Minimum mass

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Left: A representation of a star orbited by a planet. All the movement of the star is along the viewer's line-of-sight; Doppler spectroscopy will give a true value of the planet's mass.
Right: In this case none of the star's movement is along the viewer's line-of-sight and the Doppler spectroscopy method will not detect the planet at all.

In astronomy, minimum mass is the lower-bound calculated mass of observed objects such as planets, stars, binary systems, [1] nebulae, [2] and black holes.

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Minimum mass is a widely cited statistic for extrasolar planets detected by the radial velocity method or Doppler spectroscopy, and is determined using the binary mass function. This method reveals planets by measuring changes in the movement of stars in the line-of-sight, so the real orbital inclinations and true masses of the planets are generally unknown. [3] This is a result of sin i degeneracy.

If inclination i can be determined, the true mass can be obtained from the calculated minimum mass using the following relationship:

Exoplanets

Orientation of the transit to Earth

A view of inclination that would appear flat upon the green plane from Earth. Orbital elements.svg
A view of inclination that would appear flat upon the green plane from Earth.

Most stars will not have their planets lined up and orientated so that they eclipse over the center of the star and give the viewer on earth a perfect transit. It is for this reason that when we often are only able to extrapolate a minimum mass when viewing a star's wobble because we do not know the inclination and therefore only be able to calculate the part pulling the star on the plane of celestial sphere.

For orbiting bodies in extrasolar planetary systems, an inclination of 0° or 180° corresponds to a face-on orbit (which cannot be observed by radial velocity), whereas an inclination of 90° corresponds to an edge-on orbit (for which the true mass equals the minimum mass). [4]

Planets with orbits highly inclined to the line of sight from Earth produce smaller visible wobbles, and are thus more difficult to detect. One of the advantages of the radial velocity method is that eccentricity of the planet's orbit can be measured directly. One of the main disadvantages of the radial-velocity method is that it can only estimate a planet's minimum mass (). This is called Sin i degeneracy. The posterior distribution of the inclination angle i depends on the true mass distribution of the planets. [5]

Radial velocity method

However, when there are multiple planets in the system that orbit relatively close to each other and have sufficient mass, orbital stability analysis allows one to constrain the maximum mass of these planets. The radial velocity method can be used to confirm findings made by the transit method. When both methods are used in combination, then the planet's true mass can be estimated.

Although radial velocity of the star only gives a planet's minimum mass, if the planet's spectral lines can be distinguished from the star's spectral lines then the radial velocity of the planet itself can be found, and this gives the inclination of the planet's orbit. This enables measurement of the planet's actual mass. This also rules out false positives, and also provides data about the composition of the planet. The main issue is that such detection is possible only if the planet orbits around a relatively bright star and if the planet reflects or emits a lot of light. [6]

The term true mass is synonymous with the term mass, but is used in astronomy to differentiate the measured mass of a planet from the minimum mass usually obtained from radial velocity techniques. [7] Methods used to determine the true mass of a planet include measuring the distance and period of one of its satellites, [8] advanced astrometry techniques that use the motions of other planets in the same star system, [7] combining radial velocity techniques with transit observations (which indicate very low orbital inclinations), [9] and combining radial velocity techniques with stellar parallax measurements (which also determine orbital inclinations). [10]

Use of sine function

Unit circle: the radius has length 1. The variable t measures the angle referred to as th in the text. Unit circle.svg
Unit circle: the radius has length 1. The variable t measures the angle referred to as θ in the text.

In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos(θ) and sin(θ), respectively. The point's distance from the origin is always 1.

Animation showing how the sine function (in red)
y
=
sin
[?]
(
th
)
{\displaystyle y=\sin(\theta )}
is graphed from the y-coordinate (red dot) of a point on the unit circle (in green) at an angle of th. Circle cos sin.gif
Animation showing how the sine function (in red) is graphed from the y-coordinate (red dot) of a point on the unit circle (in green) at an angle of θ.

Stars

With a mass only 93 times that of Jupiter (MJ), or .09 M, AB Doradus C, a companion to AB Doradus A, is the smallest known star undergoing nuclear fusion in its core. [11] For stars with similar metallicity to the Sun, the theoretical minimum mass the star can have, and still undergo fusion at the core, is estimated to be about 75 MJ. [12] [13] When the metallicity is very low, however, a recent study of the faintest stars found that the minimum star size seems to be about 8.3% of the solar mass, or about 87 MJ. [13] [14] Smaller bodies are called brown dwarfs, which occupy a poorly defined grey area between stars and gas giants.

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HD 169830 is a star in the southern constellation of Sagittarius. It has a yellow-white hue and is dimly visible to the naked eye with an apparent visual magnitude of +5.90. The star is located at a distance of 120 light years from the Sun based on parallax. It is drifting closer with a radial velocity of −17.3 km/s, and is predicted to come as close as 20.7 ly (6.4 pc) in 2.08 million years. HD 169830 is known to be orbited by two large Jupiter-like exoplanets.

<span class="mw-page-title-main">Gliese 436</span> Star in the constellation Leo

Gliese 436 is a red dwarf located 31.9 light-years away in the zodiac constellation of Leo. It has an apparent visual magnitude of 10.67, which is much too faint to be seen with the naked eye. However, it can be viewed with even a modest telescope of 2.4 in (6 cm) aperture. In 2004, the existence of an extrasolar planet, Gliese 436 b, was verified as orbiting the star. This planet was later discovered to transit its host star.

HD 1237 is a binary star system approximately 57 light-years away in the constellation of Hydrus.

