Moon and Tides – Indicators of the Real Earth’s Spin Velocity

What is period of the Earth’s rotation around its axis?

As a rule, answer to this question is that that one turn around the axis of the Earth requires, approximately, 24 hours and that this period is cold the mean solar day. Either, more precisely, some four minutes shorter stellar day.

In fact, neither solar, nor the stellar day stands for the Earth’s full spin period. Earth turns around its axis considerably more slowly. The only frame of reference, in which its intrinsic rotation is clearly defined, while its orbital motions are excluded, has to be the one related with the Moon.

The full spin period of our planet is about 50 minutes longer lunar day – the time between one lunar zenith to the next. The term tidal day is also in use in oceanography.

Earth-Moon binary system

Consider two celestial bodies forming a gravitational binary system.

In astronomy, a commonly accepted criterion dividing a planet-satellite from the double-planet system is based on the location of the barycenter of these two objects. If their center of masses is not situated under the surface of either body, then one may refer to this pair as a double-planet system. In accordance with that criterion, since the Earth – Moon barycenter is 4671 km distant from the Earth’s center, the Moon represents the satellite of the Earth.

The late Isaac Asimov proposed a distinction between planet-moon and double-planet systems founded on, what he called, a “tug-of-war” value: if the interaction between the star and and the smaller celestial body is greater than interaction between two bodies, than the double planet takes the place, otherwise, the smaller body represents satelite of the planet. In the case of the Earth’s Moon, the Sun actually wins the tug of war, since its gravitational effect on the Moon is more than twice that of Earth’s, so Asimov reasoned that the Earth and the Moon must form a double-planet system. That is the reason why the resulting Earth’s orbit and the corresponding Moon’s orbit are everywhere concave toward the Sun.

Avoiding these semantic classifications, here we refer only to the incontestable fact that Earth and the Moon turn around their common center of mass, i.e., that they form a gravitational binary system (within the Sun –Earth – Moon three body system, of course).

Two celestial bodies, with masses m_1 and m_2 form a gravitational binary system if the kinetic energies of relative motions of eider bodies with respect to the common center of masses are always numerically smaller than the potential of their gravitational interaction

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In this expression, G represents the gravitational constante, and D_{1-2} – the distance between two bodies.

If the condition (1) is satisfied, orbits of the bodies are two similar ellipses. Their lines of apsides coincide and the common center of masses C lies in their inversely disposed foci (Fig. 1).

Motions of the bodies forming a binary system

Fig. 1 – Motions of the bodies forming a binary system

Let us consider Earth and the Moon now. We introduce two systems of coordinates: xCy (x coincides with the line of apsides) and \varsigmaC\eta (\eta coincide with the Earth-Moon direction). The second one represents the lunar frame of reference.

The average velocity of the Moon with respect to Earth is 1,022 km/s. We shall compare this quantity with the capture velocity v_c of the Moon obtained from the maximum (numerical minimum) of the potential.

Introducing

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into expression (1), one obtains the capture velocity of the Moon v_c = 1,40 km/s . This result is in accordance with the fact that Earth and the Moon form the binary system.

Provisionally accepting that the barycenter C is the origin of an inertial reference frame xCy (Fig. 1), we can write down the following expressions in this frame

m_1 \vec{r_1} + m_2 \vec{r_2} = 0, (2)

and consequentially

m_1 \vec{v_1} + m_2 \vec{v_2} = 0, (3)

m_1 \vec{a_1} + m_2 \vec{a_2} = 0, (4)

Absolute values proportions of all the kinematic characteristics are inversely proportionate to the masses ratio. Because of that, periods of the orbital rotations have to be equal. So, these two bodies move in the ideal resonance 1:1.

The Moon’s and Earth’s orbits

Parameters of the elliptical orbits:

a – semi major axis, c – semi distance of the foci, e = c/a – eccentricity and
b = \sqrt{a^2 - c^2} = a\sqrt{1 - e^2} – the semi minor axis.

Since max D_{E-M} = D_{A1-A2} = 406720 km and the Moon’s orbit eccentricity e = 0,0549 [3],

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a_M = 380868 km, b_M=380249 km, c_M=20910 km,

a_E = 4685 km, b_E=4678 km, c_E=257 km.

