Earth's gravitational field

The gravitational field of the Earth - the field of gravity, due to the Earth and the centrifugal force caused by its daily rotation. Characterized by the spatial distribution of gravity and gravitational potential .

To solve practical problems, the potential of the earth's attraction (without taking into account the centrifugal force and the influence of other celestial bodies) is expressed as a series ^[1]

V(r,\phi ,\lambda )={\frac {GM}{r}}\left[1+\sum _{n=1}^{\infty }\left({\frac {a}{r}}\right)^{n}\sum _{m=0}^{n}P_{nm}\sin \phi \left(C_{nm}\cos m\lambda +S_{nm}\sin m\lambda \right)\right],

{\ displaystyle V (r, \ phi, \ lambda) = {\ frac {GM} {r}} \ left [1+ \ sum _ {n = 1} ^ {\ infty} \ left ({\ frac {a } {r}} \ right) ^ {n} \ sum _ {m = 0} ^ {n} P_ {nm} \ sin \ phi \ left (C_ {nm} \ cos m \ lambda + S_ {nm} \ sin m \ lambda \ right) \ right],}

V (r, \ phi, \ lambda) = {\ frac {GM} {r}} \ left [1+ \ sum _ {{n = 1}} ^ {\ infty} \ left ({\ frac {a} {r}} \ right) ^ {n} \ sum _ {{m = 0}} ^ {n} P _ {{nm}} \ sin \ phi \ left (C _ {{nm}} \ cos m \ lambda + S _ {{nm}} \ sin m \ lambda \ right) \ right],

Where

r,\phi ,\lambda

{\ displaystyle r, \ phi, \ lambda}

r, \ phi, \ lambda

- polar coordinates,

G

{\ displaystyle G}

G

- gravitational constant,

M

{\ displaystyle M}

M

- the mass of the Earth,

G M

{\ displaystyle GM}

GM

= 398 603⋅10 ⁹ m ³ · s ⁻² ,

a

{\ displaystyle a}

a

- the Earth’s major semi-axis.

Acceleration of gravity

In non - inertial reference systems, the acceleration of gravity is numerically equal to the force of gravity acting on the object of a single mass.

The acceleration of free fall on the surface of the Earth g (usually pronounced “Zhe” ) varies from 9.780 m / s² at the equator to 9.832 m / s² at the poles ^[2] . The standard (“normal”) value adopted in the construction of the system of units is g = 9.80665 m / s² ^[3] ^[4] . The standard value g was defined as “average” in some sense on the entire Earth, it is approximately equal to the acceleration of free fall at a latitude of 45.5 ° at sea level . In approximate calculations, it is usually taken equal to 9.81; 9.8 or 10 m / s².

In the media and popular science literature, g is often used as a non-systemic unit of gravity, used, for example, to estimate the magnitude of overloads during training of pilots and cosmonauts , as well as the strength of other celestial bodies on the Earth with other celestial forces . bodies ).

Getting the value of g from the law of world wideness

According to the law of world wideness , the force of earthly gravity acting on a body is determined by the formula

F=G{\frac {m_{1}m_{2}}{r^{2}}}=\left(G{\frac {m_{1}}{r^{2}}}\right)m_{2}

{\ displaystyle F = G {\ frac {m_ {1} m_ {2}} {r ^ {2}}} = \ left (G {\ frac {m_ {1}} {r ^ {2}}} \ right) m_ {2}}

F=G{\frac {m_{1}m_{2}}{r^{2}}}=\left(G{\frac {m_{1}}{r^{2}}}\right)m_{2}

,

where r is the distance between the center of the Earth and the body (see below), m ₁ is the mass of the Earth and m ₂ is the mass of the body.

In addition, according to the second Newton's law , F = ma , where m is the mass and a is the acceleration,

F=m_{2}g

{\ displaystyle F = m_ {2} g}

From the comparison of the two formulas it is clear that

g=G{\frac {m_{1}}{r^{2}}}

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

Thus, in order to find the value of the acceleration of gravity g at sea level, it is necessary to substitute in the formula the values of the gravitational constant G , the mass of the Earth (in kilograms) m ₁ and the radius of the Earth (in meters) r :

g=G{\frac {m_{1}}{r^{2}}}=(6.67384\times 10^{-11}){\frac {5.9722\times 10^{24}}{(6.371\times 10^{6})^{2}}}=9.8196{\mbox{m}}\cdot {\mbox{s}}^{-2}

{\ displaystyle g = G {\ frac {m_ {1}} {r ^ {2}}} = (6.67384 \ times 10 ^ {- 11}) {\ frac {5.9722 \ times 10 ^ {24}} {( 6.371 \ times 10 ^ {6}) ^ {2}}} = 9.8196 {\ mbox {m}} \ cdot {\ mbox {s}} ^ {- 2}}

It should be noted that this formula is valid for a spherical body under the assumption that its entire mass is concentrated in its center. This allows us to use the radius of the earth for r .

There are significant uncertainties in the values of r and m ₁ , as well as the values of the gravitational constant G , which is difficult to measure accurately.

If G , g and r are known, then solving the inverse problem will allow us to obtain the magnitude of the mass of the Earth.

