Halliday & Resnick: Physics: Chapter 15

Integration and three-dimensional vector algebra from Calculus text

Law of Universal Gravitation

where

• F is the gravitational force
• M is the mass of the large body
• m is the mass of the smaller body
• G is the gravitational constant defined as (6.67259 ± 0.00085) ×10⁻¹¹ kg⁻² m³/sec² by the Committee on Data for Science and Technology (CODATA)

From this equation it is easy to see that the force of gravity decreases with increasing elevation. For example using the previous example, a point at 10,000 m has a F of only (978.946 cm/sec²)×m.

Considering two bodies A and B whose distance apart makes their dimensions negligibly small. The gravitational force that B exerts on A is

However the earth with respect to another body near it cannot be considered to have negligible dimensions. Thus we must think of it as composite of small massive elements, and use calculus to compute the attractive force between and object near the earth and the earth. If we assume that s is the mass density of the B, and separate the earth into many infinitesimally small volumes of db, then the force of B acting on A is found by triple integration as

The only problem with this function is that the density function s(r) of the earth is not well known, and the earth is not spherical.
 

Effects on Gravity Caused by Angular Velocity

where w is the angular velocity of the object of mass m. From the previous chapter, we know that the earth's orbit is 365.2564 solar days = 366.2564 sidereal days, or 1 mean sidereal day = 1 mean solar day - 3^m55.90^s = 86,164.10^s. Thus, a complete revolution of the earth can be computed as 23^h56^m04.10^s = 24^h × 365.2564/366.2564. Now it is possible to compute an average value of w at the equator as

Using an average radius of the earth of 6,378,137 m = 637,813,700 cm, the centripetal force at the equator is

Note, the centrifugal force is only 0.35% of the attractive force of the earth. For the remainder of the discussion the combined force of gravity and angular velocity will be called the force of gravity, or gravity force.

However, as seen in the figure, the angular velocity of a point is dependent on the latitude of the point since the distance p from the spin axis changes. In fact, at the poles the angular velocity is zero and thus the force of gravity is a maximum. Conversely, the angular velocity of the earth is a maximum at the equator, and thus the force of gravity is a minimum.

In addition to angular velocity reducing the force of gravity at the equator, the shape of the Earth plays an additional role. Since the earth is an oblate spheroid, the distance from the center of mass to the surface of the earth is greater at the equator than at the poles. Since F is inversely proportional to r², the magnitude of F will be less at the equator than at the poles.

The force of gravity is thus computed as

                                  (3)

where db is an infinitesimally small volume element.

Measurements of Gravity

Units of Gravity Measurements

• 1 gal = 1 cm/sec²
• 1000 gals = 1 kgal = 1 kilogal
• 0.001 gals =  1 mgal = 1 milligal
• 0.000001 gals = 1 mgal = 1 microgal

From the proceeding discussion, it can be seen that gravity varies in relation to both height and latitude of the point. Values for g have been measured all over the world with variations as great as 5 gals. Ignoring the angular velocity of the Earth, the reasons for these variations include (1) oblateness of the earth, (2) elevation of the point, and (3) uneven density distributions.

Today's instruments can measure gravity to a mgal. The "Blue Book" specifications for measuring gravity are controlled by the National Geodetic Survey. This manual discusses the instruments used to measure gravity.

Law of the Pendulum

or, more precisely as (see Hosmer, 1930)

1. Absolute determination in which both T and l are measured, and from which g can be calculated. Measurement device times the falling of a mass using a laser. Accuracy about 0.01 – 0.001 mgal. The equipment is bulky, heavy and expensive.
2. Relative determination in which T is measured at two station, and the ratio of these corresponding values of g are determined. This second method avoids the determination of l which is practically difficult. Measurement device uses a mass at the end of a spring that stretches more at points of higher gravity. Accuracy of about 0.01 mgal in five minutes.

so

Absolute determinations of g are much more difficult than relative determinations. Relative determinations can be made with accuracy since the time of oscillation can be determined with little effect from personal errors. Thus most determinations of g are made by relative methods which are all referenced to on reliable determination of g in an area.

