Orbits

1) Orbital data: There are main three orbits in satellite communication.1) Geostationary orbit.2) Low earth orbit/medium earth orbit.3) Elliptical orbits.

Satellite orbits:There is only one main force acting on a satellite when it is in orbit,

and that is the gravitational force exerted on the satellite by the Earth. This force is constantly pulling the satellite towards the centre of the

Earth.

A satellite doesn't fall straight down to the Earth because of its velocity. Throughout a satellites orbit there is a perfect balance

between the gravitational force due to the Earth, and the centripetal force necessary to maintain the orbit of the satellite.

The formula for centripetal force is: F = (mv2)/rThe formula for the gravitational force between two bodies of mass M

and m is (GMm)/r2

The most common type of satellite orbit is the geostationary orbit. This is described in more detail below, but is a type of orbit where the

satellite is over the same point of Earth always. It moves around the

Earth at the same angular speed that the Earth rotates on its axis.

We can use our formulae above to work out characteristics of the orbit.

(mv2/r) = (GMm)/r2

=> v2/r = (GM)/r2

Now, v = (2πr)/T.

=> (((2πr)/T)2)/r = (GM)/r2

=> (4π2r)/T2 = (GM)/r2

=> r3 = (GMT2)/4π2

We know that T is one day, since this is the period of the Earth. This is 8.64 x 104 seconds.

We also know that M is the mass of the Earth, which is 6 x 1024 kg.Lastly, we know that G (Newton's Gravitational Constant) is 6.67 x 10-11

m3/kg.s2

so we can work out r.

r3 = 7.57 x 1022

Therefore, r = 4.23 x 107 = 42,300 km.

So the orbital radius required for a geostationary, or geosynchronous orbit is 42,300km. Since the radius of the Earth is 6378 km the height

of the geostationary orbit above the Earth's surface is ~36000 km.

There are many different types of orbits used for satellite telecommunications, the geostationary orbit described above is just one of them. Outlined below are the most commonly used satellite orbits. The orbits are sometimes described by their inclination - this

is the angle between the orbital plane and the equatorial plane.

1) Geostationary Orbit:

The most common orbit used for satellite communications is the geostationary orbit (GEO). This is the orbit described above – the rotational period is equal to that of the Earth. The orbit has zero inclination so is an equatorial orbit (located directly above the equator). The satellite and the Earth move together so a GEO satellite appears as a fixed point in the sky from the Earth.

The advantages of such an orbit are that no tracking is required from the ground station since the satellite appears at a fixed position in the sky. The satellite can also provide continuous operation in the area of visibility of the satellite. Many communications satellites travel in geostationary orbits, including those that relay TV signals into our homes.

However, due to their distance from Earth GEO satellites have a signal delay of around 0.24 seconds for the complete send and receive path. This can be a problem with telephony or data transmission. Also, since they are in an equatorial orbit, the angle of elevation decreases as the latitude or longitude difference increases between the satellite and earth station. Low elevation angles can be a particular problem to mobile communications.

Application: Geostationary orbit is used for direct broadcast television.

2) Low Earth Orbit/Medium Earth Orbit:

A low earth orbit (LEO), or medium earth orbit (MEO) describes a satellite which circles close to the Earth. Generally, Leos have altitudes of around 300 – 1000 km with low inclination angles, and MEOs has altitudes of around 10,000 km.

A special type of LEO is the Polar Orbit. This is a LEO with a high inclination angle (close to 90degrees). This means the satellite travels over the poles.

LEO Orbit Polar Orbit

    Satellites that observe our planet such as remote sensing and weather satellites often travel in a highly inclined LEO so they can capture detailed images of the Earth’s surface due to their closeness to Earth. A satellite in a Polar orbit will pass over every region of Earth so can provide global coverage. Also a satellite in such an orbit will sometimes appear overhead (unlike a GEO which is only overhead to ground stations on the equator). This can enable communication in urban areas where obstacles such as tall buildings can block the path to a satellite. Lastly, the transmission delay is very small.

