Sun calculates to be less massive for planets which are further out - sun mass anomaly

"tadchem" wrote in message
oups.com...

The problem is that the 'circumference' of an ellipse can only be
evaluated with an elliptic integral - a calculation that required
laborious number-crunching in the pre-digital era.
http://en.wikipedia.org/wiki/Ellipse
http://astronomy.swin.edu.au/~pbourk...y/ellipsecirc/

True, but there exists approximation formulae that are
more than accurate enough for all practical purposes.
Take Ramanujan's formulas for instance -- His second
approximation formula is exquisitely accurate, even
for ellipses of fairly high eccentricities.

Greg Neill a écrit :
"srp" wrote in message ...

a écrit :


Therefore, I suspect that the values given in the reference were cooked
or calculated so that they would have the same value down to the 8th
digit for a constant mass of the sun. This means the values for the
orbital period and velocity were calculated based starting on a
constant mass of the sun using Kepler's third law. This is the only way
I can think of that the mass could come up so close for something that
cannot be very accurately measured.

You are absolutely right. This was and still is the only way possible
to get any figure for Solar system masses, including Earth's mass.
They all can only be best fit approximate estimates.


No, since we send small spacecraft to other planets they act
as test masses that can directly probe the gravitational pull
of the planet. Same with Earth satellites.

This implies that the total mass of a body has any influence on the
trajectory, which seems to me not to be the case. A body of any
mass whatsoever can be put on Earth's orbit about the Sun.

Who calculated the various planetary masses and the Sun's mass
initially? What method did he use, and how was the precision of
calculations confirmed with respect to physical reality?

André Michaud

"srp" wrote in message ...
Greg Neill a écrit :
"srp" wrote in message ...

a écrit :


Therefore, I suspect that the values given in the reference were cooked
or calculated so that they would have the same value down to the 8th
digit for a constant mass of the sun. This means the values for the
orbital period and velocity were calculated based starting on a
constant mass of the sun using Kepler's third law. This is the only way
I can think of that the mass could come up so close for something that
cannot be very accurately measured.

You are absolutely right. This was and still is the only way possible
to get any figure for Solar system masses, including Earth's mass.
They all can only be best fit approximate estimates.


No, since we send small spacecraft to other planets they act
as test masses that can directly probe the gravitational pull
of the planet. Same with Earth satellites.

This implies that the total mass of a body has any influence on the
trajectory, which seems to me not to be the case. A body of any
mass whatsoever can be put on Earth's orbit about the Sun.

Actually, both masses *do* influence the trajectory. If you
look at the differential equation for the trajectory, you
will find a term that contains the sum of the two masses.
It's really only a problem when the trajectory is being
calculated in a frame of reference that's not coincident with
the center of mass of the larger body. So, for example, if you
wanted to describe the trajectories of two equally massive
bodies co-orbiting, you would want to use the sum of the two
masses as the gravitational parameter for the system (usually
designated with the greek letter mu).

When one mass is very much smaller than the other, its mass can
usually be ignored.


Who calculated the various planetary masses and the Sun's mass
initially? What method did he use, and how was the precision of
calculations confirmed with respect to physical reality?

The ratios of the masses were known via Kepler's laws, so I
suppose that you could say that Kepler first had a handle on
them. Newton derived Kepler's laws from his Theory of
Gravitation in a form that took into account the sum of the
masses involved.

Greg Neill a écrit :
"srp" wrote in message ...

Greg Neill a écrit :

"srp" wrote in message ...


a écrit :


Therefore, I suspect that the values given in the reference were cooked
or calculated so that they would have the same value down to the 8th
digit for a constant mass of the sun. This means the values for the
orbital period and velocity were calculated based starting on a
constant mass of the sun using Kepler's third law. This is the only way
I can think of that the mass could come up so close for something that
cannot be very accurately measured.

You are absolutely right. This was and still is the only way possible
to get any figure for Solar system masses, including Earth's mass.
They all can only be best fit approximate estimates.


No, since we send small spacecraft to other planets they act
as test masses that can directly probe the gravitational pull
of the planet. Same with Earth satellites.

This implies that the total mass of a body has any influence on the
trajectory, which seems to me not to be the case. A body of any
mass whatsoever can be put on Earth's orbit about the Sun.


