eliptical orbits

Tue Apr 8 07:06:00 EDT 2008

Hi

The shape of an elliptical orbit depends on the energy and angular momentum
of the satellite. 

A mathematical analysis reveals that the ellipses two semi axes are given by

  and    , where  for a bound satellite.

The special case of a circular orbit dictates certain requirements on  and
and this is not as easy to obtain as an elliptical orbit. A demonstration is
possible with a 'conical pendulum' made to orbit under a central force. 

Keith Irvine

AIC, Auckland

  _____  

From: phys-teach-talk-bounces at nzip.org.nz
[mailto:phys-teach-talk-bounces at nzip.org.nz] On Behalf Of Phil Butler
Sent: Tuesday, April 08, 2008 9:43 AM
To: Rob Campbell; Terry Moffat; phys-teach-talk at nzip.org.nz
Subject: RE: eliptical orbits

Hi all

One of the nicest hand waving explanations is Newton's own figure which I
have attached. I seem to recall that this email list doesn't take diagrams
so it may be stripped off :-( 

The essence is to imagine standing on a tall hill (or the Auckland Sky
Tower) and throwing a ball harder and harder horizontally.

Ignore air resistance.

The first ball might hit the ground (earth) 100m away. 

The second ball might hit the ground 100km away. 

The third ball might hit the ground 1000km away, well over the horizon and
where the curvature of the earth is important. 

The fourth ball, if you throw it with exactly the right velocity, will just
miss the earth all round, and come and hit you on the back of the head. It
will have gone in a circular orbit.

The fifth ball, with an even harder throw, goes in elliptical orbit, with
you as the closest point.

Note finally that if the earth was a small diameter, or if your tower or
mountain was very high, that balls 1, 2 and 3, would travel in elliptical
orbits with you at their furthest point.

Phil

-----Original Message-----
From: phys-teach-talk-bounces at nzip.org.nz
[mailto:phys-teach-talk-bounces at nzip.org.nz] On Behalf Of Rob Campbell
Sent: Monday, 7 April 2008 10:34 p.m.
To: 'Terry Moffat'; phys-teach-talk at nzip.org.nz
Subject: RE: eliptical orbits

Hi!

Newton's explanation remains the simplest one that fully explains this, but

I won't attempt maths in a text-only email!

So, here's the hand-waving version.

Earth satellites like the Moon or the ISS 'fall' toward Earth.  They never

arrive because there's a sideways component to their velocity.  If the ISS

really does move toward Earth, it loses PE and gains sideways KE.  The

sideways motion tends to draw it tangentially away from Earth - which

doesn't prevent it coming closer (it does.)

As it approaches Earth, tangential velocity builds up, slowing then

reversing the Space Station's fall.  It starts to rise, gaining PE as it

does so, and losing KE.  At its furthest point from Earth, its orbital

velocity is lowest; at the nearest point, its orbital velocity is highest.

This generates an ellipse, though at 10pm I can't illustrate this in words!

A moment's thought will show that an ellipse is reasonable; 1/2 hour's

thought with pen and paper will prove it.

There's no difference in principle between a satellite orbiting Earth and a

planet orbiting the Sun - just one of scale.  Planets tend to follow

elliptical orbits approximating to circles, maybe on account of their fiery

origins.

An ellipse is a squashed circle (apologies, mathematicians) with a major and

minor axis rather than a constant radius.  Really, a circle is the special

case ellipse where the two axes are equal.

Hope that helps,

Rob Campbell

-----Original Message-----

From: phys-teach-talk-bounces at nzip.org.nz

[mailto:phys-teach-talk-bounces at nzip.org.nz] On Behalf Of Terry Moffat

Sent: Monday, 7 April 2008 8:22 a.m.

To: phys-teach-talk at nzip.org.nz

Subject: eliptical orbits

Hi,

can someone give me a simple reason why planets move in eliptical orbits.

does it have something to do with the fact that the system is moving through

space?

ta  Terry

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