Variations in Gravity due to Latitude

where g_f is gravity of the point at latitude f, g_e is the value of gravity at the equator, and g_p is the gravity at the pole. By measuring g_f at two locations, one near the equator and one near the pole, the values for g_e and g_p can be determined. Neglecting the variations in attraction at different parts of the surveys, this equation can be derived as follows:

At the equator, the centrifugal force is c_e = w²r and at the pole, c_p = 0. Also, at the equator

At latitude φ, x = r cos φ, and c_φ = w²r cos φ = c_e cos φ. The vector component of c_φ directly opposed to G is c_e cos² φ. Thus

Substituting this equation into (3) yields

And substituting in equation (4) yields

or

Such that, β can be determined that yields the best agreement with all observed values of g. Helmert, developed a value for β was in 1884, that yielded the equation

where g₀ is the value for gravity on the ellipsoid. In 1901, he modified this equation to be

in which the number 0.000007 is a coefficient found theoretically from assumptions regarding the internal structure of the earth. In Special Publication No. 40 from the National Geodetic Survey, a study was made of observations in the U.S., Canada, Europe and India. The formula resulting from this investigation was

In 1980 the IAG defined the normal gravity field as

γ_φ = 978.0327(1 + 0.0052790414 sin²φ + 0.0000232718 sin⁴φ + 0.0000001262 sin⁶φ) gal

This equation is considered accurate to about 0.7 μgal.

Since this time other forms of this equation have been empirically derived. Their development is discussed in the normal gravity lesson, and the accepted international equation is given the in the section on the normal gravity field below.
 
 

Variations in Gravity due to Height

If we assign 1 to m in Equation (1), we can see that

Differentiating g with respect to r gives the gravity gradient in the direction of r as

Since a change in the radial distance dr is nearly a change in height dH, the above equation can be expressed as

However this equation does not account for the earth's angular velocity which lowers gravity by an amount of

when the mean value of cos²φ is taken as 0.6.

This combination of the two values results in an equation known as the free air anomaly since it compensates for the elevation of a point, and results in the final equation of

Note that this equation can also be derived from differential calculus in the form

Since there is mass between the point at elevation H and a specific datum. This gravitational attraction caused by this mass must be compensated. This compensation is known as the Bouguer (pronounced Boo-gay) plate reduction. This reduction moves the mass between the earth's surface and datum to infinity, and then reduces the point to the datum. The formula for this correction is

where r is the density of the plate, r_m the mean density of the earth, R_m the mean radius of the earth, g_m the mean gravity value of the earth, and H is in meters. For positive elevations, this correction is always subtracted from the measured gravity to remove the effect of the mass between the point and the datum.

Other reductions that can be applied to gravity values include the topographic and isostatic reductions. These corrections are seldom used in geodetic surveying work. You can learn more about these corrections by reviewing the lesson on gravity reductions. In most typical situations in surveying only the Bouguer plate reduction and free-air reductions are applied to gravity values.
 

	Gravity value at P (in mgal) Units of H are in meters.	gP
1.	Using the Bouguer plate reduction, remove all mass above the geopotential surface WQ which contains Q and subtract their attraction from g at P.	-0.1119(HP -HQ)
2.	Since the gravity station P is now in "free-air", apply a free-air reduction, moving the gravity station from P to Q.	+0.3086(HP -HQ)
3.	Finally restore the removed mass to its former position and algebraically add its attraction to g at Q.	-0.1119(HP -HQ)
	Gravity value at Q (in mgal)	gQ = gP + 0.0848(HP-HQ)

It is important to note that this same equation can be obtained by using the Bruns formula. The Bruns formula is important in determining Helmert's orthometric heights.
 
 

The Normal Gravity Field

Note that sometimes corrections are made to the observed gravity to adjust its value to a common datum. Common corrections are made for the oblateness of the spheroid, and the height of the point. This is similar to the weather service correcting atmospheric pressure readings for elevations. That is, the radio broadcast pressure has a "sea-level" correction to compensate for the changes in pressure due the elevation of the point (approximately 1 inch Hg = 1000 ft in altitude). Thus if the free-air correction is applied to observed gravity, this is known as the free air gravity anomaly. Although, the reference gravity g can be selected arbitrarily, the field usually used for comparison is known as the normal gravity field which is based on the latitude and height of the point above a selected ellipsoid of revolution. Read an interesting article on gravity and gravity anomalies.

Attempts have been made to define a reference gravity field that limits the size of the gravity anomaly. An approximate formula for the normal gravity field was accepted by the IAG in 1980 as

This formula has an accuracy of 0.7 μgal
 
 

Gravity Potential

To determine the potential energy of a body U(r) at some distance r from the center of attraction, we must compute the work done against gravity in bringing the body from some infinitesimal distance along a radial line to some point. This is expressed as

Note that the potential energy decreases as the body is moved to an infinite distance.

