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AstronomyAnswerBook: Tides


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1. Conservation Laws ... 2. Evolution ... 3. Model Results ... 4. The past ... 5. The Near Future ... 6. The Very Distant Future ... 7. High Tide and Low Tide ... 8. Spring Tide and Neap Tide ... 9. Average Time Between Successive High Tides/Low Tides

This page answers questions about tides and tidal forces. The questions are:

[503]

The Moon recedes from the Earth at an average rate of 3.7 cm per years. It takes energy to get further from the Earth, so apparently the Moon is getting energy from somewhere. How does that work?

(See the Calculation Page about the Evolution of the Orbit of the Moon if you want to know how I did the calculations upon which this text is based.)

1. Conservation Laws

The Law of Conservation of Angular Momentum says that the total amount of angular momentum in a closed system is constant. The Law of Conservation of Energy says that the total amount of energy of a closed system is constant. This means that energy or angular momentum cannot just appear or disappear: they must always come from somewhere else or go to somewhere else.

Friction is an important cause of the conversion of kinetic energy (tied to directed motion) to heat. If you brake your bicycle or car then you cause friction between the brakes and the wheels, which reduces the speed difference between the brakes and the wheels and heats up the brakes and wheels. The kinetic energy has then been converted into heat in the brakes and wheels, which disappears into the surroundings, partly in the form of heat radiation. It is a lot more difficult to convert heat into kinetic energy.

The Earth and the Moon exert tidal forces on each other. Tidal forces occur when (slightly) different forces of gravity work on different parts of an object. The tidal forces between the Earth and the Moon try to make the spin periods of the Earth and the Moon equal to each other and equal to the orbital period of the Earth and the Moon. The spin period of the Moon is already equal to its orbital period, but the spin period of the Earth (the sidereal day) is still a lot shorter than the orbital period (the sidereal month). The tidal forces are gradually slowing down the rotation of the Earth.

If the spin rate of the Earth decreases, then the angular momentum and kinetic energy due to the spin of the Earth also decrease. The differences must have gone somewhere else, because the conservation laws say that energy and angular momentum can't just disappear. A part of the "lost" energy is converted into heat by friction, and part (all?) of that heat disappears into space as infrared radiation.

Heat radiation carries energy but no angular momentum, so energy can but angular momentum cannot disappear into space that way. The angular momentum that disappears from the spinning of the Earth can only go into the spin of the Moon or to the orbital motion of the Earth and Moon.

Tidal forces have caused the spin period and orbital period of the Moon to be equal (the Moon always shows us the same side), so they can keep them equal, too. The orbital period follows from the distance between the Earth and the Moon, so then there are only two free parameters left: the spin period of the Earth and the distance between the Earth and the Moon.

If the spin period of the Earth increases, then its angular momentum decreases, which means that the orbital angular momentum of the Moon (which is far greater than the spin angular momentum of the Moon) must increase, which means that the distance of the Moon must increase.

Fig. 1: Periods in the Earth-Moon System
Fig. 1: Periods in the Earth-Moon System

Figure 1 shows which spin period \(T_1\) of the Earth and orbital period \(P\) (along the vertical axis) of the Earth and Moon belong to each distance \(r\) (along the horizontal axis) between the Earth and the Moon. The solid line shows the sidereal spin period of the Earth, and the dotted line the sidereal orbital period of the Earth and the Moon (which is also equal to the sidereal spin period of the Moon). The little squares show the current situation. The triangles show the points at which the orbital period is equal to the spin period of the Earth. These are for \(r = 13938 \text{ km}\) (period equal to 4 hours and 50 minutes) and \(r = 554523 \text{ km}\) (period equal to 47 days 7 hours).

The spin period \(T_1\) of the Earth is increasing, because the spinning of the Earth is slowing down due to the tidal forces. The figure shows that this is only possible if the distance \(r\) between the Earth and the Moon increases.

Fig. 2: Energy in the Earth-Moon System
Fig. 2: Energy in the Earth-Moon System

Figure 2 shows which total energy \(E\) is in the Earth-Moon system if the Earth and Moon are at distance \(r\) from each other, if the Moon shows always the same side to the Earth, and if in addition the spin period of the Earth is as shown in figure 1 for that distance. The solid line goes with the scale on the left-hand side, and the dotted line with the scale on the right-hand side, which has been expanded by a factor of 10 compared to the left-hand scale. The squares show the current situation. The triangles show the points at which the orbital period is equal to the spin period of the Earth, just like in figure 1.

