copyright 2002 by Linda "Sweetwind" Tam
with research assistance from Howard "Kir" Yune
Contents:
In the last issue of Sendings, I laid the groundwork for an understanding of what tides might be like on Abode, the World of Two Moons. I discussed the concept of harmonic analysis, and created some tide charts for Earth, using harmonic analysis of only three constituents, which looked pretty similar to actual Earth tide charts. Today I'll apply harmonic analysis to Abode and we'll see what we get with four constituents. Remember, in the analysis for Earth we used one constituent for the Moon's tide-raising effect: the Principal Lunar Semidiurnal Constituent. For Abode, we'll need a constituent like that for Mother Moon as well as for Daughter Moon - that's why Abode takes four main constituents instead of Earth's three. Since the word "lunar" in the constituent's name comes from Latin, and means "pertaining to the moon," I replaced it with the Latin derived words "maternal" and "filial" (pertaining to the mother and to the child, respectively) to create the names of the two Abodean lunar constituents: the Principal Maternal Semidiurnal Constituent and the Principal Filial Semidiurnal Constituent.
For the portion of the tides raised by the Daystar, I'll still use the term Principal Solar Semidiurnal Constituent, and for the portion of the tides due to a locality's declination, I'll still use the term Lunisolar Diurnal Constituent, as on Earth.
Now for the big question: what will these four constituents look like? I can calculate the ones for Earth, because I know the masses of the Earth, the Sun, and the Moon and their distances from each other. I don't have any of that information about the Abodean system. I'll have to make some guesses. I listed all of the guesswork I did in the "Assumptions" sidebar (and now would be the perfect time for you to go and read it. Go ahead, I'll wait...).
Given my Assumption #1, the Principal Solar Semidiurnal Constituent will be exactly the same amplitude (the height of the waves, up and down, on the chart) and "speed" (left-to-right size of the waves on the chart) for Abode as it is for Earth. Furthermore, the Principal Maternal Semidiurnal Constituent will have the same amplitude as Earth's Principal Lunar Semidiurnal Constituent, and given Assumption #7, it will have nearly the same speed.
The Lunisolar Diurnal Constituent on Abode, as on Earth, depends upon the location where you want to predict the tides. It's different for Seattle than for San Diego; different for Blue Mountain than for the Forevergreen. I'll just use the same one that I used for Poughkeepsie in Part 1.
Well, that takes care of everything except the Principal Filial Semidiurnal Constituent. As usual, I have procrastinated the toughest one for last ;-). Actually, determining the speed is easy, given Assumption #7; it's finding the amplitude of the constituent that will take some work. I can calculate it in terms of the amplitude of the Principal Maternal Semidiurnal Constituent, if I know the relative mass and distance of Daughter Moon compared to Mother Moon. That's where Assumptions #2 through #6 come in.
First, the matter of the moons' size. Since I assume that Daughter Moon is about half the size
of Mother Moon in Abode's sky (#5), does that mean that Daughter Moon's diameter is half
Mother Moon's diameter? No, because Daughter Moon is closer than Mother Moon. For Daughter
Moon to appear about half the size of Mother Moon, it must actually be much smaller than
half the size. Imagine standing on Abode's surface and looking at two full moons. (See
Figure 1. That's supposed to be an eyeball on the left, there.)
Now, as you learned when you
were just a cub, the diameter of Mother Moon is given by the equation
dM = ( n/360 ) 2 Pi D
(This is based on the fact that a circle's circumference is equal to 2 Pi times the radius.
For a snippet of the circle of the Abodean sky, we just multiply the entire circumference
by the ratio of the snippet to the whole, or n/360 degrees.)
Similarly, the diameter of Daughter Moon is
dD = ( nY/360 ) 2 Pi D X
Since all we changed in the equation to get Daughter Moon's diameter was to multiply by Y
(to account for the apparent size) and X (to account for the distance), it follows that
Daughter Moon's diameter is XY times Mother Moon's diameter. Since XY = 1/2 times 0.54,
Daughter Moon's diameter is just 0.27 times that of Mother Moon.