Actually, both masses *do* influence the trajectory. If you
look at the differential equation for the trajectory, you
will find a term that contains the sum of the two masses.
It's really only a problem when the trajectory is being
calculated in a frame of reference that's not coincident with
the center of mass of the larger body. So, for example, if you
wanted to describe the trajectories of two equally massive
bodies co-orbiting, you would want to use the sum of the two
masses as the gravitational parameter for the system (usually
designated with the greek letter mu).

Hmm yes. But the masses are useful to determine how much energy
is required to set them into the specific closed orbits we want
to set them onto. But once on closed gravitationl orbits, whatever
the masses involved, they then are on the closed orbit. And
it seems to me that the reverse process is not as obvious to
establish. Knowing the parameters of an orbit gives no clue
as to the magnitude of the masses, it seems to me.

With Kepler's third law, we don't even need the mass of the Sun
to calculate the Solar system's planetary orbits.

If you remove that mass from the divisor in G, and also remove
it from the Force equation, you still get the Earth orbit if
you put the mass of the earth in the force equation.

If you then reduce the mass of the Earth to the mass of an
electron, you still get Earth's orbit. Seems pretty circular
to me, no pun intended.

What of the planets, that were set on their closed orbits before
we were here? And the Sun's mass?

What is the bottom rung of the solar masses ladder?

When one mass is very much smaller than the other, its mass can
usually be ignored.


Who calculated the various planetary masses and the Sun's mass
initially? What method did he use, and how was the precision of
calculations confirmed with respect to physical reality?

The ratios of the masses were known via Kepler's laws, so I
suppose that you could say that Kepler first had a handle on
them.

Then my question becomes how was that ratio established ?

Newton derived Kepler's laws from his Theory of
Gravitation in a form that took into account the sum of the
masses involved.

Presently, it seems to me that he built his theory of gravitation
from Kepler's third law.

I really would like to know how the ratio was established,
and the Sun's mass set.

André Michaud

"srp" wrote in message ...
Greg Neill a écrit :

Actually, both masses *do* influence the trajectory. If you
look at the differential equation for the trajectory, you
will find a term that contains the sum of the two masses.
It's really only a problem when the trajectory is being
calculated in a frame of reference that's not coincident with
the center of mass of the larger body. So, for example, if you
wanted to describe the trajectories of two equally massive
bodies co-orbiting, you would want to use the sum of the two
masses as the gravitational parameter for the system (usually
designated with the greek letter mu).

Hmm yes. But the masses are useful to determine how much energy
is required to set them into the specific closed orbits we want
to set them onto. But once on closed gravitationl orbits, whatever
the masses involved, they then are on the closed orbit. And
it seems to me that the reverse process is not as obvious to
establish. Knowing the parameters of an orbit gives no clue
as to the magnitude of the masses, it seems to me.

In the simplest case, a spacecraft of negligible mass (relatively
speaking) takes up a circular orbit about a planet. The radius
of the orbit is determined by some means (say by radar observations
or laser altimetry), and the period by stellar observations.
Then, the mass of the planet can be accurately determined via
Newton's version of Kepler's third:

T^2 = 4*pi^2 * a^3 / [G*M]

where T = obital period
a = orbital radius
M = the mass of the planet

If the mass of the spacecraft is *not* negligible w.r.t. the
body being orbited, then replace M with M + m.

With Kepler's third law, we don't even need the mass of the Sun
to calculate the Solar system's planetary orbits.

True, but then you're relying on ratios. If you use Newton's
version (above), the masses are explicit.


If you remove that mass from the divisor in G, and also remove
it from the Force equation, you still get the Earth orbit if
you put the mass of the earth in the force equation.

If you then reduce the mass of the Earth to the mass of an
electron, you still get Earth's orbit. Seems pretty circular
to me, no pun intended.

What of the planets, that were set on their closed orbits before
we were here? And the Sun's mass?

What is the bottom rung of the solar masses ladder?

When one mass is very much smaller than the other, its mass can
usually be ignored.


Who calculated the various planetary masses and the Sun's mass
initially? What method did he use, and how was the precision of
calculations confirmed with respect to physical reality?

The ratios of the masses were known via Kepler's laws, so I
suppose that you could say that Kepler first had a handle on
them.