<span class="mw-page-title-main">HD 28185 b</span> Gas giant orbiting HD 28185

HD 28185 b is an extrasolar planet 128 light-years away from Earth in the constellation of Eridanus. The planet was discovered orbiting the Sun-like star HD 28185 in April 2001 as a part of the CORALIE survey for southern extrasolar planets, and its existence was independently confirmed by the Magellan Planet Search Survey in 2008. HD 28185 b orbits its sun in a circular orbit that is at the inner edge of its star's habitable zone.

109 Piscium is a yellow hued G-type main-sequence star located about 108 light-years away in the zodiac constellation of Pisces. It is near the lower limit of visibility to the naked eye with an apparent visual magnitude of 6.27. The star is moving closer to the Earth with a heliocentric radial velocity of −45.5 km/s. It has one known exoplanet.

<span class="mw-page-title-main">Gliese 876 c</span> Gas giant orbiting Gliese 876

Gliese 876 c is an exoplanet orbiting the red dwarf Gliese 876, taking about 30 days to complete an orbit. The planet was discovered in April 2001 and is the second planet in order of increasing distance from its star.

<span class="mw-page-title-main">Upsilon Andromedae c</span> Extrasolar planet in the Andromeda constellation

Upsilon Andromedae c, formally named Samh, is an extrasolar planet orbiting the Sun-like star Upsilon Andromedae A every 241.3 days at an average distance of 0.83 AU. Its discovery in April 1999 by Geoffrey Marcy and R. Paul Butler made this the first multiple-planet system to be discovered around a main-sequence star, and the first multiple-planet system known in a multiple star system. Upsilon Andromedae c is the second-known planet in order of distance from its star.

Upsilon Andromedae d, formally named Majriti, is a super-Jupiter exoplanet orbiting within the habitable zone of the Sun-like star Upsilon Andromedae A, approximately 44 light-years away from Earth in the constellation of Andromeda. Its discovery made it the first multiplanetary system to be discovered around a main-sequence star, and the first such system known in a multiple star system. The exoplanet was found by using the radial velocity method, where periodic Doppler shifts of spectral lines of the host star suggest an orbiting object.

<span class="mw-page-title-main">Methods of detecting exoplanets</span>

Any planet is an extremely faint light source compared to its parent star. For example, a star like the Sun is about a billion times as bright as the reflected light from any of the planets orbiting it. In addition to the intrinsic difficulty of detecting such a faint light source, the light from the parent star causes a glare that washes it out. For those reasons, very few of the exoplanets reported as of January 2024 have been observed directly, with even fewer being resolved from their host star.

<span class="mw-page-title-main">Doppler spectroscopy</span> Indirect method for finding extrasolar planets and brown dwarfs

Doppler spectroscopy is an indirect method for finding extrasolar planets and brown dwarfs from radial-velocity measurements via observation of Doppler shifts in the spectrum of the planet's parent star. As of November 2022, about 19.5% of known extrasolar planets have been discovered using Doppler spectroscopy.

<span class="mw-page-title-main">HD 154345 b</span> Jupiter-Like exoplanet orbiting the star HD 154345 b

HD 154345 b is a Jupiter-mass extrasolar planet orbiting the star HD 154345.

HD 66428 is a G-type main sequence star located approximately 174 light-years away in the constellation of Monoceros. This star is similar to the Sun with an apparent magnitude of 8.25, an effective temperature of 5705 ± 27 K and a solar luminosity 1.28. Its absolute magnitude is 11.1 while its U-V color index is 0.71. It is considered an inactive star and it is metal-rich . This star has a precise mass of 1.14552 solar masses. This precision comes from the Corot mission that measured asteroseismology.

HD 154345 is a star in the northern constellation of Hercules. With an apparent visual magnitude of +6.76 it is a challenge to view with the naked eye, but using binoculars it is an easy target. The distance to this star is 59.6 light years based on parallax, but it is drifting closer with a radial velocity of −47 km/s. At least one exoplanet is orbiting this star.

HD 117207 is a star in the southern constellation Centaurus. With an apparent visual magnitude of 7.24, it is too dim to be visible to the naked eye but can be seen with a small telescope. Based upon parallax measurements, it is located at a distance of 105.4 light-years from the Sun. The star is drifting closer with a radial velocity of −17.4 km/s. It has an absolute magnitude of 4.67.

This page describes exoplanet orbital and physical parameters.

HD 30177 b is an extrasolar planet located approximately 181.6 light-years away in the constellation of Dorado, orbiting the star HD 30177.

HD 143361 b is an exoplanet located approximately 224 light-years away in the constellation of Norma, orbiting the 9th magnitude G-type main sequence star HD 143361. This planet has a minimum mass of 3.0 times that of Jupiter. Because the inclination was initially unknown, the true mass was not known. This planet orbits at a distance of 2.0 AU with an orbital eccentricity of 0.18.

HD 30562 b is an extrasolar planet which orbits the F-type main sequence star HD 30562, located approximately 85.4 light years away in the constellation Eridanus.

HD 204313 b is an extrasolar planet which orbits the G-type main sequence star HD 204313, located approximately 155 light years away in the constellation Capricorn. This planet orbits the star at a distance of 3.082 astronomical units and takes 1931 days or 5.29 years to revolve around the star. It has a minimum mass four times that of Jupiter. However the radius is not known since this planet was not detected by the transit method or direct imaging. Instead, this planet was detected by the radial velocity method using the CORALIE Echelle spectrograph mounted on the 1.2 meter Euler Swiss Telescope located at La Silla Observatory in Atacama Desert, Chile on August 11, 2009.

In astronomy, the binary mass function or simply mass function is a function that constrains the mass of the unseen component in a single-lined spectroscopic binary star or in a planetary system. It can be calculated from observable quantities only, namely the orbital period of the binary system, and the peak radial velocity of the observed star. The velocity of one binary component and the orbital period provide information on the separation and gravitational force between the two components, and hence on the masses of the components.

References

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