Planetary Stations

[Jim Eshelman]

We've talked about this before, but left it open pending coming up with a definition that makes sense. The problem is that how one measures a planetary station from looking at an ephemeris isn't usually the same for Mercury as it is for Pluto.

Having looked at this for months, trying to figure it out, I've concluded that the best simple measurement is proximity in time to the point of the station. I'd be happy if a station were measured as follows:

If using whole days, count the date of the station and one (maybe two) calendar days either side. (Possibly give a user setting for how many days either side of a station. In that case, I'd default at one - a three-day stretch altogether - and allow a larger number.)

If using something more granular, that easily measures hours, all one or two 24-hour periods either side of the exact station. (I don't know how the Swiss Ephemeris feeds you the data. This might be easier. Whole days from a point in time.) - If there is a user option, this could still be in whole days either side of the station, defaulting to 48 hours.

I think this is the basic definition that is most useful (even though all such cutoffs are somewhat arbitrary). If you want to make it an fancier, then also include any planet with an instantaneous speed of 0°00'00" (< 0.5") since it would be silly to exclude.

[Jim Eshelman]

Presuming the Swiss Ephemeris allows direct extraction of the time of a station in some useful way, I think the functional definition should be: If (1) the chart moment occurs within 36 hours either side of the station or (2) the displayed speed is 0°00'00", the planet is station.

I suggest marking this with an S immediately after the Longitude field.

[Jim Eshelman]

For perspective on what has been done in other software, here is the menu in Solar Fire. If the various options, I think "within time frame of station" is likely to produce the sanest effect.

One day, some astrologer may find a gradient way of measuring this based, most likely, on something like velocity expressed as a percentage of average speed. Conceivably, this could give a gradient score similar to that for angularity or aspect strength. (I just got this crazy idea of basing it on a cotangent curve where 0° is set at zero motion, so that at the instant of station the function is infinitely large and an instant either side it is very, very big - but it spends most of its time boringly small. If measuring against speed, then the average speed of the planet is equivalent to 90°. Possibly it should be delta-speed, change of rate of speed, rather than speed itself.)

For now, in practical terms, I still think 36 hours either side of the station is a best measurement, or perhaps let the user set the time period.

stations.png

[Mike V]

This makes sense. Solar Fire almost certainly does it because the Swiss Ephemeris doesn't have any opinion on what constitutes "stationary," so we'll have to do it the same way.

[Jim Eshelman]

This makes sense. Solar Fire almost certainly does it because the Swiss Ephemeris doesn't have any opinion on what constitutes "stationary," so we'll have to do it the same way.

Isn't there a way to pull the date/time of the station? For a quick check?

PS - Astrologers have no standard definition of what "being stationary" means, so how could SE? Also, the various models / options SF offers fight against each other a bit: What's good for Pluto isn't good for Mercury and vice versa.

It's possible a composite approach is warranted: Something like, "if within 36 hours of the moment of the station OR has a daily motion > 0°00'05'.

[Mike V]

This makes sense. Solar Fire almost certainly does it because the Swiss Ephemeris doesn't have any opinion on what constitutes "stationary," so we'll have to do it the same way.

Isn't there a way to pull the date/time of the station? For a quick check?

PS - Astrologers have no standard definition of what "being stationary" means, so how could SE? Also, the various models / options SF offers fight against each other a bit: What's good for Pluto isn't good for Mercury and vice versa.

It's possible a composite approach is warranted: Something like, "if within 36 hours of the moment of the station OR has a daily motion > 0°00'05'.

There is no existing API that I am aware of (and there seems to be none that Google is aware of) that allows us to retrieve the date/time of a station; we have to calculate it.

I had a thought on this. Who says that we have to apply the same rules for all planets? For example, we can arbitrarily decide to consider a Mercury station when Mercury's speed is less than 0°00'05'... but for Pluto, we can find the hour (or minute, or whatever) when it switches between retrograde and direct (regardless of speed), and consider the 36 hours on either end of that time as being stationary. It's going to be next to impossible to use speed to determine a station for Eris (to say nothing of a body like Sedna), but we can definitely use that for much closer bodies.

What do you think?

[Jim Eshelman]

In principle, no reason to apply the same rule. We then need to figure out which works better for which planet,

On Mercury, the 5" rule wouldn't work, though. I'm pretty sure from watching over the years that the stationary window is at least the day of the station and the day before and after. The Mercury problem is that such things as minimum motion that might make sense for another planet gives too tiny a window for Mercury.

But sure... a blended approach might work.

