approaching phase angle 0° (fully illuminated phase) the curve rises to a sharp peak. This surge in brightness is called the opposition effect because for most bodies (though not Mercury) it occurs at astronomical opposition when the body is opposite from the Sun in the sky. The width of the opposition surge for Mercury indicates that both the compaction state of the regolith and the distribution of particle sizes on the planet are similar to those on the Moon.
Early visual observations contributing to the phase curve of Mercury were obtained by G. Muller in the 1800s and by A. Danjon in the mid-twentieith century. W. Irvine and colleagues used photoelectric photometry in the 1960s. Some of these early data were analyzed by G. de Vaucouleurs, summarized by D. Harris and used for predicting apparent magnitudes in the Astronomical Almanac
for several decades. Highly accurate new observations covering the widest range of phase angles to date (2 to 170°) were carried out by A. Mallama, D. Wang and R. Howard using the Large Angle and Spectrometric Coronograph (LASCO)
on the Solar and Heliospheric Observatory (SOHO)
satellite. They also obtained new CCD observations from the ground. These data are now the major source of the phase curve used in the Astronomical Almanac
for predicting apparent magnitudes.
The apparent brightness of Mercury as seen from Earth is greatest at phase angle 0° (superior conjunction with the Sun) when it can reach magnitude -2.6. At phase angles approaching 180° (inferior conjunction) the planet fades to about magnitude +5 with the exact brightness depending on the phase angle at that particular conjunction. This difference of more than 7 magnitudes corresponds to a change of over a thousand times in apparent brightness.
Venus
The relatively flat phase curve of Venus
is characteristic of a cloudy planet. In contrast to Mercury where the curve is strongly peaked approaching phase angle zero (full phase) that of Venus is rounded. The wide illumination scattering angle of clouds, as opposed to the narrower scattering of regolith, causes this flattening of the phase curve. Venus exhibits a brightness surge near phase angle 170°, when it is a thin crescent
, due to forward scattering of sunlight by droplets of sulfuric acid
that are above the planet’s cloud tops. Even beyond 170° the brightness does not decline very steeply.
The history of observation and analysis of the phase curve of Venus is similar to that of Mercury. The best set of modern observations and interpretation was reported by A. Mallama, D. Wang and R. Howard. They used the LASCO
instrument on SOHO
and ground based CCD equipment to observe the phase curve from 2 to 179°. As with Mercury, these new data are the major source of the phase curve used in the Astronomical Almanac
for predicting apparent magnitudes.
In contrast to Mercury the apparent brightness of Venus as seen from Earth does not occur at phase angle zero. Since the phase curve of Venus is relatively flat while its distance from the Earth can vary greatly, maximum brightness occurs when the planet is a crescent, at phase angle 125°, at which time Venus can be as bright as magnitude -4.9. Near inferior conjunction the planet typically fades to about magnitude -3 although the exact values depends on the phase angle. The typical range in apparent brightness for Venus over the course of one apparition is less than a factor of 10 or merely 1% that of Mercury.
Earth
The phase curve of the Earth
has not been determined as accurately as those for Mercury and Venus because its integrated brightness is difficult to measure from the surface. Instead of direct observation, earthshine reflected from the portion of the Moon not lit by the Sun has served as a proxy. A few direct measurements of the Earth’s luminosity have been obtained with the EPOXI
spacecraft. While they do not cover much of the phase curve they reveal a rotational light curve caused by the transit of dark oceans and bright land masses across the hemisphere. P. Goode and colleagues at Big Bear Solar Observatory
have measured the earthshine and T. Livengood of NASA analyzed the EPOXI data.
Earth as seen from Venus near opposition from the Sun would be extremely bright at magnitude -6. To an observer outside the Earth’s orbit on Mars our planet would appear most luminous near the time of its greatest elongation from the Sun, at about magnitude -1.5.
Mars
Only about 50° of the martian phase curve can be observed from Earth because it orbits farther from the Sun than our planet. There is an opposition surge but it is less pronounced than that of Mercury. The rotation of bright and dark surface markings across its disk and variability of its atmospheric state (including its dust storms) superimpose variations on the phase curve. R. Schmude obtained many of the Mars
brightness measurements used in a comprehensive phase curve analysis which was performed by A. Mallama.
Because the orbit of Mars is considerably eccentric its brightness at opposition can range from magnitude -3.0 to -1.4. The minimum brightness is about magnitude +1.6 when Mars is on the opposite site of the Sun from the Earth. Rotational variations can elevate or suppress the brightness of Mars by 5% and global dust storms can increase its luminosity by 25%.
The Gas Giant Planets
The outermost planets (Jupiter
, Saturn
, Uranus
and Neptune
) are so distant that only small portions of their phase curves near 0° (full phase) can be evaluated from the Earth. That part of the curve is generally fairly flat, like that of Venus, for these cloudy planets.