Gravity Anomalies

Gravity anomalies of the Earth (according to NASA GRACE - Gravity Recovery And Climate Change). Animated version ^[5] .

Gravitational anomalies in relation to geophysics - deviations of the magnitude of the gravitational field from the calculated, calculated on the basis of a mathematical model. The gravitational potential of the earth's surface, or geoid , is usually described on the basis of mathematical theories using harmonic functions ^[6] . These deviations can be caused by various factors, including:

The earth is not homogeneous , its density is different in different areas;
The Earth is not an ideal sphere , and the formula uses the average value of its radius;
The calculated value of g takes into account only the force of gravity and does not take into account the centrifugal force arising due to the rotation of the Earth;
When the body rises above the Earth's surface, the value of g decreases (“altitude correction” (see below), the Bouguer anomaly );
The earth is affected by the gravitational fields of other cosmic bodies, in particular, the tidal forces of the Sun and the Moon.

Altitude Amendment

The first amendment for standard mathematical models, the so-called high - altitude anomaly , allows to take into account the change in the value of g depending on the height above sea level ^[7] . Use the values of the mass and radius of the Earth:

r_{\mathrm {Earth} }=6.371\times 10^{6}\,\mathrm {m}

{\ displaystyle r _ {\ mathrm {Earth}} = 6.371 \ times 10 ^ {6} \, \ mathrm {m}}

m_{\mathrm {Earth} }=5.9722\times 10^{24}\,\mathrm {kg}

{\ displaystyle m _ {\ mathrm {Earth}} = 5.9722 \ times 10 ^ {24} \, \ mathrm {kg}}

The correction factor (Δg) can be obtained from the relationship between the acceleration of gravity g and the gravitational constant G :

g_{0}=G\,m_{\mathrm {Earth} }/r_{\mathrm {Earth} }^{2}=9.8196\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}

{\ displaystyle g_ {0} = G \, m _ {\ mathrm {Earth}} / r _ {\ mathrm {Earth}} ^ {2} = 9.8196 \, {\ frac {\ mathrm {m}} {\ mathrm { s} ^ {2}}}}

where:

G=6.67384\times 10^{-11}\,{\frac {\mathrm {m} ^{3}}{\mathrm {kg} \cdot \mathrm {s} ^{2}}}.

{\ displaystyle G = 6.67384 \ times 10 ^ {- 11} \, {\ frac {\ mathrm {m} ^ {3}} {\ mathrm {kg} \ cdot \ mathrm {s} ^ {2}}}. }

.

At height h above the surface of the earth, g _{h is} calculated by the formula:

g_{h}=G\,m_{\mathrm {Earth} }/\left(r_{\mathrm {Earth} }+h\right)^{2}

{\ displaystyle g_ {h} = G \, m _ {\ mathrm {Earth}} / \ left (r _ {\ mathrm {Earth} + h \ right) ^ {2}}

Thus, the altitude correction for height h can be expressed:

\Delta g_{h}=\left[G\,m_{\mathrm {Earth} }/\left(r_{\mathrm {Earth} }+h\right)^{2}\right]-\left[G\,m_{\mathrm {Earth} }/r_{\mathrm {Earth} }^{2}\right]

{\ displaystyle \ Delta g_ {h} = \ left [G \, m _ {\ mathrm {Earth}} / \ left (r _ {\ mathrm {Earth} + h \ right) ^ {2} \ right] - \ left [G \, m _ {\ mathrm {Earth}} / r _ {\ mathrm {Earth} ^ {2} \ right]}

.

This expression can be easily used for programming or inclusion in a table. Simplifying and neglecting small quantities ( h << r _Earth ), we get a good approximation:

\Delta g_{h}\approx -\,{\dfrac {G\,m_{\mathrm {Earth} }}{r_{\mathrm {Earth} }^{2}}}\times {\dfrac {2\,h}{r_{\mathrm {Earth} }}}

{\ displaystyle \ Delta g_ {h} \ approx - \, {\ dfrac {G \, m _ {\ mathrm {Earth}} {r _ {\ mathrm {Earth} ^ {2}}} \ times {\ dfrac {2 \, h} {r _ {\ mathrm {Earth}}}}}

.

Using the above numerical values above, and the height h in meters, we get:

\Delta g_{h}\approx -3.083\times 10^{-6}\,h

{\ displaystyle \ Delta g_ {h} \ approx -3.083 \ times 10 ^ {- 6} \, h}

Given the latitude and altitude amendment, we obtain:

g_{\phi ,h}=9.780327\left(1+0.0053024\sin ^{2}\phi -0.0000058\sin ^{2}2\phi \right)-3.086\times 10^{-6}h

{\ displaystyle g _ {\ phi, h} = 9.780327 \ left (1 + 0.0053024 \ sin ^ {2} \ phi -0.0000058 \ sin ^ {2} 2 \ phi \ right) -3.086 \ times 10 ^ {- 6} h}

,

Where $\ g_{\phi ,h}$ ${\ displaystyle \ g _ {\ phi, h}}$ - acceleration of gravity at latitude $\ \phi$ ${\ displaystyle \ \ phi}$ and height h . This expression can also be represented as follows:

g_{\phi ,h}=9.780327\left[\left(1+0.0053024\sin ^{2}\phi -0.0000058\sin ^{2}2\phi \right)-3.155\times 10^{-7}h\right]\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}

{\ displaystyle g _ {\ phi, h} = 9.780327 \ left [\ left (1 + 0.0053024 \ sin ^ {2} \ phi -0.0000058 \ sin ^ {2} 2 \ phi \ right) -3.155 \ times 10 ^ { -7} h \ right] \, {\ frac {\ mathrm {m}} {\ mathrm {s} ^ {2}}}

.