Then my question becomes how was that ratio established ?

I mispoke. Kepler gave the relative scale of the orbits, so that
determining the actual size of any one orbit would allow all the
others to be calculated. It was the period and radii of orbits
of the moons of the planets, and later the perturbations of
one planet on the others, that allowed the relative masses to
be determined. Cavendish's torsion balance experiment set the
value for G and allowed the actual masses to be determined.


Newton derived Kepler's laws from his Theory of
Gravitation in a form that took into account the sum of the
masses involved.

Presently, it seems to me that he built his theory of gravitation
from Kepler's third law.

No doubt he was influenced by Kepler's laws, but he starts
his derivation from the inverse square force law and
proceeds to derive Kepler from there.


I really would like to know how the ratio was established,
and the Sun's mass set.

It goes back to Cavendish. Once G is determined the mass
can be determined from Newton's version of Kepler's 3rd.

srp wrote:
Greg Neill a écrit :

"srp" wrote in message
...

Greg Neill a écrit :

"srp" wrote in message
...


a écrit :



Therefore, I suspect that the values given in the reference were
cooked
or calculated so that they would have the same value down to the 8th
digit for a constant mass of the sun. This means the values for the
orbital period and velocity were calculated based starting on a
constant mass of the sun using Kepler's third law. This is the
only way
I can think of that the mass could come up so close for something
that
cannot be very accurately measured.


You are absolutely right. This was and still is the only way possible
to get any figure for Solar system masses, including Earth's mass.
They all can only be best fit approximate estimates.



No, since we send small spacecraft to other planets they act
as test masses that can directly probe the gravitational pull
of the planet. Same with Earth satellites.

True. See below.

This implies that the total mass of a body has any influence on the
trajectory, which seems to me not to be the case. A body of any
mass whatsoever can be put on Earth's orbit about the Sun.



Actually, both masses *do* influence the trajectory. If you
look at the differential equation for the trajectory, you
will find a term that contains the sum of the two masses.
It's really only a problem when the trajectory is being
calculated in a frame of reference that's not coincident with
the center of mass of the larger body. So, for example, if you
wanted to describe the trajectories of two equally massive
bodies co-orbiting, you would want to use the sum of the two
masses as the gravitational parameter for the system (usually
designated with the greek letter mu).

Kinda true. See below.

Hmm yes. But the masses are useful to determine how much energy
is required to set them into the specific closed orbits we want
to set them onto. But once on closed gravitationl orbits, whatever
the masses involved, they then are on the closed orbit. And
it seems to me that the reverse process is not as obvious to
establish. Knowing the parameters of an orbit gives no clue
as to the magnitude of the masses, it seems to me.

With Kepler's third law, we don't even need the mass of the Sun
to calculate the Solar system's planetary orbits.

If you remove that mass from the divisor in G, and also remove
it from the Force equation, you still get the Earth orbit if
you put the mass of the earth in the force equation.

If you then reduce the mass of the Earth to the mass of an
electron, you still get Earth's orbit. Seems pretty circular
to me, no pun intended.

Not quite. The *force* equation F = -GMm r / |r|^3 contains *both*
masses. As F = ma, dividing both sides of the force equation gives
the *acceleration* equation a = -(GM) r / |r|^3. This equation
contains the mass of the *other* body, but not the mass of the body
being accelerated. The product (GM) is all that we can measure
unless we're doing a lab experiment (a la Cavendish) in which we
already know the masses involved.

What of the planets, that were set on their closed orbits before
we were here? And the Sun's mass?

What is the bottom rung of the solar masses ladder?

When one mass is very much smaller than the other, its mass can
usually be ignored.

Yes, as we do with spacecraft.

Who calculated the various planetary masses and the Sun's mass
initially? What method did he use, and how was the precision of
calculations confirmed with respect to physical reality?


The ratios of the masses were known via Kepler's laws, so I
suppose that you could say that Kepler first had a handle on
them.

Well, he *could* have gotten Jupiter's mass, as the four Galilean
satellites were discovered while he was alive. He wouldn't have had
the tools (celestial mechanics and perturbation theory) to get
anything for the other planets. See below.

Then my question becomes how was that ratio established ?