Now, that's a wee little moon! Suppose for a moment that Mother Moon was the same size as Earth's Moon, that is, 3,477 kilometers (2,160 miles) in diameter. Then Daughter Moon's diameter would be a mere 939 km (583 mi), making it about the size of the asteroid Ceres in Earth's Solar System. It would have a total surface area comparable to Argentina! To get the mass of Daughter Moon, we multiply the volume by the density. The volume of a sphere (Assumption #3) decreases as the cube of the diameter, so Daughter Moon's volume is X3Y3 times Mother Moon's. Multiply by the relative density, which we called Z (Assumption #2), and we see that Daughter Moon's mass is X3Y3Z times that of Mother Moon.
The comparative tide-raising force of two bodies can be calculated by dividing the respective
masses by the cube of the respective distances. Say Mother Moon has a mass of 1 and is at a
distance of 1, then 1/13 = 1/1 = 1. Daughter Moon, therefore, with a mass of
X3Y3Z and a distance of X, yields
tidal force = X3Y3Z/X3 =
X3Y3Z/X3 = Y3Z
Hey, that's pretty cool. In calculating the tidal force, Daughter Moon's distance completely
drops out! The tidal force is dependent only on the apparent diameter and the density. Of course,
the apparent diameter is related to the distance: the farther from Abode Daughter Moon is, the
larger its actual diameter must be to maintain the same apparent diameter. The fact that the
distance drops out of the tidal force expression (seen above) tells us that any increase in
tidal force due to increased moon size (as we postulate Daughter Moon to be farther and farther
away) is exactly compensated for by a decrease in tidal force due to the greater distance from
Abode.
So, the relative tidal force depends only on the moons' apparent diameter, and to a much lesser
extent on their density. This makes Assumption #5 all the more crucial. How does the tide-raising
force of Daughter Moon vary depending on the relative apparent diameter? Figure 2 shows the
resulting tidal force as relative apparent diameter goes from zero to 100% (sticking with
Assumption #2 and letting Z = 1).
As you might expect, at 100% of Mother Moon's apparent
diameter, Daughter Moon would have an equal tidal force. But the tidal force drops off rapidly
as the diameter decreases, giving only 12.5% tidal force at 50% of the diameter. Note that, to
have half the tidal force of Mother Moon, Daughter Moon would have to be eight-tenths the apparent
size of Mother Moon! Assumption #5 is so important that I asked my doughty research assistant,
"Kir, the last Go-Back", to compile a
data sheet of actual apparent diameters
of the two moons
when they appear together in a single panel of the comic. I asked him to use only Wendy Pini
drawings, since she is by definition closest to model :-). The ratio varies from 0.47 to 0.6,
with an average Y = 0.54.
So let's stick the numbers from the assumptions into the tidal force expression: Y3Z is (0.54)3 times 1, which is 0.157. The amplitude of the Principal Filial Semidiurnal Constituent, then, will be 15.7% of the Principal Maternal Semidiurnal Constituent's amplitude. That's much less than the tide-raising force we have attributed to the Daystar, which is almost half (46%) that of Mother Moon. The speed (period) will be shorter, since Daughter Moon goes from full to new and back again much faster than Mother Moon does.
Now it's time for some figures. Figures 3, 4, 5, and 6 show the four Abodean constituents - the tidal effects due to Mother Moon, Daughter Moon, the Daystar, and the local declination, respectively, over a two-day period. Figures 7 and 8 are the real money shots. Figure 7 is a tide chart for Earth. You can clearly see how the height of the tides varies smoothly as the Moon goes from new to full and back again. On Earth, this simple 29.5-day cycle repeats indefinitely. Figure 8 is a tide chart for Abode. Since the Abodean tides represent a combination of a 28-day cycle and a 10-day cycle, the cycle doesn't repeat for 140 days (the least common multiple of 28 and 10).