Any LEO or MEO system however, for continuous operation, requires a constellation of satellites. The satellites also move relative to the Earth so widebeam or tracking narrowbeam antennas are needed.

Application: Satellite system such as those used for satellite phone may use Low earth orbit orbiting systems. Similarly satellite systems used for navigation like Navstar or Global Positioning System (GPS) occupies a relatively low earth orbit.

3) Elliptical and circular Orbits:

A satellite orbit the earth in one of two basic types of orbit. The most obvious is a circular orbit where the distance from the earth remains the same at all times. Second types of satellite orbit are an elliptical one. Elliptical orbits are often used, particularly for

satellite that only needs to cover a portion of the earth’s surface. For any ellipse, there are two focal points, and one of these is the geocentric of the earth. Another feature of an elliptical orbit is that there are two other major points. One is where the satellite is furthest from the earth .This point is known as APOGEE. The point where it is closest to the earth is known as the PERIGEE.

A satellite in elliptical orbit follows an oval-shaped path. One part of the orbit is closest to the centre of Earth (perigee) and another part is farthest away (apogee). A satellite in this type of orbit generally has an inclination angle of 64 degrees and takes about 12 hours to circle the planet. This type of orbit covers regions of high latitude for a large fraction of its orbital

Circular orbits: Circular orbits are classified in a number of ways. Terms such as Low earth orbit, geostationary orbit and the like detail distinctive elements of the orbit. A summary of circular orbit definitions is given in the table below

Orbit name Orbit initials Orbit altitude (Km above earth surface)

Details

Low earth orbit LEO 200-1200

Medium earth orbit

MEO 1200-35790

Geosynchronous orbit

GSO 35790 Orbits once a day, but not necessarily in the same direction as the rotation of the earth. Not necessarily stationary.

Geostationary orbit

GEO 35790 Orbits once a day and moves in the same direction as the earth and therefore appears stationary above the same point on the earth’s surface can only be above the Equator.

High earth orbit HEO Above 35790

2) Semi major axis:

In geometry

, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae.

Semi-Major Axis of satellite orbit:

Smaj = ((T/(2*p))2*Mu)(1/3)

Notes:

T is the orbit period (seconds)

Mu is the product of the gravitational constant and the earth's mass.

Mu=G*Me=6.67*10-11 * 5.974 * 1024

Mu=3.986*1014 metres3/second

Ellipse:The major axis of an ellipse Is its longest diameter, a line that runs through the centre and both foci

geometry), its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. For the special case of a circle, the semi-major axis is just the radius.

The semi-major axis' length a is related to the semi-minor axis

 B through the eccentricity e and the semi-latus rectum l, as follows:

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping l fixed. Thus and tend to infinity, a faster than b.

The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis,

Hyperbola:The semi-major axis of a hyperbola Is one half of the distance between the two branches; if this is a in the x-direction the equation is: In terms of the semi-latus rectum and the eccentricity we havethe transverse axis of a hyperbola runs in the same direction as the semi-major axis.

a Semi-major axis

Size and shape of orbit

e Eccentricity

Ω Right ascension of

ascending node

Orientation of the orbital

plane in the inertial system

ω Argument of

perigee

i Inclination

To Epoch of perigee

Position of the satellite in the orbital plane

3) Orbital Speed: A satellite in orbit moves faster when it is close to the planet or other body that it orbits, and slower when it is farther away.  When a satellite falls from high altitude to lower altitude, it gains speed, and when it rises from low altitude to higher altitude, it loses speed.