Newton derived Kepler's laws from his Theory of
Gravitation in a form that took into account the sum of the
masses involved.


Presently, it seems to me that he built his theory of gravitation
from Kepler's third law.

I really would like to know how the ratio was established,
and the Sun's mass set.

André Michaud

OK folks. Back to first principles. Two masses, call them m1 and m2.
Two position vectors, call them r1 and r2, with the origin at the center
of mass of the system, so that m1 r1 + m2 r2 = 0.

r1 r2
----+--------------
m1 O m2

This means right away that r1 and r2 must be antiparallel, and that
m1 |r1| = m2 |r2|. I'm using || to denote the magnitude of the vector.
(If usenet were smarter I'd be putting the r's in boldface, as in
textbooks, because they're vectors. But I digress.)

The force on m1, call it F1, is a vector given by
F1 = (G m1 m2) (r2-r1) / |r2-r1|^3.

Likewise,
F2 = (G m1 m2) (r1-r2) / |r1-r2|^3.

As I wrote above, divide by the masses to get the accelerations:

a1 = (G m2) (r2-r1) / |r2-r1|^3, and
a2 = (G m1) (r1-r2) / |r1-r2|^3.

So if we're dealing with an inertial coordinate system whose origin is
fixed at the center of mass, the acceleration indeed does NOT depend on
the mass being accelerated.

BUT now let's look at the *relative* vector r = r2 - r1. We can get
the acceleration of r (call it a) trivially by subtracting the above
equations for a1 and a2:

a = a2 - a1 = G (m1+m2) (r1-r2) / |r1-r2|^3
= - G (m1+m2) r /|r|^3.

This has the same functional form as a1 and a2, but notice that the
equation now contains the sum of the masses. And as this equation
is for the relative vector r, it's equivalent to moving the origin of
the coordinate system from the center of mass to the location of m1.
And this is exactly what is done when we talk about the periods of the
planets. We let m1 be the Sun and m2 be the planet. So Kepler's
third law actually becomes G (m1+m2) = a^3 n^2, where n is the "mean
motion" (angular velocity) in radians per unit time; n = 2 pi / P.

All this is preliminary. I now turn to André's question, "I really
would like to know how the ratio was established, and the Sun's mass
set."

Not an easy task if all the observations are made from the earth! If
the sun and all the planets were 8 times more massive, all the orbits
could be twice as large and we couldn't tell the difference very easily.
(The periods would stay the same, as would the relative directions.)

In order to keep absolute units OUT of the equations of motion, we
instead hold the Sun's mass fixed by defining the Gaussian gravitational
constant k such that the acceleration produced by the Sun is given by
a = -k r / |r|^3
when r is measured in astronomical units and a is measured in AU/day^2.
The value of k, originally determined by Gauss and held fixed
thereafter, is exactly
k = 0.017 202 098 95 AU^3/day^2.
(The value was selected in order to make the semimajor axis of the
Earth's orbit one AU, with a period of one sidereal year. Some of the
numbers Gauss used, particularly for precession, were slightly off, and
we now know that the Earth's semimajor axis is slightly greater than
1 AU. But I digress.)

This value of k *defines* the length of the AU. Or, to put it another
way, it creates a relationship between the "gravitational constant" GM
of the Sun (measured in km^3/sec^2) and the length of the AU (in km).
Given the value for k, if we know the AU we can calculate GM_sun, and
vice versa.

Historically, the only way to get at the length of the AU was through
parallax, using the Earth's diameter as a baseline. Observations of
Mars at close oppositions were used. Observations of transits of Venus
were used, but these weren't as good as they'd hoped owing to the
effects of Venus's atmosphere. Observations of asteroid (433) Eros,
which comes closer to us than either Venus or Mars, produced the most
reliable value this way. (Jay Lieske, a now-retired JPLer who was
instrumental in developing the 1976 theory of precession, did his Ph.D.
dissertation on this very topic.)

But once we started sending spacecraft to other planets, all this
changed. We could get range measurements -- round-trip light time --
to the spacecraft, providing a distance measurement in kilometers.
This determines the length of the AU very accurately, and we can then
back out the mass of the Sun. Actually, the GM of the Sun. And we know
GM to about 10 or 11 significant digits, which is *far* better than we
can measure G itself.