Note the path taken by the body is irrelevant in this equation. Thus if the body is moved along an equipotential surface, no work is done. We can associate a scalar field by first defining gravitational potential W as the gravitational potential energy per unit mass of a body in a gravitational field. Then

where W_C(r) = ½ p_A²w². From the above equations we see that

1. W_g decreases with increasing r (inversely proportional)
2. W_c increases with increasing p as long as the body takes part in the spin of the earth

Equipotential Surfaces

That is, the potential energy is a constant anywhere on the surface.

Properties of equipotential surfaces

• The gravity field is perpendicular to an equipotential surface
1. Since the equipotential field is a curved line, the plumb line is a curved line.
2. Local radii of curvature vary smoothly form point to point, except where mass density changes suddenly, such as from the atmosphere to the ground
1. This is called a boundary value problem
3. Define the vertical surface
4. Equipotential surfaces
1. Never cross other
2. Closed surfaces
3. Continuous, without breaks
4. Do not possess any steps or sharp edges
5. Define the horizontal direction
5. Local radii of curvature
1. Vary smoothly from point to point
2. Everywhere convex (implies no concave dips in surface (that is holes)
6. Curves are somewhat irregular due to variations in density distribution
7. They exhibit both curvature and twist (torsion)
8. The spacing of the equipotential surfaces is proportional to the magnitude of the gravity field. Thus
               (7)
9. From this equation, we see that the magnitude of gravity varies along an equipotential surface W
10. The negative sign is given to indicate that potential decreases with increasing height
11. Equipotential surfaces converge at the poles.
1. g_P − g_E = 5.186 gal = 0.0053 g_E
2. If H_P g_P = H_E g_e = −ωδW = constant, then H_E > H_P. That is H_P = (1 − 0.0053)H_E = 0.9947H_E
12. A homogeneous surface would coincide with an equipotential surface
1. This is almost true with large bodies of water. Unfortunately, not all water (even is oceans) is homogeneous.

Assume two level surfaces are 200.000 meters apart at the equator, i.e. dH = 200 m, then dW = -g_E dH = 978.049×200.000 = 195,609.8 m-gal when ignoring the negative sign which indicates direction. At the poles the separation of these same two equipotential surfaces would be dH = 195,609.8/983.221 = 198.748 m (Note the negative sign was ignored). This represents a difference in separation of -1.252 m.

Note, this example shows an important property of equipotential surfaces. That is, just performing differential leveling between two points does not provide enough information about these level surfaces. We must also know the gravity at every point along the leveling path. A monumental task!

The effect on differential leveling can be seen to the right. Assume that the two lines depicted in the figure are equipotential surfaces. If a level line was measured from A₁ to A₂, and then from A₂ northward to B₂, the difference in elevation would be A₁A₂. However, if the level line was started at B₂ and continued to B₁ and then A₁, the elevation difference would be B₁B₂. Thus the level line went from A₁ to A₂ to B₂ to B₁ and finally back to A₁. The misclosure of the loop would be A₁A₂ - B₁B₂.

As another example, assume a differential leveling line going up a hill as shown in the figure to the right. The leveling surfaces created by the turning points are depicted as 1 through 3 with average surface gravity values being g_a, g_b, and g_c. Also, let dH_A2, dH₂₃, and dH_3B represent the observed differences in elevation at the surface, and dH'_A2, dH'₂₃, and dH'_3B represent the corresponding differences between the level surfaces with the intersecting average gravity values being symbolized by g'_a, g'_b, and g'_c

Since the gravity potential is constant for a equipotential surface, this means that

Upon summation, these equations yield the orthometric height, H, that is the vertical distance from A to B. Thus,

However, gravity is the catch in the above equation. The surface gravity can economically be measured, but the subsurface gravity values, g', can only be hypothesized with the Bruns formula.

THE IMPORTANT CONCEPT HERE IS THAT THE PROPER METHOD OF LEVELING INVOLVES KNOWLEDGE OF NOT ONLY THE VERTICAL DISPLACEMENT OF THE POINT, BUT ALSO GRAVITY. THAT IS, LEVEL SURFACES ARE DEFINED BY ELEVATION AND GRAVITY!

The GEOID

Typically, the geoid is associated with normal gravity field.

Geopotential number: By rearranging and integrating Equation (7), the geopotential number of a point is defined as:

Geoidal height: (a.k.a. - geoidal separation or undulation) is the separation between the geoid and reference ellipsoid, and is generally denoted as N. Thus the height of a point above the reference ellipsoid h differs from the geoid height H by the equation

where N is the geoidal height. Note this is approximate since the normal to the ellipsoid is a straight line and the orthometric height is a curved line. However, the difference in the two is negligible for all practical purposes.