A satellite in circular orbit has a constant speed which depends only on the mass of the planet and the distance between the satellite and the center of the planet.  Here are some examples of satellites in Earth orbit:

  Altitude

  r

  Speed

  Period

  Lifetime

The Moon

385,000  km

391,370  km

1.01  km/s

27.3  daysBillions of years

100,000  km

106,370  km

1.94  km/s

4  daysBillions of years

GEO35,800 

km42,170 

km3.07  km/s

1  dayMillions of years

Navstar20,200 

km26,570 

km3.87  km/s

12  hoursMillions of years

10,000  km

16,370  km

4.93  km/s

5.8  hoursMillions of years

Lageos5,900 

km12,270 

km5.70  km/s

3.8  hoursMillions of years

2,000  km

8,370  km

6.90  km/s

2.1  hours Millenia

1,000  km

7,370  km

7.35  km/s

105  minutes Millenia

Hubble 600  km6,970 

km7.56  km/s

97  minutes Decades

ISS 380  km6,750 

km7.68  km/s

92  minutes Years

200  km6,570 

km7.78  km/s

89  minutesDays or weeks

100  km6,470 

km7.84  km/s

87  minutes Minutes

Sea Level

0  km6,370 

km7.90  km/s

84  minutes Seconds

The speed (v) of a satellite in circular orbit is:

      v = SQRT(G * M / r)

where G is the universal gravitational constant (6.6726 E-11 N m2 kg-2), M is the mass of the combined planet/satellite system (Earth's mass is 5.972 E24 kg), and r is the radius of the orbit measured from the planet's center.  "SQRT" means "square root".

Using these values gives the speed in meters per second.

The period (P) of a satellite in circular orbit is the orbit's circumference divided by the satellite's speed:

      P = 2 * pi * r / v

Using values in metric units, as above, gives the period in seconds.  (pi = 3.14159...)

Gravitational pulls from the Moon and Sun are the strongest forces perturbing orbits of Earth satellites above 40,000 km.

Off-center gravitational pull from Earth's equatorial bulge is the strongest force perturbing orbits of satellites between 500 km and 40,000 km.

Atmospheric drag is the strongest force perturbing orbits of Earth satellites below 500 km.  Dense satellites with small cross-sectional area are affected less than light satellites with large area.  A very dense satellite might just make a full orbit starting at 100 km.

Satellites in elliptical orbit move faster than the circular speed while near perigee, and slower than the circular speed while near apogee.  The period of a satellite in any orbit, circular or elliptical, is given by Kepler's third law:

      P = 2 * pi * SQRT(r3 / G * M)

where r is the mean radius of the orbit -- that is, the apogee plus the perigee (measured from the planet's center) divided by two, or half the major axis of the ellipse.

Changes to the orbit of a satellite are most efficient at perigee and apogee.  A rocket burn at perigee which increases orbital speed raises the apogee.  A burn at perigee which decreases orbital speed lowers the apogee.  Likewise, a burn at apogee which increases orbital speed raises the perigee, and a burn at apogee which decreases orbital speed lowers the perigee.

Atmospheric drag on a satellite at perigee lowers the apogee, causing the orbit to become more and more circular, until the entire orbit is at the perigee altitude, and the satellite soon falls from orbit.

The state of the atmosphere is also a factor.  Increasing activity in the eleven-year solar cycle heated Earth's upper atmosphere in the late 1970s, expanding it.  This increased the drag on Skylab, which was originally at 435 kilometers, and brought it down two years earlier than expected.

Changes in the inclination of a satellite's orbital plane are most efficient at apogee.  Large inclination changes require very large expenditures of fuel, so are rarely done.  A common plane change moves geosynchronous satellites to orbit directly over the equator.

Combining a change in altitude with a change in inclination is more efficient than using a separate engine burn for each.

Whether increasing or decreasing altitude or changing orbital inclination, the most efficient engine burns are parallel to the planet's surface.  When the Space Shuttle returned a large satellite to Earth, the de-orbit burn was deliberately made in an inefficient direction to use up extra fuel and lower the spacecraft's mass below the landing safety limit.

The most efficient way into orbit from Earth is to launch directly east from the equator.  This takes advantage of Earth's rotational speed, adding it to the speed provided by the launch vehicle.  The speed at the equator is 465 meters per second, roughly 6% of the speed needed to go into orbit.