As for the masses of the planets: the historical technique was to use
the periods and orbital radii of their satellites and apply Kepler's
Third Law. This gave us a value for Jupiter's mass (in terms of solar
masses!) which agrees to 6 digits with the currently-accepted value
(determined from flybys by two Pioneers, two Voyagers, Ulysses, Cassini,
and of course from the Galileo orbiter). The results for Uranus and
Neptune turned out to have been corrupted by systematic errors in
measuring the separation of the satellites from the planet. Voyager 2
gave much better masses -- and the new masses eliminate the supposed
discrepancies in old observations of those planets. (This work was
done by Myles Standish, also of JPL.)

Mercury and Venus don't have any natural satellites, and the only way to
determine their masses before the Space Age was to observe their
perturbations on the other planets. This was not an easy task.
Nevertheless, the values in the old _Explanatory Supplement to the
Astronomical Ephemeris and The American Ephemeris and Nautical Almanac_
(1962) are correct to the number of digits they print. Now, of course,
we've had quite a few Venus orbiters and three flybys of Mercury by
Mariner 11, and MESSENGER is en route.

Pluto has Charon and now S/2005 P1 and P2. We can use Charon's period
and semimajor axis to get the *sum* of the masses of Pluto and Charon
(see above!). To get their individual masses requires observations of
Pluto relative to the Pluto/Charon center of mass -- or at least this
was true before last year's discoveries. Now that it's a four-body
system things may actually get easier. This is work in progress.

Gee, this has been a long reply, but I think I've covered everything.

-- Bill Owen

Greg Neill a écrit :
"srp" wrote in message ...

Greg Neill a écrit :


Actually, both masses *do* influence the trajectory. If you
look at the differential equation for the trajectory, you
will find a term that contains the sum of the two masses.
It's really only a problem when the trajectory is being
calculated in a frame of reference that's not coincident with
the center of mass of the larger body. So, for example, if you
wanted to describe the trajectories of two equally massive
bodies co-orbiting, you would want to use the sum of the two
masses as the gravitational parameter for the system (usually
designated with the greek letter mu).

Hmm yes. But the masses are useful to determine how much energy
is required to set them into the specific closed orbits we want
to set them onto. But once on closed gravitationl orbits, whatever
the masses involved, they then are on the closed orbit. And
it seems to me that the reverse process is not as obvious to
establish. Knowing the parameters of an orbit gives no clue
as to the magnitude of the masses, it seems to me.


In the simplest case, a spacecraft of negligible mass (relatively
speaking) takes up a circular orbit about a planet. The radius
of the orbit is determined by some means (say by radar observations
or laser altimetry), and the period by stellar observations.
Then, the mass of the planet can be accurately determined via
Newton's version of Kepler's third:

T^2 = 4*pi^2 * a^3 / [G*M]

where T = obital period
a = orbital radius
M = the mass of the planet

If the mass of the spacecraft is *not* negligible w.r.t. the
body being orbited, then replace M with M + m.


With Kepler's third law, we don't even need the mass of the Sun
to calculate the Solar system's planetary orbits.


True, but then you're relying on ratios. If you use Newton's
version (above), the masses are explicit.


If you remove that mass from the divisor in G, and also remove
it from the Force equation, you still get the Earth orbit if
you put the mass of the earth in the force equation.

If you then reduce the mass of the Earth to the mass of an
electron, you still get Earth's orbit. Seems pretty circular
to me, no pun intended.

What of the planets, that were set on their closed orbits before
we were here? And the Sun's mass?

What is the bottom rung of the solar masses ladder?


When one mass is very much smaller than the other, its mass can
usually be ignored.



Who calculated the various planetary masses and the Sun's mass
initially? What method did he use, and how was the precision of
calculations confirmed with respect to physical reality?

The ratios of the masses were known via Kepler's laws, so I
suppose that you could say that Kepler first had a handle on
them.

Then my question becomes how was that ratio established ?


I mispoke. Kepler gave the relative scale of the orbits, so that
determining the actual size of any one orbit would allow all the
others to be calculated. It was the period and radii of orbits
of the moons of the planets, and later the perturbations of
one planet on the others, that allowed the relative masses to
be determined. Cavendish's torsion balance experiment set the
value for G and allowed the actual masses to be determined.