Let's talk about Assumption #2 some more, since it is the only assumption affecting the magnitude of Daughter Moon's tidal force that has some "wiggle room." Imagine again our Moon and Ceres playing the roles of Mother Moon and Daughter Moon: unfortunately, this gives us a value of less than one for Z, since our Moon's average density is 3.3 grams per cubic centimeter, while Ceres' is 2.0 g/cc. It won't be very interesting to make Daughter Moon less dense than Mother Moon. That would just make its tidal force even less, and the tide charts will look more like Earth's. Let's see what happens as we make Daughter Moon denser. The maximum reasonable density I would postulate for Daughter Moon is twice Mother Moon's, if Mother Moon is primarily rocky and Daughter Moon is primarily metallic. Figure 9 illustrates how the tidal force changes as Daughter Moon's density increases to that limit. It's linear - that is, doubling Daughter Moon's density doubles its tidal force. Letting Z = 2 produces a Principal Filial Semidiurnal Constituent with 31.5% of the amplitude of the Principal Maternal Semidiurnal Constituent, as shown in Figure 10. Figure 11 is an Abodean tide chart generated using this constituent. This is much more satisfyingly exotic, to my eye.
You might wonder, by the way, whether the Y3Z tidal force expression applies to the Daystar as well. As a matter of fact, it does! The reason our Sun's tidal force is 46% of the Moon's, when both have about the same apparent diameter, is because the Sun's density is 1.5 g/cc, or about 46% of the Moon's. So Assumption #1 could be also be expressed as "The Daystar and Mother Moon have equal apparent diameters, and the same density ratio as the Sun and Earth's Moon."
Ocean tides on Earth can be as high as 12 meters (40 feet) depending on local geography. The addition of a second principal lunar constituent (with Z=1) adds about 10% to the tidal forces affecting Abode compared to those affecting Earth, so we might say that Abode's tides could be as high as 13 meters (44 feet). (With Z=2, that increases to 20% over Earth, or 15 meters [48 feet].) The magnitude of the tides, then, is not wildly different from ours on Earth. The primary difference is a smearing of the pattern of the tide's height changes from day to day. The tide charts shown here would not apply everywhere on Abode, of course (remember the Long Beach tide chart from Part 1, showing how the diurnal factors can dominate!). Local geography and other factors affect the tides for any given location. And always remember that my assumptions are just that: assumptions. So enjoy the charts; just don't try to sail your "waterleaf" by them.
A few words, before I close, on scientific understanding of the tides: Until the late seventeenth century here on Earth, the cause of tides was not understood. Towards the beginning of that century no less a scientist than Galileo Galilei, esteemed for his clear thinking, *ridiculed* Johannes Kepler (no slouch himself) for hypothesizing that the Moon caused tides. (Galileo theorized that water sloshing around due to the Earth's rotation caused the tides. Nice try, but no cigar.) It wasn't until 1687, when Isaac Newton published his discovery of the laws of gravity, that the forces involved could finally be worked out, and the Moon's influence was accepted as the cause of the oceans' tides. With the extra noise in the changing heights of the tides, the puzzle was probably even more difficult to solve on Abode.
Well, that about wraps up the topic of Abodean tides. If I generate even one more tide chart I will turn into a "psychic-burnout catatonic nut case." (The hardcore EQ fan will get this reference, but if you don't, try entering it into a search engine at www.elfquest.com and see what pops up.) It will be back to dinosaurs in the next issue. Nice, friendly carnivorous dinosaurs with no numbers or equations attached!
Elfquest, its logos, characters, situations, all related indicia, and their distinctive likenesses are trademarks of Warp Graphics, Inc. All rights reserved.
© 2002
linda_tam@alumni.hmc.edu
Posted on January 5, 2003
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