The apparent magnitude of Jupiter ranges from -2.9 to -1.4, Saturn from -0.5 to +1.4, Uranus from +5.3 to +6.0, and Neptune from +7.8 to +8.0. Most of these variations are due to distance. However, the magnitude range for Saturn also depends on its ring system as explained below.
The Rings of Saturn
The brightness of the Saturn system depends on the orientation of its ring system. The rings contribute more to the overall brightness of the system when they are more inclined to the direction of illumination from the Sun and to the view of the observer. Wide open rings contribute about one magnitude of brightness to the disk alone. The icy particles that compose the rings also produce a strong opposition surge. Hubble Space Telescope and Cassini spacecraft images have been analyzed in an attempt to characterize the ring particles based on their phase curves.
The Moon
The phase curve of the Moon
approximately resembles that of Mercury due to the similarities of the surfaces and the lack of an atmosphere on either body . Clementine spacecraft data analyzed by J. Hillier, B. Buratti and K. Hill indicate a lunar opposition surge. The Moon’s apparent magnitude at full phase is -12.7 while at quarter phase it is less than 10 percent as bright.
Planetary Satellites
The phase curves of many Natural satellite
s of other planets, have been observed and interpreted. The icy moons often exhibit opposition brightness surges. This behavior has been used to model their surfaces.
Asteroid
The phase curves of many asteroid
s have also been observed and they too may exhibit opposition surges. Asteroids can be physically classified in this way. The effects of rotation can be very large and have to be factored in before the phase curve is computed. An example of such a study is reported by R. Baker and colleagues.
.
may be inadequate for physical modeling based on photometry.
Phase angle (astronomy)
Phase angle in astronomical observations is the angle between the light incident onto an observed object and the light reflected from the object...
. The brightness usually refers the object’s absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
which, in turn, is its apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
at a distance of one astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....
from the Earth and Sun. The phase angle
Phase angle (astronomy)
Phase angle in astronomical observations is the angle between the light incident onto an observed object and the light reflected from the object...
equals the arc subtended by the observer and the sun as measured at the body.
The phase curve is useful for characterizing an object’s regolith
Regolith
Regolith is a layer of loose, heterogeneous material covering solid rock. It includes dust, soil, broken rock, and other related materials and is present on Earth, the Moon, some asteroids, and other terrestrial planets and moons.-Etymology:...
(soil) and atmosphere. It is also the basis for computing the geometrical albedo and the Bond albedo
Bond albedo
The Bond albedo, named after the American astronomer George Phillips Bond , 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...
of the body. In ephemeris
Ephemeris
An ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...
generation, the phase curve is used in conjunction with the distances from the object to the Sun and the Earth to calculate the apparent magnitude.
MercuryMercury (planet)Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
The phase curve of MercuryMercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
is very steep which is characteristic of a body on which bare regolith
Regolith
Regolith is a layer of loose, heterogeneous material covering solid rock. It includes dust, soil, broken rock, and other related materials and is present on Earth, the Moon, some asteroids, and other terrestrial planets and moons.-Etymology:...
(soil) is exposed to view. At phase angles exceeding 90° (crescent
Crescent
In art and symbolism, a crescent is generally the shape produced when a circular disk has a segment of another circle removed from its edge, so that what remains is a shape enclosed by two circular arcs of different diameters which intersect at two points .In astronomy, a crescent...
phase) the brightness falls off especially sharply. The shape of the phase curve indicates a mean slope on the surface of Mercury of about 16° which is slightly smoother than that of the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
. Approaching phase angle 0° (fully illuminated phase) the curve rises to a sharp peak. This surge in brightness is called the opposition effect because for most bodies (though not Mercury) it occurs at astronomical opposition when the body is opposite from the Sun in the sky. The width of the opposition surge for Mercury indicates that both the compaction state of the regolith and the distribution of particle sizes on the planet are similar to those on the Moon.
Early visual observations contributing to the phase curve of Mercury were obtained by G. Muller in the 1800s and by A. Danjon in the mid-twentieith century. W. Irvine and colleagues used photoelectric photometry in the 1960s. Some of these early data were analyzed by G. de Vaucouleurs, summarized by D. Harris and used for predicting apparent magnitudes in the Astronomical Almanac
Astronomical Almanac
The Astronomical Almanac is an almanac published by the United States Naval Observatory and Her Majesty's Nautical Almanac Office, containing solar system ephemeris and catalogs of selected stellar and extragalactic objects....
for several decades. Highly accurate new observations covering the widest range of phase angles to date (2 to 170°) were carried out by A. Mallama, D. Wang and R. Howard using the Large Angle and Spectrometric Coronograph (LASCO)
LASCO Large Angle and Spectrometric Coronagraph
The Large Angle and Spectrometric Coronagraph is one of a number of instruments aboard the Solar and Heliospheric Observatory satellite...
