Gravitational acceleration

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In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. At a fixed point on the Earth's surface, all bodies accelerate in vacuum at the same rate, regardless of the masses or compositions of the bodies;^[1] the measurement and analysis of these rates is known as gravimetry.

At different points on Earth's surface, the free fall acceleration ranges from 9.764 m/s² to 9.834 m/s^2[2] depending on altitude, latitude, and longitude. A conventional standard value is defined exactly as 9.80665 m/s² (approximately 32.17405 ft/s²). Locations of significant variation from this value are known as gravity anomalies. This does not take into account other effects, such as buoyancy or drag.

Relation to the Universal Law[]

Newton's law of universal gravitation states that there is a gravitational force between any two masses that is equal in magnitude for each mass, and is aligned to draw the two masses toward each other. The formula is:

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }$

where ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are any two masses, ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle r}$ is the distance between the two point-like masses.

Using the integral form of Gauss's Law, this formula can be extended to any pair of objects of which one is far more massive than the other — like a planet relative to any man-scale artefact. The distances between planets and between the planets and the Sun are (by many orders of magnitude) larger than the sizes of the sun and the planets. In consequence both the sun and the planets can be considered as point masses and the same formula applied to planetary motions. (As planets and natural satellites form pairs of comparable mass, the distance 'r' is measured from the common centers of mass of each pair rather than the direct total distance between planet centers.)

If one mass is much larger than the other, it is convenient to take it as observational reference and define it as source of a gravitational field of magnitude and orientation given by:^[3]

${\displaystyle \mathbf {g} =-{GM \over r^{2}}\mathbf {\hat {r}} }$

where ${\displaystyle M}$ is the mass of the field source (larger), and ${\displaystyle \mathbf {\hat {r}} }$ is a unit vector directed from the field source to the sample (smaller) mass. The negative sign indicates that the force is attractive (points backward, toward the source).

Then the attraction force ${\displaystyle \mathbf {F} }$ vector onto a sample mass ${\displaystyle m}$ can be expressed as:

${\displaystyle \mathbf {F} =m\mathbf {g} }$

Here ${\displaystyle \mathbf {g} }$ is the frictionless, free-fall acceleration sustained by the sampling mass ${\displaystyle m}$ under the attraction of the gravitational source. It is a vector oriented toward the field source, of magnitude measured in acceleration units. The gravitational acceleration vector depends only on how massive the field source ${\displaystyle M}$ is and on the distance 'r' to the sample mass ${\displaystyle m}$. It does not depend on the magnitude of the small sample mass.

This model represents the "far-field" gravitational acceleration associated with a massive body. When the dimensions of a body are not trivial compared to the distances of interest, the principle of superposition can be used for differential masses for an assumed density distribution throughout the body in order to get a more detailed model of the "near-field" gravitational acceleration. For satellites in orbit, the far-field model is sufficient for rough calculations of altitude versus period, but not for precision estimation of future location after multiple orbits.

The more detailed models include (among other things) the bulging at the equator for the Earth, and irregular mass concentrations (due to meteor impacts) for the Moon. The Gravity Recovery and Climate Experiment (GRACE) mission launched in 2002 consists of two probes, nicknamed "Tom" and "Jerry", in polar orbit around the Earth measuring differences in the distance between the two probes in order to more precisely determine the gravitational field around the Earth, and to track changes that occur over time. Similarly, the Gravity Recovery and Interior Laboratory mission from 2011-2012 consisted of two probes ("Ebb" and "Flow") in polar orbit around the Moon to more precisely determine the gravitational field for future navigational purposes, and to infer information about the Moon's physical makeup.

Gravity model for Earth[]

The type of gravity model used for the Earth depends upon the degree of fidelity required for a given problem. For many problems such as aircraft simulation, it may be sufficient to consider gravity to be a constant, defined as:^[4]

${\displaystyle g=}$ 9.80665 metres (32.1740 ft) per s²

based upon data from World Geodetic System 1984 (WGS-84), where ${\displaystyle g}$ is understood to be pointing 'down' in the local frame of reference.

If it is desirable to model an object's weight on Earth as a function of latitude, one could use the following (^[4] p. 41):

${\displaystyle g=g_{45}-{\tfrac {1}{2}}(g_{\mathrm {poles} }-g_{\mathrm {equator} })\cos \left(2\,\varphi \cdot {\frac {\pi }{180}}\right)}$

where

• ${\displaystyle g_{\mathrm {poles} }}$ = 9.832 metres (32.26 ft) per s²
• ${\displaystyle g_{45}}$ = 9.806 metres (32.17 ft) per s²
• ${\displaystyle g_{\mathrm {equator} }}$ = 9.780 metres (32.09 ft) per s²
• ${\displaystyle \varphi }$ = latitude, between −90 and 90 degrees

Neither of these accounts for changes in gravity with changes in altitude, but the model with the cosine function does take into account the centrifugal relief that is produced by the rotation of the Earth. For the mass attraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at the poles due to being located farther from the mass center. When the rotational component is included (as above), the gravity at the equator is about 0.53% less than that at the poles, with gravity at the poles being unaffected by the rotation. So the rotational component of change due to latitude (0.35%) is about twice as significant as the mass attraction change due to latitude (0.18%), but both reduce strength of gravity at the equator as compared to gravity at the poles.

