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To create a program to explore the mechanics and mathematics behind earth
Background & Techniques
|A satellite with initial altitude of 1000
miles and 15000 mph
velocity will circle the earth in 100 minutes at altitudes from 1000 down to
100 miles above surface
This is a first adventure into Celestial Mechanics to convince myself that
satellites can really orbit the earth and, by extension, planets orbit the sun.
I previously thought that if an orbiting satellite dropped just a little closer to the earth
gravitational forces would increase and force it into a death spiral..
Web descriptions talk about centripetal force as if that were the another
force which held the answer. But trying to incorporate centripetal force into the explanation didn't help much
because it turns out that gravity IS the centripetal force that keeps the satellite from
flying out into space. So if there is only one major natural force acting on the
satellite to drag it down, what holds it up? The answer lies in Newton's
of motion - things like to keep moving in a straight line. So if we didn't have
gravity (and air drag) we could toss a baseball and hit the
moon (or miss it on the way by). The secret to orbiting is forward velocity,
speed at right angles to the radial force of gravity. The speed and direction of
our velocity vector is the result of the satellite falling straight down
toward the center of the earth and our tangential velocity which is
trying to keep the satellite moving in a straight line.
This horizontal component likely came from the rocket that put us into orbit
initially but in this simulation it is like a cannon firing from a really tall
tower . If the forward (tangential) speed is high enough, we'll miss the earth
as we fall toward it and continue on past at which point gravity works to turn
us around and drag us back. So if we don't give the satellite a big enough
initial push, it will fall to earth somewhere before it get back around to the
start point. If we give it too big a push, it will fly off into space
never to return. The speed at which this happens is called, naturally
enough, the "escape velocity" and is a function of altitude. In
theory as initial velocities increase orbits start
when the satellite just clears the earth on the far side and as starting velocities increase, become
circular and then increasingly
eccentric (longer and skinnier) until escape velocity is reached..
Escape velocity is approximately 1.4 ( square root of 2 ) times the circular orbit
The math can be kind of intimidating but the technique I adopted is called Euler
"symplectic" which has the advantage on conserving energy and therefore keeping the the orbit stable. The method loops, moving the satellite in
small steps for each chosen time step. Each new position is calculated
in two steps.. First we'll calculate a new velocity vector based on where
we were after the last step at in which direction we were moving.
We'll then use that new velocity vector to calculate a new position. The most
helpful reference I found
is a PDF titled "Orbits" available from
http://www.mathworks.com/moler/exm/chapters/orbits.pdf The equations
involve vectors which are implemented here in the Cartesian (x,y) domain since
we'll need x and ,y coordinates anyway for plotting.
Here are some program notes:
- The satellite's altitude above earth and its initial velocity and
direction define the each case. Satellite masses are assumed to be much
smaller than the earth's mass so that the "two-body" effect can be ignored.
That is to say, the orbiting satellite has no measurable effect on the
- The display is 2 dimensional and arbitrarily starts with the satellite
displayed above the earth. The angle of the initial velocity is specified
relative to the tangent to the earth and increase the way as trig functions;
counterclockwise. (This is opposite from navigational bearing
measurements which increase clockwise.
- Positive initial velocities are assumed to be directed in the
counterclockwise direction assuming the we are hovering over the North Pole
looking down . This is the same direction that the earth rotates which
makes it the preferred direction for real satellites to take advantage of a free
boost and makes geosynchronous orbits possible.
- Gravity is assumed to be from a point at the center of the earth and
doesn't know about the earth size. "Just for fun," the program has a "porous
earth" option which allows slower satellites to penetrate the earth's
surface and complete their orbit about the earth's center..
- We try to draw orbits to scale relative to the earth image. For large
orbits, this causes the earth image to disappear.
- English or metric unit may be specified for input and output values.
- There are two sampling rates specified: The "calculate time interval"
is the time increment used in the loop which is moving the satellite.
The time between plot updates is specified as "number of calculated
points per plot update". For large periods it is best to update plots
once every 100 or more calculated points. Small calculation time intervals
(e.g.. 1 second) improve the accuracy of the orbit, but sampling at 1
degree intervals seems adequate (divide the displayed orbit
period seconds by 360 to get a reasonable estimate of the value to use).
- It is possible to get a animation running very slowly (when small
calc-time interval and small number of points per plot is set for a large
orbit. Just click the Animate check box to deselect it and the
simulation should finish within a few seconds.
- Several predefined cases are included to get you started!
Non-programmers are welcome to read on, but may want to jump to bottom of
this page to download the executable program now.
Coming soon. (How often have you seen this on websites? ) I'm
tired this week, write with your questions.
Running/Exploring the Program
Suggestions for Further Explorations
||Simulate earth based rocket injection of
satellite into a target orbit.
|Original: December 18,2011
February 18, 2016