[Jim Eshelman]

Now that I'm answering this in the morning and not after being up almost 20 hours...

I still agree with the principle that we may not need the same rule for all planets; or, similar, that one valid approach may be to have two different rules (applied equally to all planets) and to consider it stationary if it meets either of them.

The problem, though, is that this approach still requires obtaining the exact moment of the station, which isn't readily available. Perhaps it can indeed be calculated iteratively and that would be a cool subroutine to develop with many possible uses; but (unless you're hot on its trail) that might not come quickly or without considerable work.

That's why I was looking for an approach that didn't require this - at least to get us started. I don't know exactly what results we would get from taking a fixed % of the planet's average speed, e.g., how long the station period would be and whether it would tend to reflect a spike or peak impression. (That idea seems not to be in this thread. I can't find it on a quick scan. I'm sure I mentioned it somewhere recently. [Certainty is no guarantee of accuracy.]

[Jim Eshelman]

I still think that 36 (or 48) hours either side of the station makes most sense AND is imperfect. Absent ready availability of the station epoch, I'm intrigued by the idea of taking the period that the absolute value of a planet's velocity drops below a fixed percentage of its average geocentric daily motion. This normalizes several things and still may cause problems, but I can't anticipate how large the problems are (such as gross inconsistencies in the length of the station window) without seeing it in action.

A few immediate problems are mostly solvable or reducible.

First - easy to solve - is getting each planet's average heliocentric speed.

Second: This, however, is not its average geocentric speed since geocentrically it goes back and forth with considerable (comparable) notion in both directions. This is easily solvable, too, without outright solving it: The amount of time each planet spends retrograde is close enough (not close - but I think close enough) that we can just reduce the fixed percentage we use by the same amount (say, cut it in half). If this proves too off the mark, I have at hone the average % of time (across its whole orbit) that each planet spends retrograde, so it's just one more step to modify fixed percentage easily. - This mostly brings us to the third issue.

Third: A planet's average daily motion does not remain the same in all parts of its orbit. This varies with eccentricity of the orbit so, of course, Pluto is the most extreme example of the confirmed planets. I suspect that using Pluto's orbit across its entire orbit would produce aberrant effects in one part of its orbit. In this case, I think we handle it by simply fudging the fixed value we use for its average velocity, with the idea that this will hold up well enough for several decades.

Pluto's eccentricity is 0.2488, a known outlier. Eris is worse (and TNOs in general have more eccentric orbits with Sedna being the worst by far). But also Mercury is nearly as extreme as Pluto, and (if this factor alters the station window enough) this makes Mercury a potential issue. Checking the other planets against an easily obtainable table:

Sedna 0.8549
Eris 0.4407
Pluto 0.2488
Mercury 0.2056
Haumea 0.1887
Makemake 0.1559
Mars 0.0934
Saturn 0.0541
Jupiter 0.0484
Uranus 0.0472
Neptune 0.0086
Venus 0.0068

[Jim Eshelman]

Here are the average lengths of orbital periods of each planet in sidereal years:

Eris 559.07
Pluto 247.94
Neptune 164.8
Uranus 84.0205
Saturn 29.4475
Jupiter 11.862
Mars 1.881
Venus 0.615198
Mercury 0.240846

Based on simple math, here is the average daily (heliocentric) speed of each planet. 360/orbit in years gives distance per year. Divide by 365.25 gives distance per day. Therefore, multiplying (360/365.25) by the reciprocal of the sidereal period in years should give degrees per day. (I'm doing this on a calculator without a memory function (but with a reciprocal function), then copying the result by hand. I suppose I should just make a spreadsheet and be done with it, but that's equally tedious on a single 11" screen. In other words, check my math!)

360/sidereal period in years = degrees/year
divide by 365.25 = degrees/day
x 3600 = seconds per day

Eris 6.35
Pluto 14.31
Neptune 21.53
Uranus 42.23
Saturn 120.49
Jupiter 299
Mars 1,886 = 31'26"
Venus 5,768 = 1 36'08"
Mercury 14,732 = 4 05"32"

[Jim Eshelman]

This is a good start, but we know it needs to be jiggled at least a little. For example, for most of the last half century, Pluto has been in a part of its orbit where it moves almost exactly the same amount each year as Neptune. If that were the whole story, we'd just substitute the Neptune value for the Pluto value and be done with it, but there are other issues. For example, I'm suspicious of the practical value of the Mercury numbers (but have an easy way to check it). Second, we need a generalizable formula to apply to other bodies not on the list. Third, simply need to see how these work in practice.