Comparison of the power of the earth with other celestial bodies

The table shows the values of the acceleration of free fall on the surface of the Earth, the Sun , the Moon , the planets of the Solar System , a number of satellites and asteroids . For giant planets, the term "surface" means the visible surface, and for the sun, the upper limit of the photosphere . The data in the table do not take into account the effect of centrifugal force on the rotation of the planets and in fact mean the values of the desired quantities near the poles of the planets. For reference, the time of the object falling on a given celestial body from a 100-meter height and the maximum speed achieved in this case (air resistance is not taken into account) is indicated.

Heavenly body	The force of gravity compared to earth	Acceleration of free fall on the surface, m / s ²	Notes	Fall time from 100-meter height / Speed achieved
The sun	27.90	274.1		0.85 sec	843 km / h
Mercury	0.3770	3.7		7.4 seconds	98 km / h
Venus	0.905	8,872		4.8 seconds	152 km / h
Land	one	9,80665	^[eight]	4.5 seconds	159 km / h
Moon	0.1657	1,625		11.1 seconds	65 km / h
Mars	0.3795	3,728		7.3 seconds	98 km / h
Ceres	0.029	0.028		26.7 seconds	27 km / h
Jupiter	2.640	25.93		2.8 seconds	259 km / h
And about	0.182	1,789		10.6 seconds	68 km / h
Europe	0.134	1,314		12.3 seconds	58 km / h
Ganymede	0.145	1.426		11.8 seconds	61 km / h
Callisto	0.126	1.24		12.7 seconds	57 km / h
Saturn	1,139	11.19		4.2 seconds	170 km / h
Titanium	0.138	1,352		12.2 seconds	59 km / h
Uranus	0.917	9.01		4.7 seconds	153 km / h
Titania	0.039	0.379		23.0 seconds	31 km / h
Oberon	0.035	0.347		24.0 seconds	30 km / h
Neptune	1,148	11.28		4.2 seconds	171 km / h
Triton	0.079	0.779		16.0 seconds	45 km / h
Pluto	0.063	0.62		18.1 seconds	40 km / h
Eris	0.0814	0.8	(approx.)	15.8 seconds	46 km / h

Notes

↑ Mironov, 1980 , p. 52-56.
“Free fall of bodies. Acceleration of gravity"
↑ Declaration of weight; conventional value of g _n (English) . Resolution of the 3rd CGPM (1901) . Bipm . The appeal date is November 11, 2015.
↑ V.M. Dengub, V.G. Smirnov. Units of quantities Dictionary - reference. M .: Publishing house of standards, 1990, p. 237.
IA NASA / JPL / GRACE Global Gravity Animation University of Texas Center for Space Research PIA12146 (Unc.) . Photojournal . NASA Jet Propulsion Laboratory. The appeal date is December 30, 2013.
↑ VL Panteleev. "Theory of the Earth" (course of lectures)
↑ Fowler, CMR The Solid Earth: An Introduction to Global Geophysics. - 2. - Cambridge, UK : Cambridge University Press , 2005. - P. 205–206. - ISBN 0-521-89307-0 .
↑ This value excludes the influence of centrifugal force due to the rotation of the Earth and, therefore, is greater than the standard value of 9.80665 m / s ² .

Links

Literature

Mironov V.S. Gravity course. - L .: Nedra, 1980. - 543 p.

[_fa21093388ec9286-1] Mironov, 1980 , p. 52-56.

[2] “Free fall of bodies. Acceleration of gravity"

[3] Declaration of weight; conventional value of g _n (English) . Resolution of the 3rd CGPM (1901) . Bipm . The appeal date is November 11, 2015.

[4] V.M. Dengub, V.G. Smirnov. Units of quantities Dictionary - reference. M .: Publishing house of standards, 1990, p. 237.

[5] IA NASA / JPL / GRACE Global Gravity Animation University of Texas Center for Space Research PIA12146 (Unc.) . Photojournal . NASA Jet Propulsion Laboratory. The appeal date is December 30, 2013.

[6] VL Panteleev. "Theory of the Earth" (course of lectures)

[Fowler-7] Fowler, CMR The Solid Earth: An Introduction to Global Geophysics. - 2. - Cambridge, UK : Cambridge University Press , 2005. - P. 205–206. - ISBN 0-521-89307-0 .

[8] This value excludes the influence of centrifugal force due to the rotation of the Earth and, therefore, is greater than the standard value of 9.80665 m / s ² .