The S-shaped curve of orbital speeds in the graphic above is caused by the choice of origin for the logarithmic altitude scale.  The origin is 15 km below Earth's surface, to better show differences at low altitudes.  Putting it at Earth's center, to coincide with the mathematical equivalent of the origin of Earth's gravitational field, makes the curve simpler, with just one bend, as at right.

4) Orbital period: The orbital period is the time taken for a given object to make one complete orbit about another object.

When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.

There are several kinds of orbital periods for objects around the Sun (or other celestial objects):

1) The sidereal period is the temporal cycle that it takes an object to make one full orbit around the Sun, relative to the stars. This is considered to be an object's true orbital period.

2) The synodic period is the temporal interval that it takes for an object to reappear at the same point in relation to two other objects (linear nodes), i.e. the Moon relative to the Sun as observed from Earth returns to the same illumination phase. The synodic period is the time that elapses between two successive conjunctions with the Sun-Earth line in the same linear order. The synodic period differs from the sidereal period due to the Earth orbit around the Sun, with one less synodic period than lunar orbit per solar orbit.

3) The draconitic period is the time that elapses between two passages of the object at its ascending node, the point of its orbit where it crosses the ecliptic from the southern to the

northern hemisphere. It differs from the sidereal period because the node is a planar coincidence rather than a linear coincidence, and the object's line of nodes typically precesses or recesses slowly in relation to orbital cycle.

4) The anomalistic period is the time that elapses between two passages of the object at its perihelion, the point of its closest approach to the Sun. It differs from the sidereal period because the object's semimajor axis typically advances slowly.

5) The Earth tropical period or year, finally, is the time that elapses between two alignments of the axis of rotation with the Sun, also viewed as two passages of the object at right ascension zero. One Earth year has a shorter interval than solar orbit (sidereal period) because the inclined axis and equatorial plane slowly precesses (rotates in sidereal terms), realigning before orbit completes with an interval equal to the inverse of the precession cycle (about 25,770 years).

Relation between sidereal and synodic period

Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.

Using the abbreviations

E = the sidereal period of Earth (a sidereal year, not the same as a tropical year) P = the sidereal period of the other planet S = the synodic period of the other planet (as seen from Earth)

During the time S, the Earth moves over an angle of (360°/E)S (assuming a circular orbit) and the planet moves (360°/P)S.

Let us consider the case of an inferior planet, i.e. a planet that will complete one orbit more than Earth before the two return to the same position relative to the Sun.

and using algebra we obtain

For a superior planet one derives likewise:

Generally, knowing the sidereal period of the other planet and the Earth, P and E, the synodic period can easily be derived:

which stands for both an inferior planet or superior planet.

The above formulae are easily understood by considering the angular velocities of the Earth and the object: the object's apparent angular velocity is its true (sidereal) angular velocity minus the Earth's, and the synodic period is then simply a full circle divided by that apparent angular velocity.

Table of synodic periods in the Solar System, relative to Earth:

   Sid. P. (a)  

Syn. P. (a)  

Syn. P. (d)

Mercury      0.241

  0.317   115.9

Venus      0.615

  1.599   583.9

Earth       1     —     —

Moon      0.0748  

  0.0809  29.5306

Mars      1.881

  2.135   780.0

4 Vesta      3.629

  1.380   504.0

1 Ceres      4.600

  1.278   466.7

10 Hygiea

      5.557

  1.219   445.4

Jupiter      11.87

  1.092   398.9

Saturn      29.45

  1.035   378.1

Uranus      84.07

  1.012   369.7

Neptune      164.9

  1.006   367.5

134340 Pluto

      248.1

  1.004   366.7

136199       557   1.002   365.9

Eris90377 Sedna

      12050

  1.00001

  365.1

In the case of a planet's moon, the synodic period usually means the Sun-synodic period. That is to say, the time it takes the moon to complete its illumination phases, competing the solar phases for an observer on the planet's surface —the Earth's motion does not determine this value for other planets, because an Earth observer is not orbited by the moons in question. For example, Deimos' synodic period is 1.2648 days, 0.18% longer than Deimos' sidereal period of 1.2624 d.