Earth – Moon binary system

Fig. 2 – Earth – Moon binary system

Relative velocity of the Moon with respect to Earth is 1,022 km/s. From the eq. (3) it follows out that the “absolute” Earth’s and Moon’s velocities in their small orbits are

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Taking 379.730 km as the average distance from the Moon to the mass center C [3], we can evaluate the circumference of its orbit ~ 2.385.916 km. This haevenly body completes one turn along this path in 2,363 229 \times 10^6 s = 27,35 mean solar days.

The same time requires Earth to cover the circumference of its own small orbit.

During one mean solar day, Earth and the Moon make ~ 0,23 rad = 13, 16^0 around the mass center C (Fig. 2).

The spin periods

The answer to the question what is the period of the Earth’s rotation around its axis will be, as a rule, eider that it is the 24 hours long mean solar day – rotational period relative to the Sun, or some four minutes shorter (23 hours, 56 minutes and 4,09 seconds) stellar, or, sidereal day – period relative to the fixed stars.

Let us quote some of sources in which the stellar day was represented as the period of the Earth’s rotation.

Period of Rotation of the Earth

  • Moche, Dinah. Astronomy. 4th ed. Wiley, 1993: 24.”23 hours, 56 minutes, 4 seconds long”86,164 s.
  • “Earth.” Microsoft Encarta 97 Encyclopedia. Microsoft, 1996.”… the earth rotates on its axis every 23 hours, 56 minutes and 4.1 seconds based on the solar year.”86,164.1 s
  • Daily, Robert. Earth. USA, 1994: 20.”… it takes earth 23 hours, 56 minutes and 4.09seconds, a period of time we call a day.”86,164.09 s.
  • “Earth.“ Encyclopedia Britannica. Chicago: Encyclopedia Britannica, 1998: 320.”… the earth spins on its axis and rotates completely once every 23 hours, 56 minutes and 4 second.”86,164 s.
  • Earth’s Rotation. Liftoff to Space Exploration. NASA. Marshall Space Flight Center.”The actual value is 23 hours, 56 minutes and 4 seconds.”86,164 s.

As seen, together with the noun “day” usually goes its qualifier defining the frame of reference in consideration. “Sun” represents a relative and “fixed stars”, the absolute reference frame.

Our planet performs, at least, five different rotations: spin, precession together with its axis, orbital motion around the Earth-Moon baricenter, turning together with this baricenter around the Sun and finally, rotation with the solar system around the center of the galaxy. The planes of all these rotations are completely different.

The solar day represents the period of the first three and the stellar day of all these finite rotations, united in one.

That is the reason why the stellar day is shorter than the solar one.

In fact, neither of these days corresponds to the Earth’s full spin period.

Does anybody adjoints the angular velocity arised because of the path curvature to the number of revolutions of the motor car engine? Eider the corresponding angular velocity to the spin of the gyrocompasse, or gyrostabiliser?

As known, our planet represents a giant gyroscope and by all means, one has to be aware of its spin pace.

Generally, study of the planet’s motion is only possible if all the rotations were separated.

Since Earth and the Moon move in 1:1 resonance along their orbits around their common mass center, their intrinsic rotations are clearly defined and their orbital motions excluded, only in the lunar frame of reference \varsigmaC\eta (Figs. 1 and 2).

The full spin period of our planet is the time interval between one lunar zenith to the next, i.e., between two successive intersections of the Earth – Moon direction with the same meridian, about 24 h 50 min and 28 s, that is, 89 028 mean solar seconds. Its name is the lunar day and it is in use in oceanography. This day is ~ 3, 32 % longer the sidereal one. The real Earth’s spin velocity is roughly 4, 6 % smaller than its (total and averaged) velocity in the absolute frame of reference.

The similar situation corresponds to the Moon’s spin. Neglecting its libration the Moon does not turn around its axis and this fact is quite clear in the \varsigmaC\eta reference frame. Observer on the Earth always perceives the same face of the Moon.

However, in the xCy reference frame, the Moon (neglecting precession of its orbit) accomplishes one full rotation around its axis in 27, 35 mean solar days. Intrinsic and orbital rotations of the Moon are in 1:1 resonance.

Approximatelly, this period corresponds to the stellar or sidereal month.

