Newtons Law of Motion and Universal Gravitation
At some point during the lifetime of most space vehicles or satellites, we must change one or more of the orbital elements. For example, we may
need to transfer from an initial parking orbit to the final mission orbit, rendezvous with or intercept another spacecraft, or correct the orbital
elements to adjust for the perturbations discussed in the previous section. Most frequently, we must change the orbit altitude, plane, or both. To
change the orbit of a space vehicle, we have to change its velocity vector in magnitude or direction. Most propulsion systems operate for only a short
time compared to the orbital period, thus we can treat the maneuver as an impulsive change in velocity while the position remains fixed. For this
reason, any mane
Newtons Laws of Motion describe a particle and forces acting on it.
Here are the laws…
* 1st Law- With no forces acting a object rest will remain at rest and a object in motion will remain in motion in a straight line.
* 2nd Law- If a force is applied there will be a change in velocity proportional to the magnitude and in the direction of the force applied. This can
be represented as the following equation: F=ma, where F is force, m is mass, and a is the acceleration.
* 3rd Law- Commonly stated; For every action there is an equal and opposite reaction.
Law of Universal Gravitation…
* Two particles having masses represented here as m1 and m2 and separated by distance of r are attracted to each other by forces equal and opposite
directed along the line joining the particles. This is directly analogous to the Sun and Earth in orbit.
The following expresses the action in terms of equation:
Courtesy of braeunig.us
G is the universal constant of gravitation and has the value of 6.67259x10-11 N-m2/kg2 (3.4389x10-8 lb-ft2/slug2).
3. Celestial Motions and The Launch of A Space Vehicle
Keplers Laws of Planetary Motion
* All planets move around the Sun in elliptical orbits with the Sun at one focus.
Let's now look at the force that the Earth exerts on an object. If the object has a mass m, and the Earth has mass M, and the object's distance
from the center of the Earth is r, then the force that the Earth exerts on the object is GmM /r2 . If we drop the object, the Earth's gravity will
cause it to accelerate toward the center of the Earth. By Newton's second law (F = ma), this acceleration g must equal (GmM /r2)/m.
At the surface of the Earth this acceleration has the valve 9.80665 m/s2 (32.174 ft/s2).
Many of the upcoming computations will be somewhat simplified if we express the product GM as a constant, which for Earth has the value 3.986005x1014
m3/s2 (1.408x1016 ft3/s2). The product GM is often represented by the Greek letter .
* A line joining any planet to the Sun sweeps out in equal areas at equal times.
* The square of a period of any planet about the Sun is proportional to the cube of the planets mean distance from the Sun.
These laws can be deduced from Newton's laws of motion and law of universal gravitation. Indeed, Newton used Kepler's work as basic information
in the formulation of his gravitational theory.
As Kepler pointed out, all planets move in elliptical orbits, however, we can learn much about planetary motion by considering the special case of
circular orbits. 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
center of mass of this system of two bodies lies along the line joining them at a point C such that mr = MR. 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 2r must equal M 2R. 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.
If one body has much greater mass than the other (Sun to Earth) its distance to the center of mass is shorter than that of the other body. Think of
‘condensation’ of mass.
Launch of A Space Vehicle
In order to launch a spacecraft into orbit you must have a period of powered flight to accelerate the craft out of the atmosphere and achieve orbital
velocity. Powered flight concludes at a burnout (with current chemical based rocket technology) at which time the craft begins free flight and is
subject only to gravitational forces of the primary. If the craft moves further out it will eventually be subject to the gravitational forces of the
Moon, Sun, or other body.
Courtesy of braeunig.us
4. Orbital Perturbations, Maneuvers, and Escape Velocity
A space vehicle's orbit may be determined from the position and the velocity of the vehicle at the beginning of its free flight.
Perturbations, or forces acting on a craft that corrupt a crafts orbit are a issue that must be dealt with by any craft going into orbit.
Third-body perturbations are from the gravitational forces of the Sun and Moon.
Perturbations due to non-spherical Earth are caused by the fact that the Earth, or any body in space is not a perfect sphere.
Perturbations from Atmospheric Drag are caused by the drag forces when moving through a planets atmosphere. In Low Earth Orbit a craft
is still subjected to the drag of the thin Earth atmosphere at that high altitude, this can cause a crafts orbit to decay and spiral back towards the
primary (of course only if an atmosphere exists). If a craft comes within 160-120 km of the Earths surface it will come crashing down within a few
days. Final disintegration will happen at about 80 kilometers above Earths surface. Above 600 km the drag is so weak that orbits for satellites
usually last for more than ten years, usually long past their lifetime.
The region above 90 km is the Earth's thermosphere where the absorption of extreme ultraviolet radiation from the Sun results in a very rapid
increase in temperature with altitude. At approximately 200-250 km this temperature approaches a limiting value, the average value of which ranges
between about 600 and 1,200 K over a typical solar cycle. Solar activity also has a significant affect on atmospheric density, with high solar
activity resulting in high density. Below about 150 km the density is not strongly affected by solar activity; however, at satellite altitudes in the
range of 500 to 800 km, the density variations between solar maximum and solar minimum are approximately two orders of magnitude. The large variations
imply that satellites will decay more rapidly during periods of solar maxima and much more slowly during solar minima.
Perturbations from solar radiation are caused by periodic solar events.
[edit on 5/5/2009 by jkrog08]