It is very difficult to convert heat or other internal kinds of energy into kinetic energy or into potential energy, so the total amount \(E\) of kinetic and potential energy in the Earth-Moon system can in practice only decrease. We saw before that the decelerating spin rate of the Earth goes with an increasing distance between the Earth and the Moon, and we see here that declining total energy goes with that, too, and that fits. The potential energy increases (because the Moon gets further from the Earth), but the total kinetic energy in the rotation and orbital motion of the Earth and in the rotation and orbital motion of the Moon decreases more, so the sum of kinetic and potential energy still decreases as expected.

2. Evolution

Orbits of moons and planets tend to be influenced by the following effects:

  1. The tidal distortion of the planet by the moon lags behind the direction to the moon and causes a torque that makes the orbital period of the moon and the spin period of the planet more equal.
  2. Because angular momentum is conserved, the change in angular momentum in the planet's spin is reflected in an opposite change in angular momentum in the orbit or in the moon's spin. If the spin period is shorter than the orbital period (as it usually is), and if the moon rotates synchronously with its orbit (as is usually the case) then the planet's spin is slowed down by the moon's tides, and the moon's orbit gets wider.
  3. Because the tidal effects drop off strongly with distance, they are strongest when the moon is closest to the planet and affect the moon's apoapse more than its periapse, leading to an increase in eccentricity.
  4. If the change in the planet's spin rate is slow enough, then the planet's spin axis moves toward a situation in which it is perpendicular to the plane of its orbit around the Sun.
  5. Even when a moon is locked in synchronous rotation with its orbit, it may experience tidal effects. If the orbit is not perfectly circular, then the moon's distance to the planet varies with time, leading to slight flexing of the moon and energy loss. This tends to make the orbit more circular.
  6. In addition, a moon travels a non-circular orbit with varying speed, leading to librations (apparent rocking to and fro as seen from the planet), which generate flexing and energy loss. This, too, tends to make the orbit more circular.

These descriptions in terms of planets and moons also hold between the Sun and the planets.

In the Earth-Moon system, if the current rate at which the Earth's rotation period increases (about 20 seconds per million years) were to stay the same until the Earth's rotation period and the length of the month synchronized, then it would take (46 days divided by 20 seconds times a million years equals) about 200 thousand million years to reach synchronization, which is about 15 times longer than the current estimate for the age of the Universe.

Because the Moon is slowly moving away from the Earth its tidal force on the Earth (and therefore also the deceleration of the Earth's rotation) decreases with time, and I expect the deceleration to also become smaller when the rotation period and orbital period are more closely matched. In addition, the efficiency of tidal forces in slowing down the rotation of the Earth is currently much greater than usual (see section 4). It will probably take much longer than 200 thousand million years before the day and the month are synchronized.

3. Model Results

Fig. 3: Evolution of the Distance of the Moon
Fig. 3: Evolution of the Distance of the Moon
I have constructed a simple model for the greatest contribution to the tides on the Earth. The evolution of the lunar orbit, day, and month according to that model is illustrated by the following series of figures. Figure 3 shows the distance \(r\) of the Moon (in units of 1 Mm = 1000 km = 621 mi) for time \(t\) in units of millions of years since today. Today's situation is shown by the small square. The solid line fits with the current recession speed of the Moon (model A, 3.7 cm per million years today), and the dashed line goes with a reduced recession speed (model B, 1.1 cm per million years today).

Fig. 4: Evolution of the Length of the Sidereal Day
Fig. 4: Evolution of the Length of the Sidereal Day

Figure 4 shows how the length \(t_1\) (measured in hours of today) of the day depends on the time \(t\). The solid line shown the length of the sidereal day for model A, the dashed line the sidereal day for model B, and the dotted lines the corresponding lengths of the synodical day (solar day). Today's situation is again shown by small squares.

