[SPACE] Tidal braking

[dataweaver]

Originally Posted by Agemegos

When calculating the height of the tide that a planet raises on its moon, or that a moon raises on its planet:

T = 1.6 million * (M * D-to-the-fourth) / (P * R-cubed)

Where M is the mass of the object raise the tide in Earth masses, D is the diameter of the object the tides are on in Earth diameters. P is the mass of the object the tides are on in Earth masses, and R is the distance between them in Earth diameters.

Note that P is used elsewhere to represent orbital periods; so you should probably choose another letter to represent the mass of the object that the tides are on. I'd be inclined to use a lowercase m. So:

Lunar tides: T = 1.6 million * (M * D-to-the-fourth) / (m * R-cubed)
Satellite Orbital Period: P = 0.0588 * square root of (R-cubed/(M+m))

T: tidal forces; 1 represents Earth-like tides.
P: orbital period of satellite in days
M: mass of satellite in Earth masses
m: mass of planet in Earth masses
D: diameter of planet in Earth diameters
R: radius of satellite's orbit in Earth diameters

Solar tides: T = 0.3 * (M * D-to-the-fourth) / (m * R-cubed)
Planetary Orbital Period: P = square root of (R-cubed/M)

T: tidal forces; 1 represents Earth-like tides.
P: orbital period in years
M: mass of star in solar units
D: diameter of planet in Earth diameters
m: mass of planet in Earth masses
R: radius of planet's orbit in AUs.

I've included the associated orbital period calculations here because they become important later on.

Originally Posted by Agemegos

In the case of an object under the tidal influence of more than one neighbour, add the T values together for a total (maximum) tidal range (ie. spring tides). But keep a record of the separate T values, because you will need them in the next step.

To determine the rotation rate of a planet or moon, roll 3d and divide by 120. The result is the initial rate of rotation of the body. Call this value I.

Now square each of the T values to which this world is subject, and add all of the squares together. Call this value S. Then calculate the present rate of rotation, w, as follows.

W = I - [(0.016 * A * P/D^5) * S]

You skimmed over the fact that planets generally have axial tilts, and thus the torque applied by tidal forces would tend to fight the axial tilt as well as the rotational speed. Because of this, planets that start with high axial tilts will tend to have their axial rotation slowed less than planets that start with low axial tilts. Also, it means that older systems will tend to have smaller axial tilts than younger systems will.

Likewise, the same tidal forces that tend to slow a planet's rotation will also tend to drive satellites into higher orbits, eventually letting them break away from the planet. Younger systems will tend to have more satellites and in tighter orbits than older systems will.

Originally Posted by Agemegos

If W is zero or less, the planet or moon is tide-locked, and its day-length is equal to its orbital period. Otherwise, the period of the planets' (or moon's) rotation is

Daylength =1/W,

in hours. If this is longer than its orbital period, the planet or moon is tidelocked. Change its period of rotation to be equal to its orbital period.

First: Tidal forces don't drive W toward zero. Rather, solar tides drive W toward 1/(8766 P), where P is the Planetary Orbital Period in years; and lunar tides drive W toward 1/(24 P), where P is the Satellite Orbital Period in days. If W is at or between these values, the various tidal forces will be in competition. In other words, a satellite will end up delaying or preventing a solar tide-lock, depending on how powerful the satellite's tides are.

Of course, the solar tides will be working to drive the satellite away from the planet at the same time that they're working to tide-lock the planet, with the twin side effects of lengthening the Satellite Orbital Period and weakening the lunar tidal effects. It may well be that planets never reach a point where lunar "tidal braking" puts up any significant opposition to solar tidal braking, because the solar tides will have driven the potential competition away first. If this is the case, it's likely that tide-locked worlds can't have satellites larger than asteroids, if that. Regardless, a tide-locked world will not be able to have a satellite with tidal forces equal to or stronger than the star's; the planet would have tide-locked to the satellite instead of the star.

Second, a nitpick: Daylength doesn't equal 1/W; Rotational Period equals 1/W. The length of a day is computed from the Rotational Period on page 118, under "Local Calendar".

[Agemegos]

Originally Posted by dataweaver

Note that P is used elsewhere to represent orbital periods; so you should probably choose another letter to represent the mass of the object that the tides are on. I'd be inclined to use a lowercase m.

Fair enough. I just have a paranoid fear of typographers' changing a lowercase m to an uppercase M for appearance's sake. It would be just like them to do so because the editor had declared the style of the work to use capitals in all formulas, and to do so without consulting the author and without realising that it profoundly changed the meaning of the equation.