Newton derived Kepler's laws from his Theory of
Gravitation in a form that took into account the sum of the
masses involved.

Presently, it seems to me that he built his theory of gravitation
from Kepler's third law.


No doubt he was influenced by Kepler's laws, but he starts
his derivation from the inverse square force law and
proceeds to derive Kepler from there.


I really would like to know how the ratio was established,
and the Sun's mass set.


It goes back to Cavendish. Once G is determined the mass
can be determined from Newton's version of Kepler's 3rd.

Very ingteresting conversation. Thanks for your input.

André Michaud

In article ,
Greg Neill wrote:
With Kepler's third law, we don't even need the mass of the Sun
to calculate the Solar system's planetary orbits.

True, but then you're relying on ratios. If you use Newton's
version (above), the masses are explicit.

Provided you can measure absolute distances. But getting to absolute
distances was tricky in the pre-spaceflight era. Classical observational
astronomy is all angles, which only gives you ratios of distances, and the
planets were too far away for triangulation using Earthly baselines. The
problem wasn't really satisfactorily solved until the development of
interplanetary radar, shortly before the first planetary probes, made it
possible to get an absolute measurement of the distance to Venus. (It
turned out that the best previous values were in error by about half a
part per thousand.) Spacecraft improved that further; in particular,
precision tracking of the Viking landers calibrated the size of the solar
system down to the meter level.

...It was the period and radii of orbits
of the moons of the planets, and later the perturbations of
one planet on the others, that allowed the relative masses to
be determined. Cavendish's torsion balance experiment set the
value for G and allowed the actual masses to be determined.

Poorly, it turns out, because G is only known to about five digits -- it
is very hard to measure well, because gravity is such a weak force. If
you want to be accurate, you work with G*M, which is routinely measured to
7-8 digits by tracking of spacecraft encounters. (Spacecraft can be
tracked *very* precisely, much more precisely than planets.) For orbits
work, it's always G*M that matters anyway -- there is no need to convert
to actual masses.

The Sun's G*M is 1.32712440018e20 m^3/s^2, but its mass is known to less
than half that many digits, thanks to the uncertainty in G.
--
spsystems.net is temporarily off the air; | Henry Spencer
mail to henry at zoo.utoronto.ca instead. |

"srp" wrote in message
...
| Greg Neill a écrit :
| "srp" wrote in message
...
|
| Greg Neill a écrit :
|
|
| Actually, both masses *do* influence the trajectory. If you
| look at the differential equation for the trajectory, you
| will find a term that contains the sum of the two masses.
| It's really only a problem when the trajectory is being
| calculated in a frame of reference that's not coincident with
| the center of mass of the larger body. So, for example, if you
| wanted to describe the trajectories of two equally massive
| bodies co-orbiting, you would want to use the sum of the two
| masses as the gravitational parameter for the system (usually
| designated with the greek letter mu).
|
| Hmm yes. But the masses are useful to determine how much energy
| is required to set them into the specific closed orbits we want
| to set them onto. But once on closed gravitationl orbits, whatever
| the masses involved, they then are on the closed orbit. And
| it seems to me that the reverse process is not as obvious to
| establish. Knowing the parameters of an orbit gives no clue
| as to the magnitude of the masses, it seems to me.
|
|
| In the simplest case, a spacecraft of negligible mass (relatively
| speaking) takes up a circular orbit about a planet. The radius
| of the orbit is determined by some means (say by radar observations
| or laser altimetry), and the period by stellar observations.
| Then, the mass of the planet can be accurately determined via
| Newton's version of Kepler's third:
|
| T^2 = 4*pi^2 * a^3 / [G*M]
|
| where T = obital period
| a = orbital radius
| M = the mass of the planet
|
| If the mass of the spacecraft is *not* negligible w.r.t. the
| body being orbited, then replace M with M + m.
|
|
| With Kepler's third law, we don't even need the mass of the Sun
| to calculate the Solar system's planetary orbits.
|
|
| True, but then you're relying on ratios. If you use Newton's
| version (above), the masses are explicit.
|
|
| If you remove that mass from the divisor in G, and also remove
| it from the Force equation, you still get the Earth orbit if
| you put the mass of the earth in the force equation.
|
| If you then reduce the mass of the Earth to the mass of an
| electron, you still get Earth's orbit. Seems pretty circular
| to me, no pun intended.
|
| What of the planets, that were set on their closed orbits before
| we were here? And the Sun's mass?
|
| What is the bottom rung of the solar masses ladder?
|
|
| When one mass is very much smaller than the other, its mass can
| usually be ignored.
|
|
|
| Who calculated the various planetary masses and the Sun's mass
| initially? What method did he use, and how was the precision of
| calculations confirmed with respect to physical reality?
|
| The ratios of the masses were known via Kepler's laws, so I
| suppose that you could say that Kepler first had a handle on
| them.
|
| Then my question becomes how was that ratio established ?
|
|
| I mispoke. Kepler gave the relative scale of the orbits, so that
| determining the actual size of any one orbit would allow all the
| others to be calculated. It was the period and radii of orbits
| of the moons of the planets, and later the perturbations of
| one planet on the others, that allowed the relative masses to
| be determined. Cavendish's torsion balance experiment set the
| value for G and allowed the actual masses to be determined.
|
|
| Newton derived Kepler's laws from his Theory of
| Gravitation in a form that took into account the sum of the
| masses involved.
|
| Presently, it seems to me that he built his theory of gravitation
| from Kepler's third law.
|
|
| No doubt he was influenced by Kepler's laws, but he starts
| his derivation from the inverse square force law and
| proceeds to derive Kepler from there.
|
|
| I really would like to know how the ratio was established,
| and the Sun's mass set.
|
|
| It goes back to Cavendish. Once G is determined the mass
| can be determined from Newton's version of Kepler's 3rd.
|
| Very ingteresting conversation. Thanks for your input.
|
| André Michaud

