How to calculate planet orbits?

Can anyone direct me to a website which explains how one might
calculate the orbits of the planets from observations in order
to confirm that orbits are in fact ellipses rather than some
other proposed shape? A general description would be more
useful than the detailed mathematics, but both would be good.
If no-one has a website, could you just give me an outline of
what would be involved? I'm more interested in an easy way to
do it with modern techniques than in how Kepler did it.

-- Jeff, in Minneapolis

Subtract 1 from my e-mail address above for my real address.
..

"Jeff Root" wrote in message
om...
Can anyone direct me to a website which explains how one might
calculate the orbits of the planets from observations in order
to confirm that orbits are in fact ellipses rather than some
other proposed shape? A general description would be more
useful than the detailed mathematics, but both would be good.
If no-one has a website, could you just give me an outline of
what would be involved? I'm more interested in an easy way to
do it with modern techniques than in how Kepler did it.

-- Jeff, in Minneapolis

How about starting with the Earth's orbit?

We can see, by the progression of the constellations
in the night sky through the year, that the Earth
goes around the Sun (or vice versa, but let's not go
down that route... assume we know that the Earth
goes around the Sun).

Make precise measurements of the size of the Sun's
disk throughout the year and use the information
to show that the orbit is elliptical -- we can
derive the eccentricity and relative dimensions of
the orbit from the disk size information. In
particular, the data should fit an equation of the
form:

r = p/(1 + e*cos(q))

where r is the radius
p is a scale parameter
e is the eccentricity
q is the true anomaly

Greg Neill replied to Jeff Root:

How about starting with the Earth's orbit?

Exactly where I wanted to start!

We can see, by the progression of the constellations
in the night sky through the year, that the Earth
goes around the Sun (or vice versa, but let's not go
down that route... assume we know that the Earth
goes around the Sun).

Make precise measurements of the size of the Sun's
disk throughout the year and use the information
to show that the orbit is elliptical --

I thought of that. I don't know but can figure out on my own
what the change in size should be, to determine how difficult
it would be to measure. I wonder whether the change would be
enough to produce useable values, given no access to the McMath
Solar Telescope, or the like.

we can derive the eccentricity and relative dimensions of
the orbit from the disk size information. In particular,
the data should fit an equation of the form:

r = p/(1 + e*cos(q))

where r is the radius
p is a scale parameter
e is the eccentricity
q is the true anomaly

I wildly surmise that p would be the same for all observations
if the setup used to make the measurements is unchanging, but I
can't guess whether it would then become 1 or if the value needs
to somehow be determined separately.

The date of perihelion and length of the year would be easy
enough to determine, but there are still unknowns on both sides
of the equation: radius and eccentricity.

I don't know what to do with the equation.

And I need to go beyond Earth to determine the ellipticality
of other planets' orbits, too.

-- Jeff, in Minneapolis

..

"Jeff Root" wrote in message
m...
Greg Neill replied to Jeff Root:


Make precise measurements of the size of the Sun's
disk throughout the year and use the information
to show that the orbit is elliptical --

I thought of that. I don't know but can figure out on my own
what the change in size should be, to determine how difficult
it would be to measure. I wonder whether the change would be
enough to produce useable values, given no access to the McMath
Solar Telescope, or the like.

we can derive the eccentricity and relative dimensions of
the orbit from the disk size information. In particular,
the data should fit an equation of the form:

r = p/(1 + e*cos(q))

where r is the radius
p is a scale parameter
e is the eccentricity
q is the true anomaly

Okay. At perihelion q = 0, and at aphelion q = 180 degrees.

So the perihelion and aphelion orbital radii a

rp = p/(1 + e) = a(1 - e) a is the semimajor axis

ra = p/(1 - e) = a(1 + e)

If the diameter of the Sun is D and the distance is r then
the angle subtended by the sun is (for small angles):

w = D/r

The ratio angles, perihelion to aphelion, is

R = (D/rp)/(D/ra) = ra/rp = (1 + e)/(1 - e)

For the Earth the eccentricity is about 0.0167 . So the
ratio becomes:

R = 1.034

That's about a 3.4% difference in size, which should be
easily measurable on a projection of the Sun's image,
which can be a couple of feet in diameter if desired.

So working in reverse, from the measured ratio we
can find the eccentricity.


I wildly surmise that p would be the same for all observations
if the setup used to make the measurements is unchanging, but I
can't guess whether it would then become 1 or if the value needs
to somehow be determined separately.

Note that p dropped out of the equation for the eccentricity
determination.


The date of perihelion and length of the year would be easy
enough to determine, but there are still unknowns on both sides
of the equation: radius and eccentricity.

I don't know what to do with the equation.

And I need to go beyond Earth to determine the ellipticality
of other planets' orbits, too.


Once you've got the Earth's orbital shape pinned down
(eccentricity and orientation), use its measure as the
unit (i.e. astronomical unit AU) and determine the other
orbits in terms of this scale. The value of p in the
first equation is related to the eccentricity and
semimajor axis by

p = a(1 - e^2)

So if a is set at 1AU, p for the Earth follows, and you've
got an equation for the Earth's orbit: radius vs true
anomaly. The true anomaly can be determined for a given
moment by observing the constellation locations.

Now you need to take all your observations of the planets,
keeping in mind that you're working from a moving platform,
the Earth. But you can now 'reduce' your observations
thanks to knowing how your platform is moving. It becomes
an exercise in geometry. It might be easiest to employ a
modern orbit-fitting procedure like Herget's method for
optical observations.

Jeff Root wrote:
Greg Neill replied to Jeff Root:

How about starting with the Earth's orbit?

Exactly where I wanted to start!

We can see, by the progression of the constellations
in the night sky through the year, that the Earth
goes around the Sun (or vice versa, but let's not go
down that route... assume we know that the Earth
goes around the Sun).

Make precise measurements of the size of the Sun's
disk throughout the year and use the information
to show that the orbit is elliptical --

I thought of that. I don't know but can figure out on my own
what the change in size should be, to determine how difficult
it would be to measure. I wonder whether the change would be
enough to produce useable values, given no access to the McMath
Solar Telescope, or the like.

It's easy to see that the orbit cannot be a circle centered in the
Sun. This was known to Jesuit astronomers using pinhole projection
(on a really large and precise scale) within the generation after
Galileo. In fact, they were able to show that the results fit
Kepler's equal-area law rather than the sort of motion that the
Ptolemaic system would induce with uniform but off-center
circular motion. (See Heilbron's book _The Sun in the Church_).

I still giggle at these poor scholastics deciding who got to tell
the Pope.

Bill Keel

On 27 Oct 2003 06:28:12 -0800, (Jeff Root) wrote:

Can anyone direct me to a website which explains how one might
calculate the orbits of the planets from observations in order
to confirm that orbits are in fact ellipses rather than some
other proposed shape? A general description would be more
useful than the detailed mathematics, but both would be good.
If no-one has a website, could you just give me an outline of
what would be involved? I'm more interested in an easy way to
do it with modern techniques than in how Kepler did it.

Well, the method of Laplace is available online:

http://www.tamuk.edu/math/scott/stars/docs/laplace.pdf

and/or:

http://www.scottkurowski.com/mcad/laplaceorbits.html

As for a general description, your local library is probably a better
resource.

Basically, modern methods of orbit determination work by presuming the
orbit to be a conic section, as this can be derived directly from
Newtonian physics. If you want to prove this to be true by looking at
actual orbits, you have to go back to earlier methods, such as that of
Kepler...

Al Moore