Space mathematics

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The purpose of this article is to introduce non-scientists, non-engineers, and other laymen to the basic mathematics of space exploration. The math presented here is simple arithmetic and algebra, easily comprehended by the average high school student or non-technically trained adult. It is hoped that this material will help to stimulate further interest in the rapidly expanding world of space exploration, astronautics, and astrodynamics.

Contents

[edit] Newton's laws of motion

All rockets and space vehicles operate according to three laws of motion enunciated by Sir Isaac Newton in the late 17th century. These three rules are stated here:

  • An object at rest or in motion remains so unless acted on by a force. The motion is in a straight line.
  • Applying a force to any object gives it an acceleration (or deceleration) proportional to its mass.
  • For every action there is an equal and opposite reaction.

The first law tells us that a spaceship on course for the Moon, say, will continue coasting in a more-or-less straight line until the braking rockets are fired. Of course, the gravity fields of Earth and Moon also affect its motion, but for short distances, the motion appears perfectly straight.

The second law is most easily remembered by memorizing the simple equation, F = ma. This says that an acceleration applied to a mass requires a force. It also says that a force F applied to an object with mass m gives that object an acceleration a.

F = m a.
Force equals mass times acceleration.

Sir Isaac's third law explains the operation of rockets on the launchpad and in space. The "action" spoken of is the acceleration of hot exhaust gases from the nozzle of the rocket engine, while the "reaction" is the acceleration of the launch vehicle or spacecraft in the opposite direction.

[edit] Weight, mass, and acceleration

All physical objects in the known Universe have a property called mass, which is distinct from weight in physics jargon usage. The mass of an object is just the amount of matter it contains. Mass is given in kilograms or slugs or pounds, to distinguish it from weight, which is the force of gravity, given in metric newtons or pounds-force. (Stating that you have a weight of so many kilograms is a use of the word weight with a different meaning.) The mass of any object is the same everywhere in the Universe, whereas the force that object exerts due to gravity will change according to the local acceleration of gravity. In space, of course, everything is weightless, but still has the same mass.

There is a useful equation, easily memorized, which states that an object's weight on any planet or moon equals its mass times the local acceleration of gravity. It is written

W = mg
Weight equals mass times gravitational acceleration.

and is actually the equivalent of Newton's second law, F = ma, since this weight is a force, and gravity is expressed as an acceleration

[edit] See also

[edit] Orbits

[edit] Kepler's laws of planetary motion

  • All the planets orbit on elliptical paths, with the Sun at one focus.
  • A line drawn from the Sun to a planet sweeps out equal areas in equal times.
  • The square of a planet's period is proportional to the cube of its average distance from the Sun.

[edit] 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.

[edit] Escape trajectories

Once the circular orbital velocity is known, the escape trajectory velocity is easily found by multiplying by the square root of 2:

\ v = \sqrt 2\sqrt{\frac {GM} {r}\ } = \sqrt{\frac {2GM} {r}\ }.

[edit] References

  • Myrl H. Ahrendt. The Mathematics of Space Exploration. Holt, Rinehart, and Winston, Inc., New York, 1965.
  • Maxwell W. Hunter, II. Thrust into Space. Holt, Rinehart, and Winston, Inc., New York, 1968.
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