# Manual Celestial Mechanics. Part II

Other variations in the motion were accounted for by offsetting the centres of the deferent for each planet from Earth by a short distance. By choosing the combination of speeds and distances appropriately, Ptolemy was able to predict the motions of the planets with considerable accuracy. His scheme was adopted as absolute dogma and survived more than 1, years until the time of Copernicus. Nicolaus Copernicus assumed that Earth was just another planet that orbited the Sun along with the other planets.

His belief that planetary motion had to be a combination of uniform circular motions forced him to include a series of epicycles to match the motions in the noncircular orbits. The epicycles were like terms in the Fourier series that are used to represent planetary motions today.

A Fourier series is an infinite sum of periodic terms that oscillate between positive and negative values in a smooth way, where the frequency of oscillation changes from term to term. They represent better and better approximations to other functions as more and more terms are kept. Copernicus also determined the relative scale of his heliocentric solar system, with results that are remarkably close to the modern determination.

Although this model is mathematically equivalent to the heliocentric model of Copernicus, it represents an unnecessary complication and is physically incorrect.

## Celestial Mechanics Part by Sternberg Shlomo

Celestial mechanics. We shall neglect the forces between planets, considering only a planet's interaction with the sun. These considerations apply equally well to the motion of a satellite about a planet. Let's examine the case of two bodies of masses M and m moving in circular orbits under the influence of each other's gravitational attraction. The large body of mass M moves in an orbit of constant radius R and the small body of mass m in an orbit of constant radius r , both having the same angular velocity. For this to happen, the gravitational force acting on each body must provide the necessary centripetal acceleration.

Since these gravitational forces are a simple action-reaction pair, the centripetal forces must be equal but opposite in direction. That is, m 2 r must equal M 2 R. The specific requirement, then, is that the gravitational force acting on either body must equal the centripetal force needed to keep it moving in its circular orbit, that is.

If one body has a much greater mass than the other, as is the case of the sun and a planet or the Earth and a satellite, its distance from the center of mass is much smaller than that of the other body. If we assume that m is negligible compared to M , then R is negligible compared to r. Thus, equation 4. This is a basic equation of planetary and satellite motion. It also holds for elliptical orbits if we define r to be the semi-major axis a of the orbit.

Click here for example problem 4. In this case the size of the secondary cannot be ignored. The distance R is no longer negligible compared to r and, therefore, must be carried through the derivation. Equation 4.

## Celestial mechanics

More commonly the equation is written in the equivalent form. The semi-major axis used in astronomy is always the primary-to-secondary distance, or the geocentric semi-major axis. For example, the Moon's mean geocentric distance from Earth a is , kilometers. On the other hand, the Moon's distance from the barycenter r is , km, with Earth's counter-orbit R taking up the difference of 4, km. Kepler's second law of planetary motion must, of course, hold true for circular orbits.

In such orbits both and r are constant so that equal areas are swept out in equal times by the line joining a planet and the sun. For elliptical orbits, however, both and r will vary with time. Let's now consider this case. Figure 4. The area swept out by the radius vector in a short time interval t is shown shaded. This expression becomes more exact as t approaches zero, i. The rate at which area is being swept out instantaneously is therefore. For any given body moving under the influence of a central force, the value r 2 is constant. Let's now consider two points P 1 and P 2 in an orbit with radii r 1 and r 2 , and velocities v 1 and v 2. Since the velocity is always tangent to the path, it can be seen that if is the angle between r and v , then. Multiplying through by r , we have. Thus, letting P 1 and P 2 be these two points we get. Let's now look at the energy of the above particle at points P 1 and P 2. Conservation of energy states that the sum of the kinetic energy and the potential energy of a particle remains constant.

Applying conservation of energy we have. From equations 4.