Just to get some representative (not definitive) numbers - just to see if we're on the right track - here are outer planet distances between two consecutive solar returns - my current and next ones. (Not expected to match exactly, but to give a general idea.)

Eris 0 13'17" = 797" > 2.2"/dy
Pluto 1 44'07" = 6,247" > 17.1"/dy
Neptune 2 17"07" = 8,227" > 22.5"/dy
Uranus 4 17'38" = 15,458" > 42.3"/dy
Saturn 12 45'38" = 45,938" > 126"/dy
Jupiter 37 52'27" = 136,347" >373"/dy

From this, we see several things of interest. First, Eris is in a part of it's orbit where it is moving about a third of the pace it averages, and this will stay this way throughout the lifetime of anyone reading this: We might as well take 2.18"/dy as its actual average speed. Pluto is moving currently about halfway between its normal average and the average speed of Neptune. Considering the years in which most people using this program were born and will live, I suggest setting the value to match Neptune's and annotate the source code that this is a kludge to match this part of Pluto's orbit and will need to be tweaked (or a better method found) in future decades. Neptune, Uranus, and Saturn are almost exactly their average. Jupiter, in this particular year, is running faster than its average, though Jupiter is a low eccentricity orbit so I'm not inclined to screw with it.

Of the outer planets, it gives us only Pluto and Eris (the highest eccentricity orbits) that need modification. This suggests that other TNOs either need to have a current "working value" calculated (current actual one year motion as an example) when their elements are defined, or some other solution I haven't thought of.

This gives us the following working table (so far) of average helio daily motion.

Eris 6.35 2.2"
Pluto 14.31 21.53
Neptune 21.53
Uranus 42.23
Saturn 120.49
Jupiter 299
Mars 1,886 = 31'26"
Venus 5,768 = 1 36'08"
Mercury 14,732 = 4 05"32"

[Jim Eshelman]

Next, let's tackle the Mercury problem.

For a representative example, let's use Marion's chart. She SHOULD show Mercury retrograde and SHOULD NOT show Saturn retrograde. We can use her chart as a case study in how Mercury behaves (sometimes) surrounding its station.

Marion was born 9h48m before a Mercury direct station. At the moment she was born, Mercury's velocity was -1'48". Here was Mercury's instantaneous velocity at 12 and 24 hours intervals before and after the exact station (and an extra day for comparison):

48 hours before: -8'42"
36 hours before: -6'34"
24 hours before: -4'24"
12 hours before: -2'12"
Exact station: 0'00"
12 hours after: +2'13"
24 hours after: +4'27"
36 hours after: +6'42"
48 hours after: +8'57"

These figures are remarkably symmetrical. They should, in fact, be similarly symmetrical before and after all stations, though (without testing other stations - such as all Mercury stations over a year or more - I can't say that the actual pace of velocity is the same.

Using this one example as a starting point, though, Mercury's velocity 36 hours either side of the station maxxed at 6'42" = 402". We certainly want it this far out. (24 hours was 4'27" = 267".) We probably don't want it 48 hours out which was just under 9' = 540" (about twice the value). You can even see that the speed is picking up briskly - the delta of the curve is changing faster - with a 50% increase between 24 and 36 hours and a 134% increase between 36 and 48 hours.

Unfortunately for math purposes, Mercury's heliocentric average daily motion is 14,732". This means that 267" (24-hour point) is 1.5% of that and 402" (36-hour point) is 2.7%. Can this number possibly be transferrable to the other planets?

Using, say 3% would give us the following speculative table of planet daily velocities to be counted as stationary:

Eris 0.07"
Pluto 0.65"
Neptune 0.65"
Uranus 1.27"
Saturn 3.6"
Jupiter 9"
Mars 56.58" = ~1'
Venus 173" = ~3'
Mercury 442" = ~7.3'

This isn't bad at all! (I'm not yet saying it's great - but by no means is it bad.) Mercury is right at where we found it at 36 hours out (that was forced). The others feel like they are at least in the neighborhood of being right.

I'm going to each planet's station for this calendar year (2024) and see if it's velocity is anywhere near these values 36 hours after the station.

[Jim Eshelman]

Pulling all stations this year plus Venus, one Mars, and one Jupiter from next year, then calculating velocity 36 hours after.

MERCURY
There were, of course, abundant Mercury stations this year (like every year). Theory: 7.3'. Observed: 14', 15', 12', 8', 12', 10', 13'. These are all larger than expected. This suggests that the Mercury number should be larger - perhaps 12'? - which would be ironic since the smaller number gave the theory that seems reasonably good for other planets.