Fig. 5: Evolution of the Length of the Sidereal Month
Fig. 5: Evolution of the Length of the Sidereal Month

Figure 5 shows how the length \(P\) of the month depends on the time \(t\). The solid line shown the length of the sidereal month for model A, the short-dashed line shows the sidereal month for model B, and the dotted lines show the length of the corresponding synodical months (from Full Moon to Full Moon), all of them measured in units of 1 solar day of today (86400 seconds). The long-dashed lines show the length of the synodical months measured in units of the solar days of that time. Today's situation is again shown by small squares. The solar day increases in length relatively faster than the length of the synodical month does, so the number of solar days in a synodical month decreases in the future, even though the length of the synodical month measured in a fixed unit of time decreases in the future.

4. The past

Geological research has shown that some lunar rocks are about 4000 million years old, and no indications have been found that the Moon has had any traumatic experiences (such as a close encounter with the Earth) during the last 3000 million years or so. However, according to model A (Figure 3), the Moon was practically touching the Earth about 1400 million years ago. These things don't fit, because according to geology the Moon hasn't been close to the Earth during the last 3000 million years.

The explanation for this problem is that the tides are currently unusually efficient at slowing down the spinning of the Earth, because at the moment the spin period of the Earth is quite close to the main sloshing period of the Earth's oceans. Far into the past and far into the future the tides are less efficient at slowing down the Earth. In model B the tides are about 0.3 times as efficient as in model A, and then the Moon can remain sufficiently far from the Earth even when they were formed (estimated at about 5000 million years ago).

5. The Near Future

The figures show that, according to the model, during the coming 6000 million years, the Moon will recede to about 470 thousand kilometers from the Earth (about 90 thousand kilometers more than today), the day will lengthen to about 47 hours, and the month will grow to about 36 days sidereal and 40 days synodical. The synodical month (currently about 29.5 days long) will be exactly 30 current days long in about 110 million years, and will have a length of exactly 29 solar days of the time in about 240 million years.

Because the tides are currently temporarily unusually efficient at slowing down the rotation of the Earth, the changes in the lengths of the day and the month will probably go a lot slower than model A suggest. Model B could be a better fit in the long term, and it says that the Moon will get to only 425 thousand km during the next 6000 million years, and the day gets only up to 32 hours long, and the month up to 32 days sidereal and 35 days synodical. According to model B, the synodical month will be exactly 30 current days long in about 386 million years, and exactly 29 future solar days long in about 840 million years. Clearly, the Earth's rotation will not synchronize with the Moon's orbital period before the Sun turns into a red giant (which is estimated to be in about 5 thousand million years).

6. The Very Distant Future

Fig. 6: Evolution of the Month in the Far Distant Future
Fig. 6: Evolution of the Month in the Far Distant Future
If the Sun never becomes a red giant, and the Earth and Moon can continue plying their orbit around the Sun forever like they do today, then we can extend models A and B far beyond 6000 million years. Figure 6 shows the number of solar days \(P\) in a synodical month (from Full Moon to Full Moon) displayed against time \(t\) measured in millions of years since today, for models A (solid line) and B (dashed line). The solar day and synodical month get ever closer together (\(P\) decreases towards the value 1), but only after about 84 thousand million years (according to model A) or 295 thousand million years (model B) does the synodical month decrease to below 2 solar days.

The dotted lines in Figure 6 belong to approximations

\begin{align} P_A & ≈ 85 - 7.5 \ln(t) \\ P_B & ≈ 94.4 - 7.5 \ln(t) \end{align}

if \(t\) is measured in millions of years after today, and \(P_A\) and \(P_B\) are measured in solar days of the future.

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7. High Tide and Low Tide

There are usually two high tides and two low tides each day because the forces due to the gravity of the Earth and Moon pull water towards the point on Earth that is closest to the Moon and also towards the point on Earth that is furthest from the Moon, so there are two high tide regions: one on the side that faces the Moon, and one on the opposite side; and there is a low tide region that circles the Earth in between those two high tide regions. It takes roughly a day for the Moon to return to about the same position in the sky, so during that day the two high tides and the two low tides in between the two high tides pass through your location.

The tides are the result of two forces. The first of those two forces is the force of gravity from the Moon, which pulls the water toward the Moon. Let's call the point of the Earth that is nearest to the Moon point N, and the point of the Earth that is furthest from the Moon point F, and the center of the Earth point C. C lies midway between N and F. N is closest to the Moon and F is furthest from the Moon, so the force of gravity from the Moon is greatest at point N and weakest at point F.

