Gravitational acceleration
In physics, gravitational acceleration is the free fall acceleration of an object in vacuum — without any drag. This is the steady gain in speed caused exclusively by the force of gravitational attraction. At given GPS coordinates on the Earth's surface and a given altitude, all bodies accelerate in vacuum at the same rate.[1] This equality is true regardless of the masses or compositions of the bodies.
At different points on Earth surface, the gravitational speed gain ranges from 9.764 m/s^{2} to 9.834 m/s^{2}[2] depending on altitude and latitude, with a conventional standard value of exactly 9.80665 m/s^{2} (approximately 32.17405 ft/s^{2}). 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:
where and are any two masses, is the gravitational constant, and is the distance between the two point-like masses.
Using the integral form of Gauss' Law this formula can be extended to any pair of objects of which one is extremely 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]
where '' is the mass of the field source (larger), and is a unit vector directed from the field source to the sample (samller) mass. The negative sign just indicates that the force is attractive (points backward, toward the source).
Then the attraction force vector onto a sample mass '' can be expressed as:
Here is the friction-less, free-fall acceleration sustained by the sampling mass '' 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 '' is and on the distance 'r' to the sample mass ''. It does not depend on the magnitude of the vanishing-small sampling 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 (GRAIL) 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]
- 9.80665 metres (32.1740 ft) per s^{2}
based upon data from World Geodetic System 1984 (WGS-84), where 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):
where
- = 9.832 metres (32.26 ft) per s^{2}
- = 9.806 metres (32.17 ft) per s^{2}
- = 9.780 metres (32.09 ft) per s^{2}
- = 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 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:
where:
M is the mass of the attracting object, 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/s^{2}, when GM = 3.986×10^{14} m^{3}/s^{2}, and R=6.371×10^{6} m. The centripetal radius is r = R cos(φ), and the centripetal time unit is approximately (day / 2π), reduces this, for r = 5×10^{6} metres, to 9.79379 m/s^{2}, which is closer to the observed value.
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
References
- Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus to Einstein and beyond (3rd ed.). Rutgers University Press. p. 113. ISBN 978-0-8135-2908-0.
- Hirt, C.; Claessens, S.; Fecher, T.; Kuhn, M.; Pail, R.; Rexer, M. (2013). "New ultrahigh-resolution picture of Earth's gravity field". Geophysical Research Letters. 40 (16): 4279–4283. Bibcode:2013GeoRL..40.4279H. doi:10.1002/grl.50838.
- Fredrick J. Bueche (1975). Introduction to Physics for Scientists and Engineers, 2nd Ed. USA: Von Hoffmann Press. ISBN 978-0-07-008836-8.
- Brian L. Stevens; Frank L. Lewis (2003). Aircraft Control And Simulation, 2nd Ed. Hoboken, New Jersey: John Wiley & Sons, Inc. ISBN 978-0-471-37145-8.
- Richard B. Noll; Michael B. McElroy (1974), Models of Mars' Atmosphere [1974], Greenbelt, Maryland: NASA Goddard Space Flight Center, SP-8010.