planetary sciences Planetary science (or more rarely, planetology) is the scientific study of planets (including Earth), celestial bodies (such as moons, asteroids, comets) and planetary systems (in particular those of the Solar System) and the processes of their ...

, the moment of inertia factor or normalized polar moment of inertia is a

dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

quantity that characterizes the radial distribution of

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...

inside a

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotope ...

. Since a

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...

must have dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.

Definition

For a planetary body with

principal moments of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...

A, the moment of inertia factor is defined as
: \frac,
where ''C'' is the polar moment of inertia of the body, ''M'' is the

of the body, and ''R'' is the mean

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of the body. For a sphere with uniform density,

C/MR^2=0.4

. For a differentiated planet or satellite, where there is an increase of density with depth,

C/MR^2<0.4

. The quantity is a useful indicator of the presence and extent of a

planetary core A planetary core consists of the innermost layers of a planet. Cores may be entirely solid or entirely liquid, or a mixture of solid and liquid layers as is the case in the Earth. In the Solar System, core sizes range from about 20% (the Moon ...

, because a greater departure from the uniform-density value of 0.4 conveys a greater degree of concentration of dense materials towards the center.

Solar System values

The

Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...

has by far the lowest moment of inertia factor value among

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...

bodies; it has by far the highest central density (, compared to ~13 for

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

) and a relatively low average density (1.41 g/cm³ versus 5.5 for Earth).

Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...

has the lowest value among the

gas giant A gas giant is a giant planet composed mainly of hydrogen and helium. Gas giants are also called failed stars because they contain the same basic elements as a star. Jupiter and Saturn are the gas giants of the Solar System. The term "gas giant" ...

s in part because it has the lowest bulk density (). Ganymede has the lowest moment of inertia factor among solid bodies in the Solar System because of its fully differentiated interior, a result in part of

tidal heating Tidal heating (also known as tidal working or tidal flexing) occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either (or both) the surface ocean or interior of a planet or satellite. When an object ...

due to the Laplace resonance, as well as its substantial component of low density water

ice Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaq ...

Callisto Callisto most commonly refers to: *Callisto (mythology), a nymph *Callisto (moon), a moon of Jupiter Callisto may also refer to: Art and entertainment *''Callisto series'', a sequence of novels by Lin Carter *''Callisto'', a novel by Torsten Kro ...

is similar in size and bulk composition to Ganymede, but is not part of the orbital resonance and is less differentiated. The

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

is thought to have a small core, but its interior is otherwise relatively homogenous.

Measurement

The polar moment of inertia is traditionally determined by combining measurements of spin quantities ( spin precession rate and/or

obliquity In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbit ...

) with

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

quantities (coefficients of a

spherical harmonic In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...

representation of the gravity field). These

geodetic Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...

data usually require an orbiting

spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, p ...

to collect.

Approximation

For bodies in

hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...

, the Darwin–Radau relation can provide estimates of the moment of inertia factor on the basis of shape, spin, and gravity quantities.

Role in interior models

The moment of inertia factor provides an important constraint for models representing the interior structure of a planet or satellite. At a minimum, acceptable models of the density profile must match the volumetric mass density and moment of inertia factor of the body.

Gallery of internal structure models

File:Sun poster.svg, The

(''C/MR²'' = 0.070) File:Saturn diagram.svg,

(''C/MR²'' = 0.22) File:Ganymede diagram.svg, Ganymede (''C/MR²'' = 0.3115) File:Earth poster.svg,

(''C/MR²'' = 0.3307) File:Callisto diagram.svg,

(''C/MR²'' = 0.3549) File:Moon diagram.svg, The

(''C/MR²'' = 0.3929)

Notes

References

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