Note that for satellites, orbits are decoupled from the rotation of the Earth so the orbital period is not necessarily one day, but also that errors can accumulate over multiple orbits so that accuracy is important. For such problems, the rotation of the Earth would be immaterial unless variations with longitude are modeled. Also, the variation in gravity with altitude becomes important, especially for highly elliptical orbits.

The Earth Gravitational Model 1996 (EGM96) contains 130,676 coefficients that refine the model of the Earth's gravitational field (^[4] p. 40). The most significant correction term is about two orders of magnitude more significant than the next largest term (^[4] p. 40). That coefficient is referred to as the ${\displaystyle J_{2}}$ term, and accounts for the flattening of the poles, or the oblateness, of the Earth. (A shape elongated on its axis of symmetry, like an American football, would be called prolate.) A gravitational potential function can be written for the change in potential energy for a unit mass that is brought from infinity into proximity to the Earth. Taking partial derivatives of that function with respect to a coordinate system will then resolve the directional components of the gravitational acceleration vector, as a function of location. The component due to the Earth's rotation can then be included, if appropriate, based on a sidereal day relative to the stars (≈366.24 days/year) rather than on a solar day (≈365.24 days/year). That component is perpendicular to the axis of rotation rather than to the surface of the Earth.

A similar model adjusted for the geometry and gravitational field for Mars can be found in publication NASA SP-8010.^[5]

The barycentric gravitational acceleration at a point in space is given by:

${\displaystyle \mathbf {g} =-{GM \over r^{2}}\mathbf {\hat {r}} }$

where:

M is the mass of the attracting object, ${\displaystyle \scriptstyle \mathbf {\hat {r}} }$ is the unit vector from center-of-mass of the attracting object to the center-of-mass of the object being accelerated, r is the distance between the two objects, and G is the gravitational constant.

When this calculation is done for objects on the surface of the Earth, or aircraft that rotate with the Earth, one has to account for the fact that the Earth is rotating and the centrifugal acceleration has to be subtracted from this. For example, the equation above gives the acceleration at 9.820 m/s², when GM = 3.986×10¹⁴ m³/s², and R=6.371×10⁶ m. The centripetal radius is r = R cos(φ), and the centripetal time unit is approximately (day / 2π), reduces this, for r = 5×10⁶ metres, to 9.79379 m/s², which is closer to the observed value.^{[citation needed]}

Comparative gravities of the Earth, Sun, Moon, and planets[]

The table below shows comparative gravitational accelerations at the surface of the Sun, the Earth's moon, each of the planets in the Solar System and their major moons, Ceres, Pluto, and Eris. For gaseous bodies, the "surface" is taken to mean visible surface: the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune), and the Sun's photosphere. The values in the table have not been de-rated for the centrifugal force effect of planet rotation (and cloud-top wind speeds for the gas giants) and therefore, generally speaking, are similar to the actual gravity that would be experienced near the poles. For reference the time it would take an object to fall 100 metres, the height of a skyscraper, is shown, along with the maximum speed reached. Air resistance is neglected.

Body	Multiple of Earth gravity	m/s2	ft/s2	Notes	Time to fall 100 m and maximum speed reached
Sun	27.90	274.1	899		0.85 s	843 km/h (524 mph)
Mercury	0.3770	3.703	12.15		7.4 s	98 km/h (61 mph)
Venus	0.9032	8.872	29.11		4.8 s	152 km/h (94 mph)
Earth	1	9.8067	32.174	[a]	4.5 s	159 km/h (99 mph)
Moon	0.1655	1.625	5.33		11.1 s	65 km/h (40 mph)
Mars	0.3895	3.728	12.23		7.3 s	98 km/h (61 mph)
Ceres	0.029	0.28	0.92		26.7 s	27 km/h (17 mph)
Jupiter	2.640	25.93	85.1		2.8 s	259 km/h (161 mph)
Io	0.182	1.789	5.87		10.6 s	68 km/h (42 mph)
Europa	0.134	1.314	4.31		12.3 s	58 km/h (36 mph)
Ganymede	0.145	1.426	4.68		11.8 s	61 km/h (38 mph)
Callisto	0.126	1.24	4.1		12.7 s	57 km/h (35 mph)
Saturn	1.139	11.19	36.7		4.2 s	170 km/h (110 mph)
Titan	0.138	1.3455	4.414		12.2 s	59 km/h (37 mph)
Uranus	0.917	9.01	29.6		4.7 s	153 km/h (95 mph)
Titania	0.039	0.379	1.24		23.0 s	31 km/h (19 mph)
Oberon	0.035	0.347	1.14		24.0 s	30 km/h (19 mph)
Neptune	1.148	11.28	37.0		4.2 s	171 km/h (106 mph)
Triton	0.079	0.779	2.56		16.0 s	45 km/h (28 mph)
Pluto	0.0621	0.610	2.00		18.1 s	40 km/h (25 mph)
Eris	0.0814	0.8	2.6	(approx.)	15.8 s	46 km/h (29 mph)

General relativity[]

In Einstein's theory of general relativity, gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, masses distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. The gravitational force is a fictitious force. There is no gravitational acceleration, in that the proper acceleration and hence four-acceleration of objects in free fall are zero. Rather than undergoing an acceleration, objects in free fall travel along straight lines (geodesics) on the curved spacetime.

See also[]

• Air track
• Gravimetry
• Gravity of Earth
• Gravitation of the Moon
• Gravity of Mars
• Newton's law of universal gravitation
• Standard gravity

Notes[]

References[]