If, on the other hand, motion of this body is considered in the solar frame of reference then the angle of the Moon’s rotation has to be reduced by the corresponding angle of the baricenter C turnover Sun. In that case, rotation of the Moon appears slower and it comes to 29, 54 mean solar days. This is the solar or synodic month.

As a paradigm, let us mention here the planet Mercury. In the stellar, that is inertial frame of reference it moves prograde, in the rotational/orbital resonace 3/2. If one subtracts two orbital rotations from this fraction, he obtains -1/2. The proper rotation of this planet is retrograde. It makes one full backward spin in two full orbital rotations.

Another indicator of the Earth’s spin pace is the rhythm of tides, that is, of the periodic rises and fallings of the large bodies of water. Namely, due to the the Moon’s tidal force, the body of the ocean water is, practically, “pulled apart” along the direction C\eta. In the zone closest to the Moon, this force raises masses of the water from the ocean bed, that is, from the Earth, while, in the farthest zone, it pulls the bed (Earth) from the water. That is the reason why the oceans tend to bulge toward and away from the Moon in these zones (Fig. 3). As seen in the same figure, the low water occurs at abour right angles to the Earth – Moon direction.

The Sun’s gravitational pull does also influence tides to some degree, of course, but the effect of it on the Earth tides is less than half that of the Moon. Particularly, large tides are experienced in the Earth’s oceans when the Sun and the Moon ar aligned with the Earth, at new and full phases of the Moon. These tides are called spring tides. Conversely, when the Moon is at the right angle toward the Earth Sun direction (first quarter and last quarter phases) tidal bulges are generally weaker and this are called neap tides.

High and low tide

Fig. 3 – High and low tide

Depending on the angle between the C\eta axis and the Earth’s spin axis (by the way, it does not coincide with the axis of the Earth’s total rotation), on the disposition of the Sun with respect to the Earth-Moon direction, but on the ocean depth and on the coast line form, as well, the period between two successive high (low) tides varies from the semidiurnal ~ 12 h 25 min to diurnal ~ 24 h 58 min tides.

Because of that, the lunar day is also named tidal day in oceanography.

Conclusion

Conventional answer to the question what is the period of the Earth’s full rotation around its axis is either that it is, the 24 hours long, mean solar day, or, about four minutes shorter, stellar day.

Our planet performs, at least, five different rotations: spin, precession together with its axis, orbital motion around the Earth – Moon baricenter, turning together with this baricenter around the Sun and finally, rotation with the solar system around the center of the galaxy.

The solar day represents the period of the first three and the stellar day, of all these finite rotations, united in one.

That is why the stellar day is shorter then the solar one.

In the solar and the stellar frames of reference it seems that Earth rotates faster, that the Moon rotates around its axis in the, so-called, ideal resonance, that Mercury has the prograde (3/2 spin-orbit resonance) rotation, while, in fact, it has the retrograde spin (-1/2 resonance) and so on.

If one wants to study the Earth’s real motion, its rotations have to be uncoupled.

The only frame of reference in which the Earth’s spin is separated from all the other rotations is the lunar system \varsigmaC\eta.

Actually, the Earth’s spin period is more than 50 minutes longer then the solar and stellar day.

References

[1] Astronomical Almanac Online (2010). United States Naval Observatory. s.v. solar time, apparent; diurnal motion; apparent place.

[2] Astronomical Information Sheet No. 58. (2006). HM Nautical Almanac Office.

[3] Fred Hoyle, Astronomy, Rathbone Books Limited, London 1962.

[4] Jean Meeus, Mathematical astronomy morsels (Richmond, Virginia: Willmann-Bell, 1997) 345–6.

[5] John Walker, Inconstant Moon: The Moon at Perigee and Apogee, www.fourmilab.ch/earthview/moon_ap_per.html

[6] J. H. Argyris, An excursion into large rotations, Comp. Meths. Appl. Mech. Engrg., 32, 85–155, 1982.

[7] Mellor, George L. (1996). Introduction to physical oceanography. Springer. p. 169. ISBN 1563962101.

[8] M. Marjanov: Gravitational Interaction of Two Real Bodies: Complex Harmony of Motions on Stable Orbits. Lecture on the Mathematical Institute SANU, Department of Mechanics, March 2007.

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