Sidereal Periods and Heliocentric Longitudes

[RingsOfSaturn22]

This question is for Jim or anyone who might know the answer.

When we have sidereal periods of a celestial body, we are told that it is a measure of the orbit in relation to a "distant star".

What is this distant start that everything is being compared to? I've been searching and searching, both through NASA's site as well as journals (and even sites like the Astronomy StackExchange), and have not found an explanation of what this "distant star" is!

Then, this further leads into how are they coming up with heliocentric longitudes? These seem to be based on the vernal point as a point of reference — thus, they have precession built in already. I'm inclined to believe these heliocentric longitudes are being found based off of geocentric observations done in right ascension, and then they are just using equations to convert them into heliocentric.

The only thing I've come across so far that has been a little help is this 1932 article by William H. Pickering entitled Planet U, and the Orbits of Saturn and Jupiter. On page 73, first paragraph, he states:

We feel that we may therefore safely use the longitude of the minimum of 1872.6 taken from the table number III published with the curve of Figure 1 in 1929. The corresponding heliocentric longitude for the former date, and not corrected for precession, is 289ª.8.

He, however, does not detail more beyond that. The number that current heliocentric ephemerides gives up for July 1872 for Saturn is off by a couple of degrees. I don't know if it's because our current ephemerides have precession already built in, or if it was because there was some error in their former readings.

If current heliocentric longitudes do indeed have precession built in, what would be the equation to correct for this? I'm assuming the equation would also have to consider the tilt of the various orbits to one another.

For some of the things I'm investigating in the Gann thread, I'd need to know these parameters. One of the things I'm curious is whether or not the heliocentric position in relation to the stars (and NOT the vernal point) has any influence on the patterns we see.

[Jim Eshelman]

This question is for Jim or anyone who might know the answer.

When we have sidereal periods of a celestial body, we are told that it is a measure of the orbit in relation to a "distant star".

That's a very casual way of saying it, probably phrased for a popular audience. Since stars have proper motion and (more importantly) the closer ones have small amounts of parallax, it's a little imprecise. - More accurately, one should measure it against precession-free space.

There's not actually a comparison star.

Then, this further leads into how are they coming up with heliocentric longitudes? These seem to be based on the vernal point as a point of reference — thus, they have precession built in already.

They then have to delete accrued precession, a simple process.

I'm inclined to believe these heliocentric longitudes are being found based off of geocentric observations done in right ascension, and then they are just using equations to convert them into heliocentric.

No, the process of calculating geocentric longitudes starts with calculating heliocentric longitudes and then adjusting for parallax. I think, for calculation speed, the actual calculations are now done from mammoth catalogues of calculated helio X, Y, Z coordinates rather than from root formulae, but in any case it goes helio first and then converted. (The only other way you could do it is with something that was ultimately heliocentric anyway, like Ptolemaic epicycles.)

In practice, I'm not clear why it would matter which way they calculate them as long as they get correct results.

If current heliocentric longitudes do indeed have precession built in, what would be the equation to correct for this?

Subtract accrued precession. The approximate rate is 50"/year, but they would use the actual rate.

I'm assuming the equation would also have to consider the tilt of the various orbits to one another.

No. Orbital tilt is irrelevant.

[quoe]For some of the things I'm investigating in the Gann thread, I'd need to know these parameters. One of the things I'm curious is whether or not the heliocentric position in relation to the stars (and NOT the vernal point) has any influence on the patterns we see.[/quote]
Just calculate Sidereal heliocentric positions. Solar Fire can do this with one extra click.

[RingsOfSaturn22]

Thank you! Just a few follow up comments.

This question is for Jim or anyone who might know the answer.

When we have sidereal periods of a celestial body, we are told that it is a measure of the orbit in relation to a "distant star".

That's a very casual way of saying it, probably phrased for a popular audience.

Unfortunately, that was the exact wording used in SEVERAL astronomy courses at universities!!!

I'm inclined to believe these heliocentric longitudes are being found based off of geocentric observations done in right ascension, and then they are just using equations to convert them into heliocentric.

No, the process of calculating geocentric longitudes starts with calculating heliocentric longitudes and then adjusting for parallax. I think, for calculation speed, the actual calculations are now done from mammoth catalogues of calculated helio X, Y, Z coordinates rather than from root formulae, but in any case it goes helio first and then converted. (The only other way you could do it is with something that was ultimately heliocentric anyway, like Ptolemaic epicycles.)

For this question, I was referring to the actual initial observations, not calculations that come afterwards. For instance, when NASA compiled J2000,B1950, or DE430, doesn't most of the observational data used to create the calculations come from geocentrically based things (including the satellites)? What is the reference point that they use for such observations?

In the 2021 version of The Astronomical Almanac, page L6, it says (in reference to DE430):

Optical, radar, laser, and spacecraft observations were analyzed to determine starting conditions for the numerical integration and values of fundamental constants such as the planetary masses and the length of the astronomical unit in meters. The reference frome for the basic ephemerides is the ICRF; the alignment onto this frame has an estimated accuracy of 1-2 milliarcseconds.

Then, on page L2, where it explains the ICRS more, it says:

Although the directions of the ICRS coordinate axes are not defined by the kinematics of the Earth, the ICRS axes (as implemented by the ICRS and Hipparcos Celestial Reference Frame) closely approximate the axes that would be defined by the mean Earth equator and equinox of J2000.0 (to within 0.1 arcsecond).