Traumatic experiences….

You skimmed over the fact that planets generally have axial tilts, and thus the torque applied by tidal forces would tend to fight the axial tilt as well as the rotational speed. Because of this, planets that start with high axial tilts will tend to have their axial rotation slowed less than planets that start with low axial tilts. Also, it means that older systems will tend to have smaller axial tilts than younger systems will.

True, I did. I was thinking of correcting the erroneous formulas in Space rather than of extending its scope.

I don't think they are going to accept a corrigendum that increases the page count.

However, it might be possible to decrease the axial tilt of planets with a high value of A * P/D^5) * S without making the rules on p. 118 too complicated.

Likewise, the same tidal forces that tend to slow a planet's rotation will also tend to drive satellites into higher orbits, eventually letting them break away from the planet. Younger systems will tend to have more satellites and in tighter orbits than older systems will.

True. Perhaps you would like to suggest corrigenda and upgrades for the 'Satellite Orbital Radius' procedures on p 116.

First: Tidal forces don't drive W toward zero.

Indeed they don't. I just felt that the easiest way to right the rules was to calculate a temporary value for W, calculate the implied period, and override it if this implied retrograde rotation. W isn't used subsequently for anything else, so there is no need to correct it.

Rather, solar tides drive W toward 1/(8766 P), where P is the Planetary Orbital Period in years; and lunar tides drive W toward 1/(24 P), where P is the Satellite Orbital Period in days. If W is at or between these values, the various tidal forces will be in competition. In other words, a satellite will end up delaying or preventing a solar tide-lock, depending on how powerful the satellite's tides are.

True. I simply feared that I could not write a rule to sum this up that was sufficiently simple.

Second, a nitpick: Daylength doesn't equal 1/W; Rotational Period equals 1/W. The length of a day is computed from the Rotational Period on page 118, under "Local Calendar".

True. I ought to have used 'S' (sidereal period) rather than 'Daylength'. Which means I shouldn't have used 'S' for that sum of squares value.

[dataweaver]

Originally Posted by Agamegos

Fair enough. I just have a paranoid fear of typographers' changing a lowercase m to an uppercase M for appearance's sake. It would be just like them to do so because the editor had declared the style of the work to use capitals in all formulas, and to do so without consulting the author and without realizing that it profoundly changed the meaning of the equation.

Traumatic experiences….

True enough. The other alternative would be a subscripted capital M - something like Mplanet.

Originally Posted by Agemegos

True, I did. I was thinking of correcting the erroneous formulas in Space rather than of extending its scope.

I don't think they are going to accept a corrigendum that increases the page count.

However, it might be possible to decrease the axial tilt of planets with a high value of A * P/D^5) * S without making the rules on p. 118 too complicated.

I don't think you're going to be able to get any changes into the book, aside from "we said X when we meant to say Y". Other kinds of changes are the sorts of things that ought to have been addressed during the playtest, and accepting them would essentially place the "final" product in a state of perpetual playtest. With e-books, this would be less of a concern; but with hardcover books, you really don't want different printings (as opposed to different editions) to differ substantially.

I think that the best you're going to manage will be to provide online house rules.

Originally Posted by Agemegos

True. Perhaps you would like to suggest corrigenda and upgrades for the 'Satellite Orbital Radius' procedures on p 116.

Unfortunately, I don't know enough about the details of the subject (i.e., the actual numbers) to do so.

Originally Posted by Agemegos

Indeed they don't. I just felt that the easiest way to right the rules was to calculate a temporary value for W, calculate the implied period, and override it if this implied retrograde rotation. W isn't used subsequently for anything else, so there is no need to correct it.

Note that the rules as written don't say anything about the rotational speed being zero; they talk exclusively about stopping just shy of retrograde motion.

Originally Posted by Agemegos

True. I simply feared that I could not write a rule to sum this up that was sufficiently simple.

Use the stellar evolution rules as guidelines:

First, determine how long it will take for the planet's rotation to become synchronized with the satellite. If the age of the system is less than that, determine the actual rotational period by finding the total change in rotational speed over the full period and applying the same fraction of that as the fraction of the full period that has elapsed. (See the rules for determining a star's luminosity for an example of how to do this.)

If the age of the system is greater than this point, use the same formula to determine how much longer it will take for the planet's rotational period to match the primary's rotational period, but subtract the square of the satellite's T from the square of the primary's T instead of adding it. If the difference of the T-squares is zero or negative, then the planet is tidally locked to the satellite. If it's positive, repeat the above age check to see if the planet has had time to become tide-locked to the sun; and if not, interpolate (as above) to determine what its current rotational speed is.