on the Solar and Heliospheric Observatory (SOHO)
Solar and Heliospheric Observatory
The Solar and Heliospheric Observatory is a spacecraft built by a European industrial consortium led by Matra Marconi Space that was launched on a Lockheed Martin Atlas IIAS launch vehicle on December 2, 1995 to study the Sun, and has discovered over 2100 comets. It began normal operations in May...
satellite. They also obtained new CCD observations from the ground. These data are now the major source of the phase curve used in the Astronomical Almanac
Astronomical Almanac
The Astronomical Almanac is an almanac published by the United States Naval Observatory and Her Majesty's Nautical Almanac Office, containing solar system ephemeris and catalogs of selected stellar and extragalactic objects....
for predicting apparent magnitudes.
The apparent brightness of Mercury as seen from Earth is greatest at phase angle 0° (superior conjunction with the Sun) when it can reach magnitude -2.6. At phase angles approaching 180° (inferior conjunction) the planet fades to about magnitude +5 with the exact brightness depending on the phase angle at that particular conjunction. This difference of more than 7 magnitudes corresponds to a change of over a thousand times in apparent brightness.
VenusVenusVenus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
The relatively flat phase curve of VenusVenus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
is characteristic of a cloudy planet. In contrast to Mercury where the curve is strongly peaked approaching phase angle zero (full phase) that of Venus is rounded. The wide illumination scattering angle of clouds, as opposed to the narrower scattering of regolith, causes this flattening of the phase curve. Venus exhibits a brightness surge near phase angle 170°, when it is a thin crescent
Crescent
In art and symbolism, a crescent is generally the shape produced when a circular disk has a segment of another circle removed from its edge, so that what remains is a shape enclosed by two circular arcs of different diameters which intersect at two points .In astronomy, a crescent...
, due to forward scattering of sunlight by droplets of sulfuric acid
Sulfuric acid
Sulfuric acid is a strong mineral acid with the molecular formula . Its historical name is oil of vitriol. Pure sulfuric acid is a highly corrosive, colorless, viscous liquid. The salts of sulfuric acid are called sulfates...
that are above the planet’s cloud tops. Even beyond 170° the brightness does not decline very steeply.
The history of observation and analysis of the phase curve of Venus is similar to that of Mercury. The best set of modern observations and interpretation was reported by A. Mallama, D. Wang and R. Howard. They used the LASCO
LASCO Large Angle and Spectrometric Coronagraph
The Large Angle and Spectrometric Coronagraph is one of a number of instruments aboard the Solar and Heliospheric Observatory satellite...
instrument on SOHO
Solar and Heliospheric Observatory
The Solar and Heliospheric Observatory is a spacecraft built by a European industrial consortium led by Matra Marconi Space that was launched on a Lockheed Martin Atlas IIAS launch vehicle on December 2, 1995 to study the Sun, and has discovered over 2100 comets. It began normal operations in May...
and ground based CCD equipment to observe the phase curve from 2 to 179°. As with Mercury, these new data are the major source of the phase curve used in the Astronomical Almanac
Astronomical Almanac
The Astronomical Almanac is an almanac published by the United States Naval Observatory and Her Majesty's Nautical Almanac Office, containing solar system ephemeris and catalogs of selected stellar and extragalactic objects....
for predicting apparent magnitudes.
In contrast to Mercury the apparent brightness of Venus as seen from Earth does not occur at phase angle zero. Since the phase curve of Venus is relatively flat while its distance from the Earth can vary greatly, maximum brightness occurs when the planet is a crescent, at phase angle 125°, at which time Venus can be as bright as magnitude -4.9. Near inferior conjunction the planet typically fades to about magnitude -3 although the exact values depends on the phase angle. The typical range in apparent brightness for Venus over the course of one apparition is less than a factor of 10 or merely 1% that of Mercury.
EarthEarthEarth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
The phase curve of the EarthEarth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
has not been determined as accurately as those for Mercury and Venus because its integrated brightness is difficult to measure from the surface. Instead of direct observation, earthshine reflected from the portion of the Moon not lit by the Sun has served as a proxy. A few direct measurements of the Earth’s luminosity have been obtained with the EPOXI
EPOXI
EPOXI is a NASA unmanned space mission led by the University of Maryland using the existing Deep Impact vehicle to begin a new series of observations. It first investigated extrasolar planets and, on November 4, 2010, it performed a close approach to the comet 103P/Hartley...
spacecraft. While they do not cover much of the phase curve they reveal a rotational light curve caused by the transit of dark oceans and bright land masses across the hemisphere. P. Goode and colleagues at Big Bear Solar Observatory
Big Bear Solar Observatory
The Big Bear Solar Observatory is an astronomical telescopic observatory with main interests in studying the physics of the Sun. The instruments and telescopes of the observatory are designed and employed specifically for studying the activities and phenomena of our solar system's star...
have measured the earthshine and T. Livengood of NASA analyzed the EPOXI data.
