astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

, minimum mass is the lower-bound calculated

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

of observed objects such as

planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...

star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...

s, binary systems,

nebula A nebula (; or nebulas) is a distinct luminescent part of interstellar medium, which can consist of ionized, neutral, or molecular hydrogen and also cosmic dust. Nebulae are often star-forming regions, such as in the Pillars of Creation in ...

e, and

black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

s. Minimum mass is a widely cited statistic for

extrasolar planet An exoplanet or extrasolar planet is a planet outside the Solar System. The first confirmed detection of an exoplanet was in 1992 around a pulsar, and the first detection around a main-sequence star was in 1995. A different planet, first detect ...

s 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 inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Earth ...

s and true masses of the planets are generally unknown. 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:

M_\text = \frac

Exoplanets

Orientation of the transit to 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 system A planetary system is a set of gravity, gravitationally bound non-stellar Astronomical object, bodies in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although ...

s, 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). 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 (

M_\text \cdot

). This is called Sin i degeneracy. The posterior distribution of the inclination angle ''i'' depends on the true mass distribution of the planets.

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 Methods of detecting exoplanets usually rely on indirect strategies – that is, they do not directly image the planet but deduce its existence from another signal. Any planet is an extremely faint light source compared to its parent star. For e ...

. 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 line A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission (electromagnetic radiation), emission or absorption (electromagnetic radiation), absorption of light in a narrow frequency ...

s 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. The term true mass is synonymous with the term

, but is used in astronomy to differentiate the measured mass of a planet from the minimum mass usually obtained from radial velocity techniques. Methods used to determine the true mass of a planet include measuring the distance and period of one of its

satellites A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scientif ...

, advanced

astrometry Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other Astronomical object, celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, th ...

techniques that use the motions of other planets in the same

star system A star system or stellar system is a small number of stars that orbit each other, bound by gravity, gravitational attraction. It may sometimes be used to refer to a single star. A large group of stars bound by gravitation is generally calle ...

, combining radial velocity techniques with transit observations (which indicate very low orbital inclinations), and combining radial velocity techniques with

stellar parallax Stellar parallax is the apparent shift of position (''parallax'') of any nearby star (or other object) against the background of distant stars. By extension, it is a method for determining the distance to the star through trigonometry, the stel ...

measurements (which also determine orbital inclinations).

Use of sine function

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

, a unit circle is the circle of radius one centered at the origin (0, 0) in the

Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...

. 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 and , respectively. The point's distance from the origin is always 1. Circle cos sin

Stars

With a mass only 93 times that of

Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...

(), or , AB Doradus C, a companion to AB Doradus A, is the smallest known star undergoing nuclear fusion in its core. 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 . 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 . Smaller bodies are called

brown dwarf Brown dwarfs are substellar objects that have more mass than the biggest gas giant planets, but less than the least massive main sequence, main-sequence stars. Their mass is approximately 13 to 80 Jupiter mass, times that of Jupiter ()not big en ...

s, which occupy a poorly defined grey area between stars and

gas giant A gas giant is a giant planet composed mainly of hydrogen and helium. Jupiter and Saturn are the gas giants of the Solar System. The term "gas giant" was originally synonymous with "giant planet". However, in the 1990s, it became known that Uranu ...

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