It is also possible to reference the geoid to a locally fitting ellipsoid such as the Clarke 1866 which was a "best-fit" of the African and North American continents, and was used for the NGVD29 datum. When this happens, the separation between the geoid and local ellipsoid is called relative geoidal height.

The geoidal height is developed using a Fourier transform function with n = m = 360 order and over 100,000 coefficients. The necessary data and code can be downloaded from the NGS at http://www.ngs.noaa.gov/PC_PROD/GEOID03/, or an interactive screen can be used to get N for a particular location. You can learn more about the geoid used in the US by visiting the NGS site at http://www.ngs.noaa.gov/cgi-bin/GEOID_STUFF/geoid03_prompt1.prl. On-line articles about the Geoid can be viewed at http://www.ngs.noaa.gov/GEOID/geolib.html.

Geoid 03 is a specialized U.S. version of the WGS 84 Earth Gravity Model (EGM96). You can learn more about his model at the National Geospatial-Information Agency (NGA) web site. A Geoid calculator can be found at http://164.214.2.59/GandG/wgsegm/egm96.html

HEIGHTS

Dynamic Height

where C_i is the geopotential number for the point. Note that the dynamic height will have length units. The reference gravity can be selected as the normal gravity for the mean earth ellipsoid for a reference latitude f_r selected such that g₀ represents an approximate average gravity value for a region. Thus this value can be thought of as a scale factor. Still, one must be careful not to interpret the dynamic height of a point as a geometrical distance between the geoid and the point. That is, even thought the dynamic heights of points may be equal, their geometrical distances above the ellipsoid are not necessarily equal. However when dynamic heights are equal, points will lie on the same equipotential surface. Thus dynamic heights are useful for projects such as hydrological studies. It can even be argued that dynamic heights should be used for super-elevations.

The NGS uses a normal gravity value γ₀, based on a latitude of 45° for the GRS80 ellipsoid. Thus g = 980.6199 gal.

The dynamic height difference ΔH_ij^D between two points i and j is defined as

An alternative formula for dynamic height differences is obtained by expressing it as a summation of leveled heights differences, Δl_ij plus a correction, or,

where the dynamic height correction DC_ij is  is given by the equation

where is the mean gravity (see Prey reduction) between setups or benchmarks i and i + 1.

Orthometric Height

where the integration is carried out along the plumb line. Substituting for dh', and denoting gravity along the plumb line by g_i' yields

If we use the mean gravity value  along the plumb line of P_i in the integral sense, then we can finally write

However since it is impractical to determine  since the density distribution within the Earth is not known, but can be mathematically expressed as

where g(z) is the actual gravity variable which has a height of z below the surface. The simplest approximation of g(z) is called thePrey reduction and is given as

where g is measured at the ground point. Substituting this equation into the previous yields

Normal Heights

where is the normal counterpart of , and is computed as

The advantage of normal heights is the that they are in traditional units of meters, and use the normal gravity value which can be computed for a specific ellipsoid. Their disadvantage is that normal height will be based on a specific reference ellipsoid which define values for m and f, and thus it is possible for a point to have multiple normal heights. This problem could be eliminated by using an international ellipsoid.
 
 

Helmert Orthometric Height

where the mean value of g_i^H of gravity is taken as g_i^H = g_i + 0.0424H_i where g_i is the gravity value of P_i on the earth's surface. The NGS supplies Helmert heights on their control data sheets for bench marks.

ORTHOMETRIC HEIGHT CORRECTIONS

Now returning to the dynamic height correction, we can develop the following orthometric height correction as follows:

Let Δl_AB be the measured height difference between points A and B. Then

So                (10),

where  is the dynamic height correction.

Now inserting Equations (10) and (11) into (9), and rearranging

where OC_AB = DC_AB + DC_A0A − DC_B0B is the orthometric height correction. Now,

and thus 

Thus, technically, the orthometric correction for a line of differential levels is given as

where γ₀⁴⁵ is the normal gravity at the datum for a latitude of 45°.

However, Bomford and Wolf have suggested an approximate formula based upon the change in latitude for each setup of the leveling line which is

where ΔH_AB is the orthometric height difference of points A and B on a level surface at height H, Δl_AB is the leveled height difference, δφ" is the difference in latitude between the backsight and foresight stations, f is the latitude of the instrument, and r is the 206,264.8"/rad.