The launch of a satellite or space vehicle consists of a period of powered flight during which the vehicle is lifted above the Earth's atmosphere and accelerated to orbital velocity by a rocket, or launch vehicle. Powered flight concludes at burnout of the rocket's last stage at which time the vehicle begins its free flight. During free flight the space vehicle is assumed to be subjected only to the gravitational pull of the Earth.

If the vehicle moves far from the Earth, its trajectory may be affected by the gravitational influence of the sun, moon, or another planet. A space vehicle's orbit may be determined from the position and the velocity of the vehicle at the beginning of its free flight. A vehicle's position and velocity can be described by the variables r, v , and , where r is the vehicle's distance from the center of the Earth, v is its velocity, and is the angle between the position and the velocity vectors, called the zenith angle see Figure 4.

If we let point P 2 represent the perigee, then equation 4. Substituting equation 4. Like any quadratic, the above equation yields two answers. The smaller of the two answers corresponds to R p , the periapsis radius. The other root corresponds to the apoapsis radius, R a. Please note that in practice spacecraft launches are usually terminated at either perigee or apogee, i.

This condition results in the minimum use of propellant. This angle is given by. This angle is called the flight-path angle , and is positive when the velocity vector is directed away from the primary as shown in Figure 4. When flight-path angle is used, equations 4. Orbit Tilt, Rotation and Orientation Above we determined the size and shape of the orbit, but to determine the orientation of the orbit in space, we must know the latitude and longitude and the heading of the space vehicle at burnout.

If , , and 2 are given, the other values can be calculated from the following relationships:. In equation 4. If is positive, periapsis is west of the burnout point as shown in Figure 4. The longitude of the ascending node, , is measured in celestial longitude, while 1 is geographical longitude.

The celestial longitude of the ascending node is equal to the local apparent sidereal time , in degrees, at longitude 1 at the time of engine burnout. Sidereal time is defined as the hour angle of the vernal equinox at a specific locality and time; it has the same value as the right ascension of any celestial body that is crossing the local meridian at that same instant.

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At the moment when the vernal equinox crosses the local meridian, the local apparent sidereal time is See this sidereal time calculator. The geodetic latitude or geographical latitude , , is the angle defined by the intersection of the reference ellipsoid normal through the point of interest and the true equatorial plane.

The geocentric latitude , ', is the angle between the true equatorial plane and the radius vector to the point of intersection of the reference ellipsoid and the reference ellipsoid normal passing through the point of interest. Declination , , is the angular distance of a celestial object north or south of Earth's equator. It is the angle between the geocentric radius vector to the object of interest and the true equatorial plane.

R is the magnitude of the reference ellipsoid's geocentric radius vector to the point of interest on its surface, r is the magnitude of the geocentric radius vector to the celestial object of interest, and the altitude h is the perpendicular distance from the reference ellipsoid to the celestial object of interest. The value of R at the equator is a , and the value of R at the poles is b.

The ellipsoid's flattening, f , is the ratio of the equatorial-polar length difference to the equatorial length. When solving problems in orbital mechanics, the measurements of greatest usefulness are the magnitude of the radius vector, r , and declination, , of the object of interest. However, we are often given, or required to report, data in other forms. For instance, at the time of some specific event, such as "orbit insertion", we may be given the spacecraft's altitude along with the geodetic latitude and longitude of the sub-vehicle point.

Problems in Celestial Mechanics (part 2)

In such cases, it may be necessary to convert the given data to a form more suitable for our calculations. The relationship between geodetic and geocentric latitude is,. The radius of the reference ellipsoid is given by,. The length r can be solved from h , or h from r , using one of the following,. And declination is calculated using,.

## astronomy - fundamentals of celestial mechanics - danby - PDF Free Download

For spacecraft in low earth orbit, the difference between and ' is very small, typically not more than about 0. Even at the distance of the Moon, the difference is not more than about 0. It is important to note that the value of h is not always measured as described and illustrated above. In some applications it is customary to express h as the perpendicular distance from a reference sphere, rather than the reference ellipsoid. The example problems presented in this web site also assume this method of measurement.