Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 31

Gravity, being the predominant force in space flight, must be understood before constructing any model for space flight dynamics. Consider two particles of masses, and , whose instantaneous positions in an inertial frame, OXYZ, are denoted by the vectors, and , respectively. The relative position of mass, , with respect to the mass, , is given by the vector . By Newton's law of gravitation, the two particles apply an equal and opposite attractive force on each other, which is directly proportional to the product of the two masses, and inversely proportional to the distance, , between them. The equations of motion of the two particles are expressed as follows by Newton's second law of motion:

(2.29)

(2.30)

being the universal gravitational constant. Adding the two equations of motion yields the important result that the centre of mass of the two particles is non‐accelerating:

(2.31)

where

(2.32)

is the position of the centre of mass. This approach can be extended to a system of particles, where the particle has the following equation of motion:

(2.33)

where locates the particle from the i^th particle (Fig. 2.2). A summation of Eq. (2.33) for all the particles produces the following result:

(2.34)

because . Thus Eq. (2.31) is seen to be valid for the ‐particles problem, where the centre of mass is located by

(2.35)

The non‐accelerating centre of mass is the result of the law of conservation of linear momentum in the absence of a net external force on the system of particles. Integrated twice with time, Eq. (2.31) shows that the centre of mass moves in a straight line at a constant velocity:

(2.36)

where are constants (the initial position and the constant velocity, respectively, of the centre of mass).

Figure 2.2 A system of particles in an inertial reference frame OXYZ.

For the convenience of notation, consider the overdot to denote the time derivative relative to the inertial reference frame. Taking a scalar product of Eq. (2.33) with , and summing over all particles, we have

(2.37)

The term on the left‐hand side of Eq. (2.37) is identified to be the time derivative of the net kinetic energy of the system given by

(2.38)

Before proceeding further, it is necessary to consider that gravity is a conservative force, because, as will be seen later, it has no influence on the net energy, , of a system. A force which depends only upon the relative position of the masses (as gravity does) is a conservative force, and can be expressed as the gradient of a scalar function, called the gravitational potential. The gradient of a scalar, U, with respect to a position vector, , is defined to be the following derivative of the scalar with respect to the given vector,

(2.39)

Thus, the gradient of a scalar with respect to a column vector is a row vector.

Consider, for example, an isolated pair of masses, . The gravitational attraction on due to is given by the force, . By Newton's law of gravitation, we have

(2.40)

where the relative position of mass from the mass is given by the vector . Let be the gravitational potential at the location of the particle, , defined by

(2.41)

The gradient of with respect to is the following:

(2.42)

On comparing Eqs. (2.40) and (2.42), we have

(2.43)

The acceleration of the mass due to the gravitational field created by the mass is therefore given by

(2.44)

and is independent of the test mass, .

The concept of gravitational potential between a pair of isolated masses, , can be extended to a system of point masses, where the net gravitational acceleration caused by point masses, , on the particle of mass is the vector sum of all the individual gravitational accelerations given by

(2.45)

or

(2.46)

where

(2.47)

is the net gravitational potential experienced by the particle due to the gravity of all the other particles.

The potential energy, , of the ‐particle system can be defined by

(2.48)

to be the net work done by the gravitational forces to assemble all the particles, beginning from an infinite separation, , where . Thus a finite separation of the particles results in a negative potential energy (a potential well), escaping from which requires a positive energy expenditure.

The gradient of with respect to gives the negative of the gravitational force, , on the particle as follows:

(2.49)

Hence the right‐hand side of Eq. (2.37) is expressed as follows:

(2.50)

A substitution of Eqs. (2.38) and (2.50) into Eq. (2.37) yields the important result that the total energy of the system is conserved:

(2.51)

or This is true for any system solely governed by gravity.

To demonstrate another constant of the ‐particle system, consider the vector product of Eq. (2.33) with , followed by summing over all particles:

(2.52)

Because , all the terms on the right‐hand side of Eq. (2.52) vanish, resulting in the following:

(2.53)

or

(2.54)

This implies that the ‐particle motion takes place in a constant (or invariant) plane containing the centre of mass. The constant vector is normal to the invariant plane, and is termed the net angular momentum of the system about the origin . This is the law of conservation of angular momentum in the absence of a net external torque about .

The conservation of linear and angular momentum, as well as the total energy of the ‐particle system, is valid for any system ruled only by gravitational forces. The conservation principles are also valid for ‐bodies of arbitrary shapes, as no restrictions have been applied in deriving those principles for the ‐particle system. A body is defined to be a collection of a large number of particles. Thus the particles can be grouped into several bodies, each translating and rotating with respect to a common reference frame. However, solving for the motion variables (linear and angular positions and velocities) of a system of bodies (referred to as the ‐body problem) requires a numerical determination of the individual gravity fields of the bodies, as well as an integration of the first‐order, ordinary differential equations governing their motion. The next section discusses how such differential equations are derived for a body. The solar system is an example of the ‐body system. Numerical approximations and simplifying assumptions are invariably employed in the solution of the ‐body problem. For example, when the separations between the centres of mass of the respective bodies given by , , are always large, the problem is approximated as that of ‐bodies of spherical shape with radially symmetrical mass distributions.