VENUS
Theory: 3'. Observed: 3'32" and 3'42" (So maybe < 4' is the key.)

MARS
Theory: 1' (57"?). Observed 1'06", 1'12", close match.

JUPITER
Theory: 9". Observed 18", 18". Very interesting. Jupiter also seems to be in a part of its orbit where it is moving slightly faster. This needs to be jiggled.

SATURN
Theory: < 4" (3.6"). Observed: 9", 8" (larger than expected, may need tweaking).

URANUS
Theory: 1" (1.27"). Observed: 4", 4" (needs to be larger than theory)

NEPTUNE
Theory: 0" (0.65"). Observed: 3", 2".

PLUTO
Theory: 0" (0.65"). Observed: 2", 2".

ERIS
Theory: 0" (0.7"). Observed: 1", 0"

[Jim Eshelman]

All, of these came out larger than expected. This means that the original Mercury example used to estimate the 3% rule was an outlier. The real value is probably closer to 5%. We could use a 5% of the average daily heliocentric motion or - as a start - hard code numbers. Ideally these should be tested against a few more years, but the final numbers may look very close to this list below. We might consider a planet stationary if the absolute value of its instantaneous velocity is no more than:

Mercury: 12' or 15'
Venus: 4'
Mars: 1'15"
Jupiter: 18"
Saturn: 9"
Uranus: 4"
Neptune: 3"
Pluto: 2"
Eris: 1"

A 4.5% rate resembles this:

Eris 0.3"
Pluto 1"
Neptune 1"
Uranus 2"
Saturn 6"
Jupiter 13"
Mars 1'25"
Venus 4.3'
Mercury 11'

Enough for now. I have a rehearsal soon and haven't showered or looked at my lines.

[Mike V]

Thank you for doing all of this.

Assuming that, for the time being, we mostly care about whether a planet is or is not "stationary" at any given time, and not identifying ahead of time the exact date and time of "the exact station," there is another way of tackling this. I managed to get some very basic code down under a similar sort of paradigm, but without precalculating these speeds. In essence, I tried to decide what constitutes a "station" and then just checked to see if such an event occurred within a valid time window (say 36 hours in either direction) of the present moment.

(The level of granularity here is 1 hour, which makes this iteration basically instantaneous)

For faster planets (geocentrically), consider a station to be when speed for a given hour falls below a given threshold - which is fully compatible with your conclusions here (and any further tweaks). For slower planets, consider a station to be when the direction of motion changes between 2 hourly checks, and ignore exact speed. Thinking about it now, I can see this failing if the planet "stutters" and gets extremely slow for a time but does not change direction.

In either case, we decide on what the length of a station is; for example, 72 total hours with the "exact station" at the center. I gave the TNOs far more than this (for example, 1 week for Pluto, 2 weeks for Sedna). All of that is changeable.

For any given chart, I just check, hour by hour, to see if the velocity or direction-change criteria were met within a 72-hour window (larger for slower planets). If so, the planet is "stationary" in that chart. The end.

I think this concept is fully compatible with yours, and is not especially difficult to integrate - I just plug in those numbers rather than the ones I used (which happened to be 5" of instantaneous velocity for most planets, and then I rely on the direct/retrograde switch for Pluto and the TNOs). If you like the concept of averaged speeds within 36 hours of the exact station, and using only this pre-calculated as the determining factor, it gets even easier: I just check the speed of the given body at the given time, and that's it.

[Jim Eshelman]

I just realized how simple it can be. No need to calculate hourly. It's more work than I thought was right, but it's simple: If we're using, say, a 36-hour window, just calculate velocity 36 hours later and earlier. If either is a different direction than the current one, there is a station in that window.

[Mike V]

Wow, you’re right. Super easy. I’ll set it to 36 hours in either direction for all bodies; let me know if you think some should have longer windows. (Although that may require at daily checks for direction changes… but it won’t be noticeable)

[Jim Eshelman]

I think (emphasis on think) 36 is a good window. More to the point, I think we won't likely know it's not until we live with it a while.

Hard coding it here is fine at first, I think - definitely better than we have now - but at some point adding this to chart options (some misc. items page) is likely warranted.

This has been useful because I've sorted out in my head a lot of detail about stations and direction changing. I think the shape of the curve and the rate of change is what describes the behavior. (Rate of change in velocity rather than velocity itself) but, without a special calculating tool, that would be hard to show. - And we don't really have to show it. Just a way to think about how the dynamics work.