Using NGS Data Sheets

where H_C(A) is the potential height of station A in units of kgals-meters, g_A is the modeled gravity value at station A in units of kgals, and H_A is the published orthometric height of the station. (Note that the factor of 1/1,000,000 is present to convert the results from mgals as given on NGS data sheet to kgals.) Following this, the difference in the potential heights is computed and divided by the average gravity value (g_A + g_B)/2 for the two bench marks. That is,

EXAMPLE

Given the following information from the control data sheets for F 137 and J 231, what is the leveled height difference between stations?

Deflection of the Vertical

The deflection of the vertical is usually recorded in terms of two components xi (ξ) in the meridian and eta (η) in the prime vertical. Essentially x is the north-south component, and h is the east-west component. The signs of the components are both taken as positive if the actual gravity vector is north (ξ) or east (η) of the geodetic vertical (ellipsoid normal) in both the northern and southern hemispheres. It is necessary to relate these components to the definitions of astronomic and geodetic latitude, longitude, and azimuth.

Deviation in the meridian, ξ, provides the difference in latitude as follows:

1. Astronomical latitude of P is 90° − YZ_A
2. Geodetic latitude of P is 90° − YZ_G

Similarly, the deflection of the vertical in the prime vertical, h, provide the difference in the longitude. From the figure, it can be shown that Z_AYZ_G is a function of the difference in the astronomic and geodetic longitudes,  Λ and λ, respectively.

Applying the sine law (spherical trigonometry) to triangle Z_AYZ_G, we get

From the figure we see that

where the subscript of the latitude has been dropped since the difference by the geodetic and astronomic latitudes is very small, and does not matter in practice. Furthermore, since the sine of a very small angle is approximately equal to the angle in radian units, the sin η ≈ η. Thus, it can be seen that

or that η equals the difference in the astronomic, Φ, and geodetic latitudes, φ, times the cosine of the latitude. From triangle N_GYN_A in the figure, it can be written

where N_A is the astronomical direction of north, and N_G is the geodetic direction of north. Again using the fact that the sine of a small angle is that angle in radian units, it can be stated that

Thus, it can be said that the correction between the astronomic azimuth A and geodetic azimuth α is the difference in the directions of north, or:

This is known as the Laplace equation. This is one of the most fundamental equations in geometric geodesy in that it establishes a link between longitude and azimuth. From this equation, it can be seen that

From this equation it can be seen that the astronomical and geodetic azimuths, A and α, respectively, and longitudes must be measured. A station with these measured quantities is known as a Laplace Station. Dividing Eq. (14) by (15) yields

The component of the deflection of the vertical in the direction of the azimuth of a line is given as

Ψ = −(ξ cos A + η sin A)       (16)

and the component at right angles to the direction (azimuth) of a line (90° + azimuth) is

ς = ξ sin A − η cos A

The above two corrections represent components of the deflection of the vertical in the direction of the line. The component in the direction of the azimuth of a line between stations i and j must corrected for the zenith angle of the observation. Thus following relations can be derived as

α_ij = Α_ij − η tanφ + (ξ sin A'_ij − η_i cos A'_ij)cot z_c                       (17)
z_c = z_ij + ξ_i cos A'_ij + η_i sin A'_ij = z_ij − Ψ_i

where A_ij is the astronomic azimuth, z_ij is the observed zenith angle, A'_ij is the observed azimuth, α_ij is the geodetic azimuth, and z_c is the reduced geodetic zenith angle. Note that equation (17) corrects for both the difference in the directions of North (N_GN_A) and the skewness in the normals at stations as observed by the zenith angle z_ij. Also note that difference in the trigonometric values for astronomic and geodetic observations are so small that either values can be used with the trigonometric functions.

The student should remember that an ellipsoid, unlike the geoid, is arbitrarily chosen in space and the value for the geodetic latitude, longitude and azimuth change with the selection of the ellipsoid. Thus the size of the deflection of the vertical at a point is not unique, but is a function of the chosen reference ellipsoid.
 

On to normal ellipsoids

Reference

• Hosmer, G.L. 1930. Geodesy Including Astronomical Observations, Gravity Measurements, and Method of Least Squares. John Wiley & Sons, Inc. New York, NY.
• Bomford, Guy. 1980. Geodesy, 4th Ed. Oxford University Press, New York, NY.
• Heiskanen, Weikko, A. and Helmut Moritz. 1967. Physical Geodesy. W.H. Freeman and Co. London, England. Library Call: QB281.H4. On Reserve.
• Torge, Wolfgang, 1991. Geodesy 2nd Ed. de Gruyter, New York, NY.