5) Eccentricity of an Orbit:

Eccentricity (e) is defined as:e2=1-(b/a)2

You may think that most objects in space that orbit something else move in circles, but that isn't the case. Although some objects follow circular orbits, most orbits are shaped more like "stretched out" circles

This animation shows the shapes of some elliptical orbits. These orbits have different eccentricities. If an ellipse has a high eccentricity, is it round like a circle or long like an oval? Click on image for full size (29K GIF)Original animation by Windows to the Universe staff (Randy Russell).

or ovals. Mathematicians and astronomers call this oval shape an ellipse.

An ellipse can be very long and thin, or it can be quite round - almost like a circle. Scientists use a special term, "eccentricity", to describe how round or how "stretched out" an ellipse is. If the eccentricity of an ellipse is close to one (like 0.8 or 0.9), the ellipse is long and skinny. If the eccentricity is close to zero, the ellipse is more like a circle.

The eccentricity of Earth's orbit is very small, so Earth's orbit is nearly circular. Earth's orbital eccentricity is less than 0.02. The orbit of Pluto is the most eccentric of any planet in our Solar System. Pluto's orbital eccentricity is almost 0.25. Many comets have extremely eccentric orbits. Halley's Comet, for instance, has an orbital eccentricity of almost 0.97!

The Sun is not at the center of an elliptical orbit. It is a little off to one side, at a point called a "focus" of the ellipse. Because of this offset the planet moves closer to and further away from the Sun every orbit. The close point in each orbit is called perihelion. The far away point is called aphelion. If an orbit has a large eccentricity, the difference between the perihelion distance and the aphelion distance will also be large. Earth is only 3% further from the Sun at aphelion than it is at perihelion. Pluto's aphelion distance from the Sun is 66% greater than its perihelion distance.

6) Rotation Period:

The Sidereal Period of Rotation vs. the Synodic Period of Rotation       As a planet rotates around its axis, the stars appear to move around a projection of the planet's axis

into space. The time required for the stars to move once around their paths is called the sidereal period of rotation, or the rotation period of the planet.

      While the planet rotates, it is also moving around the Sun. This changes the apparent position of the Sun among the stars, and as a result, it does not move around the sky in quite the same period of time

that the stars do. Depending upon whether the rotation of the planet is direct (in the same direction as its orbital motion) or retrograde (in the opposite direction as its orbital motion), the time that the Sun takes to

go once around the sky, which is called the synodic period of rotation, or the length of the day, may be longer or shorter than the sidereal period of rotation. Table 1 shows the rotation period and the length of

the day for the Moon, and the planets. As you can see, for most of the bodies, the two times are very similar, but for objects which have slow rotation periods, such as the Moon, Mercury and Venus, there is a

large difference between the two time periods.

BodyMercu

ryVenusEarthMoon

Sidereal Period

58.6467 days- 243.02 days23 hr 56 min

4.1 sec

Synodic Period = "Day"

175.940 days- 116.75 days24 hr 0 min 0

sec

MarsJupiterSaturnUranu

sNeptu

nePluto

27.322 days24 hr 37 min

22.66 sec9 hr 55 min 30

sec10 hr 32 min 35

sec- 17 hr 14.4 min

16 hr 6.6 min- 6 days 9 hr

17.6 min

29.53 days24 hr 39 min

35.24 sec9 hr 55 min 33

sec10 hr 32 min 36

sec- 17 hr 14.4 min

16 hr 6.6 min- 6 days 9 hr

17.0 min

Table 1: Comparison of Sidereal and Synodic Periods

Retrograde Rotation       All of the planets orbit, or revolve, around the Sun in the same eastward direction. Most of them also

rotate around their axes in that same direction. Venus, Uranus and Pluto, however, rotate in the opposite direction, and if we need to do any arithmetic involving their rotation, such as comparing their rotation

rate to their day length, we have to distinguish between DIRECT rotation, which is in the same direction as the orbital motion, and