So to me, it sounds like they are not using a "distant object" or even precession free space, since the reference system itself keeps wrapping back around to the vernal point. They mention a "frame bias" multiple times, and have pages upon pages of equations in Section B to correct for this. I was hoping I wouldn't have to use such complicated equations to fix this, haha.

[Jim Eshelman]

For this question, I was referring to the actual initial observations, not calculations that come afterwards. For instance, when NASA compiled

OK, but that's not how anyone is calculating sidereal periods (planet orbits). Observations (which, yes, are necessarily geocentric and then corrected for parallax to helio) were used in the original determination of planet orbits (and continue to be used to refine the calculation process), but any calculation of sidereal periods is simply done from derived formula.

...doesn't most of the observational data used to create the calculations come from geocentrically based things (including the satellites)? What is the reference point that they use for such observations?

I'm not clear what you mean by "reference point." If you mean coordinate systems, astronomers routinely use RA and declination (or altitude-azimuth that are then converted to equatorial coordinates). Positions are calculated as if viewed from Earth's center.

Although the directions of the ICRS coordinate axes are not defined by the kinematics of the Earth, the ICRS axes (as implemented by the ICRS and Hipparcos Celestial Reference Frame) closely approximate the axes that would be defined by the mean Earth equator and equinox of J2000.0 (to within 0.1 arcsecond).

For particular purposes, current observations may be converted to the equator and vernal point of a different epoch for communication purposes. This allows all sorts of comparisons that would be cumbersome otherwise. J2000.0 is the most common current standard, from what I've seen (just as it was 1950.0, Julian or Besselian, before that).

So to me, it sounds like they are not using a "distant object" or even precession free space, since the reference system itself keeps wrapping back around to the vernal point. They mention a "frame bias" multiple times, and have pages upon pages of equations in Section B to correct for this. I was hoping I wouldn't have to use such complicated equations to fix this, haha.

All quality astrological software (and probably all the lower quality astrology software) takes care of this. You don't have to do anything with it at all.

[RingsOfSaturn22]

Thank you. I'm the type the likes to know all the background processes behind the numbers, so this is helpful.

For some time, I've been wondering how astronomers even 200 years ago were getting their observed measurements. I mean, we base everything on the vernal point for our reference systems; yet, the vernal point is not anything you can see in the sky. I haven't heard it often detailed what exactly is done to ascertain you have proper coordinates. Maybe measurements from a sun dial or similar type device were used to ascertain the angle difference from the equinox on a particular day and then you gain your bearings from there? Or use a particular star that you know the hour angle of? There has to be something that you can use to independently verify that the calculations are correct.

I'm not clear what you mean by "reference point." If you mean coordinate systems, astronomers routinely use RA and declination (or altitude-azimuth that are then converted to equatorial coordinates). Positions are calculated as if viewed from Earth's center.

For celestial coordinates, we use the vernal equinox as a reference point. That's what I was referring to. That's the zero point that everything is measured against. But there should be some other "zero point" that we can use to also verify the information.

I hope that came across clearly. If not, it's no big deal. I will manage.

[Jim Eshelman]

For some time, I've been wondering how astronomers even 200 years ago were getting their observed measurements. I mean, we base everything on the vernal point for our reference systems; yet, the vernal point is not anything you can see in the sky. I haven't heard it often detailed what exactly is done to ascertain you have proper coordinates. Maybe measurements from a sun dial or similar type device were used to ascertain the angle difference from the equinox on a particular day and then you gain your bearings from there?

No, it's way easier and more precise than that. You're relying too much on what could be visually determined from Earth (which btw is a good demonstration of why ancient people often never cared about the equinoxes and took millennia to bother to calculate them).

It's all math: The equinoctial points are the two points where the ecliptic intersects the celestial equator. Visual observation went into (and, at very minute adjustment levels, probably still goes into) figuring out how to calculate these things in the first place, but - once those were identified - it's just a matter of calculating where the two great circles intersect. In fact, there's no need to do that any more since there has long been a single equation for calculating the rate of precession across time. (I used to calculate the SVP from formula on a programmable calculated but threw away the notebooks with those formulae 20-30 years ago. There's around somewhere on the Internet.)

Or use a particular star that you know the hour angle of? There has to be something that you can use to independently verify that the calculations are correct.

That one is simple and is observational. (I did it in college and amateur astronomers do it a million times a night.) You see the star and measure the RA distance from the meridian.

I'm not clear what you mean by "reference point." If you mean coordinate systems, astronomers routinely use RA and declination (or altitude-azimuth that are then converted to equatorial coordinates). Positions are calculated as if viewed from Earth's center.

For celestial coordinates, we use the vernal equinox as a reference point.

I don't now who you mean by "we." I certainly don't. It's irrelevant to the zodiac.

But I suspect you mean that it's a factor used in the process of converting equatorial coordinates to ecliptical coordinates, so I think I answered it.

[RingsOfSaturn22]

For anyone else that comes across this thread in the future, it was in relation to Derek's old thread (section 3) discussing the difference between a sidereal day and an inertial day, hence all the questions regarding the "reference point" for observations for a sidereal day, sidereal year, etc.

Also, this 2012 Physics StackExchange post goes into the same issue. Basically, it seems the reference point is an arbitrarily constructed one -- the inertial frame of reference.

@Jim, I appreciate your help!