Orbital mechanics Totally Explained

Everything about Orbital Mechanics totally explained

Orbital mechanics or astrodynamics is the study of the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation, collectively known as classical mechanics. Celestial mechanics focuses more broadly on the orbital motions of artificial and natural astronomical bodies such as planets, moons, and comets. Orbital mechanics is a subfield which focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsion. General relativity provides more exact equations for calculating orbits, sometimes necessary for greater accuracy or high-gravity situations (such as orbits close to the Sun).

Rules of thumb

The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics. The specific example discussed is of a satellite orbiting a planet, but the rules of thumb could also apply to other situations, such as orbits of small bodies around a star such as the Sun.

Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold in the absence of thrust:
- Orbits are either circular, with the planet at the center of the circle, or form an ellipse, with the planet at one focus.
- A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.
- The square of a satellite's orbital period is proportional to the cube of its average distance from the planet.
Without firing a rocket engine (generating thrust), the height and shape of the satellite's orbit won't change, and it'll maintain the same orientation with respect to the fixed stars.
A satellite in a low orbit (or low part of an elliptical orbit) moves more quickly with respect to the surface of the planet than a satellite in a higher orbit (or a high part of an elliptical orbit), due to the stronger gravitational attraction closer to the planet.
If a brief rocket firing is made at only one point in the satellite's orbit, it'll return to that same point on each subsequent orbit, though the rest of its path will change. Thus to move from one circular orbit to another, at least two brief firings are needed.
From a circular orbit, a brief firing of a rocket in the direction which slows the satellite down, will create an elliptical orbit with a lower perigee (lowest orbital point) at 180 degrees away from the firing point, which will be the apogee (highest orbital point). If the rocket is fired to speed the rocket, it'll create an elliptical orbit with a higher apogee 180 degrees away from the firing point (which will become the perigee).

The consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the same circular orbit and wish to dock, unless they're very close, the trailing craft can't simply fire its engines to go faster. This will change the shape of its orbit, causing it to gain altitude and miss its target. One approach is to actually fire a reverse thrust to slow down, and then fire again to re-circularize the orbit at a lower altitude. Because lower orbits are faster than higher orbits, the trailing craft will begin to catch up. A third firing at the right time will put the trailing craft in an elliptical orbit which will intersect the path of the leading craft, approaching from below.
To the degree that the assumptions don't hold, actual trajectories will vary from those calculated. Atmospheric drag is one major complicating factor for objects in Earth orbit. The differences between classical mechanics and general relativity can become important for large objects like planets. These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system.

Laws of astrodynamics

The fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is his differential calculus. Standard assumptions in astrodynamics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.). More accurate calculations can be made without these simplifying assumptions, but they're more complicated. The increased accuracy often doesn't make enough of a difference in the calculation to be worthwhile. Kepler's laws of planetary motion may be derived from Newton's laws, when it's assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's laws still apply, but Kepler's laws are invalidated. When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws.

Escape velocity

The formula for escape velocity is easily derived as follows. The specific energy (energy per unit mass) of any space vehicle is composed of two components, the specific potential energy and the specific kinetic energy. The specific potential energy associated with a planet of mass M is given by

- G M / r ,

while the specific kinetic energy of an object is given by

v^2/2 ,

Since energy is conserved, the total specific orbital energy

v^2/2 - G M / r ,

does not depend on the distance,

r

, from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite

r

only if this quantity is nonnegative, which implies

vgeqsqrt

where

a_p

is the semimajor axis of the planet's orbit relative to the Sun;

m_p

and

m_s

are the masses of the planet and Sun, respectively.
This simplification is sufficient to compute rough estimates of fuel requirements, and rough time-of-flight estimates, but it isn't generally accurate enough to guide a spacecraft to its destination. For that, numerical methods are required.

The universal variable formulation

To address the shortcomings of the traditional approaches, the universal variable approach was developed. It works equally well on circular, elliptical, parabolic, and hyperbolic orbits; and also works well with perturbation theory. The differential equations converge nicely when integrated for any orbit.

Perturbations

The universal variable formulation works well with the variation of parameters technique, except now, instead of the six Keplerian orbital elements, we use a different set of orbital elements: namely, the satellite's initial position and velocity vectors

x_0

and

v_0

at a given epoch

t = 0

. In a two-body simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation. Conversely, at any moment in the satellite's orbit, we can measure its position and velocity, and then use the universal variable approach to determine what its initial position and velocity would have been at the epoch. In perfect two-body motion, these orbital elements would be invariant (just like the Keplerian elements would be).
However, perturbations cause the orbital elements to change over time. Hence, we write the position element as

x_0(t)

and the velocity element as

v_0(t)

, indicating that they vary with time. The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions

x_0(t)

and

v_0(t)

Non-ideal orbits

The following are some effects which make real orbits differ from the simple models based on a spherical earth. Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they're small relative to the corresponding two-body effects.

Equatorial bulges cause precession of the node and the perigee

Tesseral harmonics (External Link

) of the gravity field introduce additional perturbations

lunar and solar gravity perturbations alter the orbits

Atmospheric drag reduces the semi-major axis unless make-up thrust is used Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behaviour can become chaotic. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such as station-keeping, ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude.

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