RETROGRADE rotation, which is in the opposite direction. To accomplish this, we define the rotation period as the time that it takes for the planet to turn once around its axis TO THE EAST, causing the stars to turn around the sky to the west. If the planet rotates in the opposite direction, causing the stars to turn around the sky in the

opposite direction, we would have to run time backwards in order to see a westward motion for the stars. As a result, the rotation period of

a planet which has a retrograde rotation is a negative number, as shown in the table for the three planets which have such a rotation.

      Keep in mind that even though some planets have retrograde ROTATION, they ALL orbit, or REVOLVE around the Sun, in the same

direction.       For additional discussion of retrograde motion in general, or

retrograde rotation in particular, refer to Retrograde Motion.

Explaining the Difference between Rotation and Day Length

      As shown in the table, the rotation period and day length are nearly identical for all of the outer planets. In fact, for most of them,

the values are so nearly identical that they are the same, to the accuracy shown here. For the Moon and the inner planets, however,

the situation is quite different. The Earth and Mars, which rotate relatively quickly, have only a few minutes difference between the

their rotation period and the day length, but for the Moon, Venus and

Mercury, the difference between the two values is quite large. For the Moon, the difference is two days -- not enough to make the day appear

tremendously different from the rotation period, but enough to be confusing, if the reason for the difference is not understood -- and for Mercury and Venus, the difference between the two values is so large

that their rotation periods appear, at first glance, to be completely unrelated to the lengths of their days.

      To explain why the day length, or synodic period of rotation, is different from the sidereal period of rotation, we consider how a given place moves around a planet, and the way in which this changes its

view of the sky, during one rotation period.       In the diagram below, the four blue dots on the right represent the

position of the planet at four times, separated from each other by a third of a rotation period. The number of rotations that the planet has made is indicated by the numbers to the right of each dot. The white

dot shows how the position of a specific place on the planet changes as the planet rotates to the east (counter-clockwise, in this diagram), and the large yellow dot on the far left represents the position of the Sun. The sizes of the Sun and planet and the angle that the planet moves through during one rotation have been exaggerated to make it easier

to see what is happening.

The Rotation Period of the Earth      To see how this works, consider the case of the Earth. The Earth

goes around the Sun in one year, or approximately 365 1/4 days. The number of degrees in a circle is 360 degrees, which is about the same as the number of days in a year, so the angle that the Sun seems to move, relative to the stars, during one rotation is approximately one

degree.       To calculate how long it takes for the Earth to rotate through an

angle of one degree, we divide the length of a day, 24 hours, or 1440 minutes, by the 360 degrees that it turns through during that rotation, obtaining a rotational speed of 4 minutes per degree. Since the Sun's

motion differs from the stars' motion by one degree, and it takes 4 minutes for the Earth to turn through one degree, it takes the Sun 4

minutes longer to go around the sky than it takes for the stars to do so, and the rotation period of the Earth is 4 minutes less than the length of its day. Since we define a day as having exactly 24 hours, the rotation

period is 23 hours 56 minutes, as shown in the table.

The Rotation Periods of the Outer Planets       For the outer planets, we can use the same sort of calculation that we just did for the Earth, taking advantage of the fact that they have much longer orbital periods, so the number of rotations in a year is

much larger, and the distance they move during one rotation is

correspondingly smaller.       For Mars, the year is nearly twice as long as ours, 686.98 days, and

the rotation period is a little longer than ours, 24 hours 37 minutes 22.66 seconds (this is often erroneously listed as the length of the day). Dividing the rotation period into the year, we find that Mars rotates 670 times in a year, moving around the Sun about half a

degree during each rotation. Since Mars rotates at about the same rate that we do, it would take about 2 minutes to make up for this half-

degree motion; so on Mars, the day must be about 2 minutes longer than the rotation period. These calculations are rounded off considerably, so the results are only approximate; but if the

calculations were accurately done, the results would be reasonably accurate, as well.