If you have multiple satellites, arrange them in order of their orbital periods, from shortest to longest. As the planet's orbital period passes each one, move its T-squared from the plus column to the minus column and repeat the process until you reach the current age of the system, until the sum in the minus column equals or exceeds the sum in the plus column (at which point the planet is tidally locked to the last satellite to move into the minus column), or until all of them are in the minus column (at which point you go to the final step of determining solar tide-locking as above).

The result is an iterative process, with one iteration for each satellite and a final iteration for the primary. Lengthy and potentially repetitive, but not particularly complicated.

Originally Posted by Agemegos

True. I ought to have used 'S' (sidereal period) rather than 'Daylength'. Which means I shouldn't have used 'S' for that sum of squares value.

'R', actually. The Local Calendar section is clear that it uses 'R' to mean the rotational period of the planet, with A meaning the apparent period of a cycle and S representing the sidereal period of the cycle. In particular, "day length" is found by setting S equal to the planet's orbital period, while "month length" is found by setting S equal to the satellite's orbital period.

Note that the Local Calendar section tends to break the rule-of-thumb of using the same letter to mean the same (sort of) thing throughout: it uses 'R' to mean "Rotational Period" instead of "Radius" and 'A' to mean "Apparent Period" instead of "Age". So the fact that you're using S to mean something not related to what the Local Calendar uses it for isn't a major crime. Given that, I'd be more inclined to use 'P' (some sort of period) instead of 'Daylength', rather than 'R' or 'S'.

[Agemegos]

Originally Posted by dataweaver

True enough. The other alternative would be a subscripted capital M - something like Mplanet.

Yeah, but SJG's typographers seem to have vetoed subscripts and superscripts. And root signs.

I'm getting flashbacks. I used to work for a freaking economic research bureau where the typesetters tried to stop the researchers from publishing equations in standard format.

I don't think you're going to be able to get any changes into the book, aside from "we said X when we meant to say Y". Other kinds of changes are the sorts of things that ought to have been addressed during the playtest, and accepting them would essentially place the "final" product in a state of perpetual playtest. With e-books, this would be less of a concern; but with hardcover books, you really don't want different printings (as opposed to different editions) to differ substantially.

Ideally, we'd like to simply replace some formulas and maybe table entries. I don't see an easy way to do it regarding rotational period, though there is space to save by cutting that table that says that bigger planets start out rotating faster.

I think that the best you're going to manage will be to provide online house rules.

Well, if SJG isn't interested in providing a fix I guess I'll keep my house rules to myself.

[Pomphis]

Originally Posted by Agemegos

Well, if SJG isn't interested in providing a fix I guess I'll keep my house rules to myself.

You know, there are more people in the world than just you and steve. :-)
And some of us others may lack the skills to do such stuff on our own, but are still interested and would appreciate it if we could read about it.

[Agemegos]

Originally Posted by Pomphis

You know, there are more people in the world than just you and steve. :-)

Indeed. But this is not a very efficient way to reach them. I'd feel a bit more encouraged if I thought my efforts were going to result in an on-line errata page, or one of those errata sheets that publishers used to slip inside the covers of printed and bound books.

One of the roots of my dissatisfaction is that I am not a physicist but an economist, so work on orbital ballistics, tidal braking, and the climatic meteorology of tide-locked worlds is well outside my comfort zone. I do my best, but I'm very conscious that I might make errors. And I have about ten or fifteen points worth of psychological disadvantages having to do with my loathing of letting people be misled and misinformed. I hate to think that I might be publishing errors, and no-one will do so much as to check my algebra.

[sir_pudding]

Originally Posted by Agemegos

Indeed. But this is not a very efficient way to reach them. I'd feel a bit more encouraged if I thought my efforts were going to result in an on-line errata page, or one of those errata sheets that publishers used to slip inside the covers of printed and bound books.

Have you submitted this to errata@sjgames.com? So that Andy can check with Jon and then maybe it will get on the errata page for Space. Just grousing about it on the forum is not a way to get things done. If you fail to use the proper channels you have no right to complain when you fail to get results.

[Agemegos]

Originally Posted by sir_pudding

Have you submitted this to errata@sjgames.com?

Yes, I did so on the 18th of August last year.

[sir_pudding]

Originally Posted by Agemegos

Yes, I did so on the 18th of August last year.

Very well. I must have misunderstood your post. It seemed you were complaining that nothing was changed based on this thread. Sorry if I was rude.

[Agemegos]

Originally Posted by sir_pudding

Sorry if I was rude.

No problem.