But of course this assumes that Newton's G is really constant from
Cavendish to solar system. ;-) Just as we have been talking about in
the other thread. All that is really known to a high degree of accuracy
for the solar system data is the product GM. After looking at this
paper that Ken mentioned, I think you might be right about G not being
constant.

http://www.arxiv.org/abs/gr-qc/0511026

More study is needed. ;-)

FrediFizzx

http://www.vacuum-physics.com/QVC/qu...uum_charge.pdf
or postscript
http://www.vacuum-physics.com/QVC/qu...cuum_charge.ps

http://www.vacuum-physics.com

In article ,
Bill Owen wrote:
Historically, the only way to get at the length of the AU was through
parallax, using the Earth's diameter as a baseline.

Actually, some other techniques were tried. You could measure Earth's
orbital velocity by looking at changes in stellar spectrum-line Doppler
shifts, which would give you the absolute size of Earth's orbit. You
could measure changes in the timing of eclipses of Jupiter's moons, due to
speed-of-light lag and the changing distance to Jupiter, and convert those
to absolute distances using the known speed of light. And you could use a
close approach of Eros in a different way, measuring the perturbation in
Eros's orbit and relating that to Earth's known G*M to establish how close
Eros came.

If memory serves, that last actually gave the best pre-spaceflight values.

But once we started sending spacecraft to other planets, all this
changed. We could get range measurements -- round-trip light time --
to the spacecraft, providing a distance measurement in kilometers.

The history was a bit more complicated than that.

The first spacecraft determination of the AU used Pioneer V -- built for a
Venus flyby, actually flown as an interplanetary-environment mission when
it missed the Venus launch window. Its tracking gave both angles (giving
a classical relative-distances orbit) and absolute distances, allowing the
two to be related. Its value for the AU agreed fairly well with the best
Eros value.

Had that value been used for the Mariner 2 Venus encounter -- the first
actual spacecraft encounter of another planet -- Mariner would have gone
past the wrong side of Venus, with its instruments looking away from the
planet. There was a computation error in the Eros data analysis, and the
calculation of Pioneer V's orbit didn't allow properly for light pressure,
and the two errors happened to roughly match.

As I noted briefly in another posting, the day was saved by the first
interplanetary radar observations, measuring planetary distances directly.
The later spacecraft determinations did improve precision further.
--
spsystems.net is temporarily off the air; | Henry Spencer
mail to henry at zoo.utoronto.ca instead. |