      For planets which are even further out, the motion around the Sun is even smaller, and the time required to compensate for it is only a few seconds. For Jupiter, the difference between the rotation period and the length of the day is about 3 seconds; and for Saturn, Uranus and Neptune, the difference is about 1 second. Even for Pluto, which

has a much longer rotation rate than the Jovian planets, the difference between the rotation period and the day is less than 40 seconds -- a

small difference, compared to its rotation period of more than six days. As a result, the day length and rotation period are about the same, and we often treat them as being the same (just as we often do, though not

as accurately, for the Earth).       The above technique works well for the planets with rotation rates which are rapid compared to their orbital periods, so that they rotate

hundreds, thousands, or even tens of thousands of times in each orbit. But for objects which rotate very few times in an orbit, the angle A is very large, which means that it takes quite a while for the planet to rotate through that extra angle, and the difference between the day

and the rotation period can become surprisingly large, as shown below, for the Moon.

The Rotation Period and Day Length of the Moon       Since the Moon moves around the Sun with the Earth, the Sun

moves the same one degree per day in the lunar sky as it does during the Earth's rotation. But while the Earth rotates in about one day, the Moon takes more than 27 days to rotate, so during one lunar rotation, the Sun moves over 27 degrees, relative to the stars. To make up for

this, the Moon has to rotate more than two days longer, as can be seen by comparing its rotation period to its day length. The 2.2 day

difference between the periods seems extreme, but the basic idea is the same as for the Earth; it's just that since the Moon rotates so

slowly, (1) the Sun moves much further during one lunar rotation than during one Earth rotation, and (2) the Moon takes longer than the

Earth to make up for any given solar motion. Each of these factors increases the four minute difference between the Earth's rotation and

day length by the 27.3 times slower motion of the Moon, for a combined increase of 27.3-squared, or nearly 750 times. So the four minute difference between day length and rotation period becomes 4

times 750, or 3000 minutes, which is a little over two days.       Unfortunately, that's not the end of the calculation, because during

the time the Moon must rotate to make up for its motion around the Sun, it continues to move around the Sun, causing another two degree difference between the solar and stellar motions, which takes another four hours to make up (4 minutes per degree for the Earth, times two

degrees to make up, times 27 times slower rotation for the Moon). And during that four hours, the Moon moves another sixth of a degree

around the Sun, so for really precise results, we'd have to keep doing this, with smaller and smaller times and angles, until the difference

was too small to bother with.

A More Accurate Calculation       For the Moon, we might be satisfied with just the two-step

calculation shown above, and not bother with smaller and smaller corrections. But if we wanted accurate results for Mercury and Venus, which rotate even more slowly, and have shorter orbital periods, we couldn't get away with just one or two corrections; half a dozen or

more might be needed, depending upon the accuracy desired. So for those planets, we need a different way of doing the calculations; but

fortunately, there is a relatively simple way of calculating the difference between rotation period and day length, which can be done as accurately as we want, by simply using accurate numbers to do the

arithmetic.       In one year, the planet rotates a certain number of times, and the

stars go around the sky that number of times. If the planet were stationary relative to the Sun, so that the Sun was fixed in the sky

relative to the stars, it would rise and set the same number of times as the stars, but during one year, the planet moves once around the Sun, and as a result, the Sun moves once to the east among the stars. If the

planet has direct rotation, as most do, so that the stars move westwards across the sky, the Sun's eastward motion relative to the

stars is backwards, so it goes around the sky one less time, and

The number of days in a year = the number of rotations - 1.