The second of those two forces is the centrifugal force due to the motion of the Earth around the common barycenter (center of gravity) of the Earth and the Moon. People often say that the Moon orbits around the Earth, but actually both the Earth and the Moon orbit around their common barycenter. The orbit of the Earth around the barycenter is much smaller than the orbit of the Moon around that barycenter, because the Earth has so much more mass than the Moon does. That's why most people think that the Moon doesn't move the Earth at all, but it does.

On a merry-go-round the centrifugal force pushes you away from the center more strongly if you are further away from the center, and the same goes for the centrifugal force linked to the barycenter of the Earth-Moon system. The barycenter of the Earth-Moon system (let's call it point B) lies between the center of the Earth and the center of the Moon, but much closer to the center of the Earth than to the center of the Moon. It lies between points N and C. So, the order of the points is N, B, C, F. The centrifugal force is greatest at F (which is furthest from barycenter B) and least at N (which is nearest to B), and points away from B.

At the center of the Earth (C), the force of gravity from the Moon and the centrifugal force are equally strong but point in opposite directions, so they even out and the Earth (on average) stays at the same distance from the Moon.

At point N, the gravity from the Moon is stronger than at C, because point N is closer to the Moon than point C is. The centrifugal force always points away from B, so at point N the centrifugal force points toward the Moon, just like the gravity of the Moon does. So there those two forces add up and point toward the Moon, which means away from the center of the Earth, which leads to a high tide at point N.

At point F, the gravity from the Moon is weaker than at C, because point F is further from the Moon than point C is. The centrifugal force is stronger than at C, because point F is further from the barycenter B than point C is. At C the centrifugal force and the gravity were equally strong, so at F the centrifugal force is stronger than the gravity of the Moon. The centrifugal force at F points away from the Moon, and the gravity at F points toward the Moon and is weaker than the centrifugal force, so the gravity cannot fully balance the centrifugal force, and some net centrifugal force remains that points away from the Moon, which means away from the center of the Earth, which leads to a high tide at point F.

So there is one high tide on the side of the Earth nearest the Moon, because there both the gravity from the Moon and the centrifugal force from the barycenter point toward the Moon, and there is one high tide on the side of the Earth furthest from the Moon, because there the centrifugal force from the barycenter (pointing away from the Moon and from the center of the Earth) wins from the gravity of the Moon (pointing toward the Moon and to the center of the Earth).

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8. Spring Tide and Neap Tide

When it is New Moon, then the Moon and the Sun are on the same side of the Earth, so then the high tides from the Moon and the high tides from the Sun add up to high tides that are higher than usual, and the low tides from the Moon and the low tides from the Sun add up to low tides that are lower than usual. Those are called spring tides.

When it is Full Moon then the Moon and the Sun are on opposite sides of the Earth, so then the high tides from the Moon (of which there are two: one on the side facing the Moon and one on the side facing away from the Moon ― see question 579) and the high tides from the Sun (of which there are two as well: one on the side facing the Sun, which is then the side facing away from the Moon, and one on the side facing away from the Sun, which is then the side facing the Moon) add up to spring tides.

When it is First Quarter or Last Quarter, then the high tides from the Moon are at right angles to the high tides from the Sun, so the high tides from the Moon coincide with the low tides from the Sun, and the high tides from the Sun coincide with the low tides from the Moon, so then the tides from the Sun (which are weaker than those from the Moon) cancel part of the tides from the Moon, which leads to neap (= weaker than average) tides.

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9. Average Time Between Successive High Tides/Low Tides

The average time between successive high tides (about two per day) and low tides (two per day) is tied to the average time between success moonrises/transits/sets (one per day). See question 588 for that average time. The tides are caused by the Sun and the Moon, but for the tides the Sun is only half as important as the Moon is. The times of high water and low water mainly follow the Moon, with a bit of variation because of the Sun. A simple rule for the times of low and high tide is that they occur a fixed time before or after the transit of the Moon (when the Moon is highest in the sky) ― but that does not yet include the effect of the Sun, and that fixed time depends on the place where you look at the tides.



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