Earth as seen from Venus near opposition from the Sun would be extremely bright at magnitude -6. To an observer outside the Earth’s orbit on Mars our planet would appear most luminous near the time of its greatest elongation from the Sun, at about magnitude -1.5.
MarsMarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
Only about 50° of the martian phase curve can be observed from Earth because it orbits farther from the Sun than our planet. There is an opposition surge but it is less pronounced than that of Mercury. The rotation of bright and dark surface markings across its disk and variability of its atmospheric state (including its dust storms) superimpose variations on the phase curve. R. Schmude obtained many of the MarsMars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
brightness measurements used in a comprehensive phase curve analysis which was performed by A. Mallama.
Because the orbit of Mars is considerably eccentric its brightness at opposition can range from magnitude -3.0 to -1.4. The minimum brightness is about magnitude +1.6 when Mars is on the opposite site of the Sun from the Earth. Rotational variations can elevate or suppress the brightness of Mars by 5% and global dust storms can increase its luminosity by 25%.
The Gas Giant PlanetsGas giantA gas giant is a large planet that is not primarily composed of rock or other solid matter. There are four gas giants in the Solar System: Jupiter, Saturn, Uranus, and Neptune...
The outermost planets (JupiterJupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
, Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
, Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...
and Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...
) are so distant that only small portions of their phase curves near 0° (full phase) can be evaluated from the Earth. That part of the curve is generally fairly flat, like that of Venus, for these cloudy planets.
The apparent magnitude of Jupiter ranges from -2.9 to -1.4, Saturn from -0.5 to +1.4, Uranus from +5.3 to +6.0, and Neptune from +7.8 to +8.0. Most of these variations are due to distance. However, the magnitude range for Saturn also depends on its ring system as explained below.
The Rings of SaturnRings of SaturnThe rings of Saturn are the most extensive planetary ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometres to metres, that form clumps that in turn orbit about Saturn...
The brightness of the Saturn system depends on the orientation of its ring system. The rings contribute more to the overall brightness of the system when they are more inclined to the direction of illumination from the Sun and to the view of the observer. Wide open rings contribute about one magnitude of brightness to the disk alone. The icy particles that compose the rings also produce a strong opposition surge. Hubble Space Telescope and Cassini spacecraft images have been analyzed in an attempt to characterize the ring particles based on their phase curves.The MoonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
The phase curve of the MoonMoon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
approximately resembles that of Mercury due to the similarities of the surfaces and the lack of an atmosphere on either body . Clementine spacecraft data analyzed by J. Hillier, B. Buratti and K. Hill indicate a lunar opposition surge. The Moon’s apparent magnitude at full phase is -12.7 while at quarter phase it is less than 10 percent as bright.
Planetary SatellitesNatural satelliteA natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
The phase curves of many Natural satelliteNatural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
s of other planets, have been observed and interpreted. The icy moons often exhibit opposition brightness surges. This behavior has been used to model their surfaces.
AsteroidAsteroidAsteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s
The phase curves of many asteroidAsteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s have also been observed and they too may exhibit opposition surges. Asteroids can be physically classified in this way. The effects of rotation can be very large and have to be factored in before the phase curve is computed. An example of such a study is reported by R. Baker and colleagues.
Exoplanets
Programs for characterizing planets outside of the solar system depend largely on spectroscopy to identify atmospheric constituents and states, especially those that point to the presence of life forms or which could support life. However, brightness can be measured for very distant Earth-sized objects that are too faint for spectroscopic analysis. A. Mallama has demonstrated that phase curve analysis may be a useful tool for identifying planets that are Earth-like. Additionally, J. Bailey has pointed out that phase curve anomalies such as the brightness excess of Venus could be useful indicators of atmospheric constituents such as water which might be essential to life in the universeExtraterrestrial life
Extraterrestrial life is defined as life that does not originate from Earth...
.
Criticisms
Inferences about regoliths from phase curves are frequently based on Hapke parameterization. However, in a blind test M. Shepard and P. Helfenstein found no strong evidence that a particular set of Hapke parameters derived from photometric data could uniquely reveal the physical state of laboratory samples. These tests included modeling the three-term Henyey-Greenstein phase functions and the coherent backscatter opposition effect. This negative finding suggests that the radiative transfer model developed by B. HapkeBruce William Hapke
Bruce Hapke is a noted American planetary scientist. An expert in bidirectional reflectance spectroscopy, Hapke has been described as the father of planetary remote sensing.- Career :...
may be inadequate for physical modeling based on photometry.