      If the planet has retrograde rotation, the stars will move across the sky to the east, instead of to the west, and the Sun's eastward motion will result in its crossing the sky one more time than the stars in one

year. However, since we treat this kind of rotation as having a negative

rotation period, the larger negative number is still one less than the number of rotations. This equation can, therefore, be applied to all the

planets, regardless of how they rotate.       The table below shows the results of using this method for all the planets, and the Moon. The most interesting result is for Mercury. Its

slow rotation and fast orbital motion cause a huge difference between the day and rotation period, which differ by a factor of three, more

than for any other planet. But what is really striking is that the rotation period is exactly 2/3 of the orbital period, and the day is exactly two orbital periods, so that one side of Mercury faces the Sun for a whole

year, then the other side, for the next year. Even more surprisingly, as discussed in The Rotation of Mercury, when Mercury is closest to the Sun, at perihelion, its orbital and rotational motion are nearly identical, so that for more than a week, the same side of the planet faces an apparently motionless Sun, almost as if it always kept the same side to the Sun, as we

used to believe.

Object

OrbitalPeriod

RotationPeriod

RotationsPer Year

DaysPer Year

Day Length

Mercury

VenusEarthMoonMars

JupiterSaturnUranu

sNeptu

nePluto

87.970 days

224.70 days

365.256* days

365.256 days

686.980 days

4332.59 days

10759.22 days

30685.4 days

60189 days

90465 days

58.6467 days- 243.02 days23 hr 56 min

4.1 sec27.322 days24 hr 37 min

22.66 sec9 hr 55 min 30

sec10 hr 32 min 35

sec- 17 hr 14.4 min

16 hr 6.6 min- 6 days 9 hr

17.6 min

1.500000-

0.92462

366.256

13.369669.59

9410476.

824492.

07-

4271789667

- 14163.

4

0.500000-

1.92462

365.256

12.369668.59

9410475.

824491.

07-

4271889666

- 14164.

4

175.940 days- 116.75 days24 hr 0 min 0

sec29.53 days

24 hr 39 min 35.24 sec

9 hr 55 min 33 sec

10 hr 32 min 36 sec

- 17 hr 14.4 min16 hr 6.6 min- 6 days 9 hr

17.0 min

* The length of the Earth's orbital period, although called a "year" in the discussion, is not the same as the calendar (or tropical) year, which is about 20 minutes shorter than the orbital period, due to the precession of the Equinoxes, and is about 365.244 days.

.

Measuring rotation:

For solid objects, such as rocky planets and asteroids, the rotation period is a single value. For gaseous/fluid bodies, such as stars and gas giant planets, the period of rotation varies from the equator to the poles due to a phenomenon called differential rotation. Typically, the stated rotation period for a gas giant (Jupiter, Saturn, Uranus, Neptune) is its internal rotation period, as determined from the rotation of the planet's magnetic field. For objects that are not spherically symmetrical, the rotation period is in general not fixed, even in the absence of gravitational or tidal forces. The moment of inertia of the object around the rotation axis can vary, and hence the rate of rotation can vary (because the product of the moment of inertia and the rate of rotation is equal to the angular momentum, which is fixed). Hyperion, a satellite of Saturn, exhibits this behaviour, and its rotation period is described as chaotic.

Summary       Each planet, as it goes around the Sun, sees the Sun move

eastward among the stars once each year, and as a result, the stars' movement around the sky, which defines the rotation period, is not the same as the day, which is defined by the Sun's movement around the

sky.       Calculating the difference between the two periods is done in one

of two ways. For planets with fast rotations or long orbital periods, estimating the Sun's daily motion relative to the stars, and how long it would take the planet to rotate through that angular motion, yields the difference between the day and the rotation period. In fact, for planets with hundreds, thousands, or tens of thousands of days in a year, this is the only way to calculate that difference without using many-digit precision. But for planets with slow rotations or show orbital periods, this method won't work, because the planet moves around the Sun during the difference between the two periods, requiring additional

corrections. For those planets, using the fact that the number of days in a planet's year is always one less than the number of rotations

yields far more accurate results