Saturday, 4 June 2011

Mekanika Orbit: Hukum-Hukum Astro Dinamika

"Hukum-Hukum Astro Dinamika Akan Menjadi Dasar Bagi Pengembangan IPTEK Antariksa Masa Depan Umat Manusia" 
*Arip Nurahma*

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 are more complicated. The increased accuracy often does not 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 is 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. The three laws are:
  1. The orbit of every planet is an ellipse with the sun at one of the foci.
  2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time.
  3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.

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 \,

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

v\geq\sqrt{2 G M / r}
The escape velocity from the Earth's surface is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the solar system from the vicinity of the Earth requires around 42 km/s velocity, but there will be "part credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.

Formulae for free orbits

Orbits are conic sections, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:
r = {a \over (1 + e \cos \theta) }.
The parameters are given by the orbital elements.

Circular orbits

Although most orbits are elliptical in nature, a special case is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit at distance r from the center of gravity of mass M is
\ v = \sqrt{\frac{GM} {r}\
where G is the gravitational constant, equal to
6.672 598 × 10−11 m3/(kg·s2)
To properly use this formula, the units must be consistent; for example, M must be in kilograms, and r must be in meters. The answer will be in meters per second.
The quantity GM is often termed the standard gravitational parameter, which has a different value for every planet or moon in the solar system.
Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by the square root of 2:
\ v = 
\sqrt 2\sqrt{\frac {GM} {r}\ } = \sqrt{\frac {2GM} {r}\ }.
Sumber: Wikipedia

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