rotation on its axis ("day")

58.65 Earth days (the Aricebo observatory used Doppler effects to measure rotation) 6.8 miles per hour (10.9 km/hr) prograde (counterclockwise when viewed from above the North Pole)

87.97 Earth days, or two-thirds of an orbit 29.9 miles per second (47.9 km/s) 3 minutes, 20 seconds to reach Mercury 35,984,076 miles (57,909,175 km), or 0.387 AU 28.58 million miles (46.0 million km), or 0.31 AU 43.38 million miles (69.82 million km), or 0.47 AU 0.2056; this is the second-highest eccentricity in the solar system, after Pluto; the next most eccentric is Mars, at only 0.093 7.01°

rotational speed at equator rotational direction sidereal period ("year") orbital velocity (average) sunlight travel time (average) average distance from the Sun perihelion aphelion orbital eccentricity orbital inclination to the ecliptic obliquity (inclination of equator to orbit) 0° (though some say it is 180°)

axis oriented toward its partner. This causes them to become slightly egg-shaped; the extra stretch is called a tidal bulge (the bulge is significant on pairs such as the Earth and Moon, but the Sun's mass is so large compared to Mercury that the Sun does not bulge). If either of the two bodies is rotating relative to the other, this tidal bulge is not stable. The rotation of the body will cause the long axis to move out of alignment with the other object, and the gravitational force will work to reshape the rotating body. Because of the relative rotation between the bodies, the tidal bulges move around the rotating body to stay in alignment with the gravitational force between the bodies. This is why ocean tides on Earth rise and fall with the rising and setting of its Moon, and the same effect occurs to some extent on all rotating orbiting bodies.

The rotation of the tidal bulge out of alignment with the body that caused it results in a small but significant force acting to slow the relative rotation of the bodies. Since the bulge requires a small amount of time to shift position, the tidal bulge of the moon is always located slightly away from the nearest point to its planet in the direction of the moon's rotation. This bulge is pulled on by the planet's gravity, resulting in a slight force pulling the surface of the moon in the opposite direction of its rotation. The rotation of the satellite slowly decreases (and its orbital momentum simultaneously increases). This is in the case where the moon's rotational period is faster than its orbital period (the time required to make a complete orbit) around its planet. If the opposite is true, tidal forces increase its rate of rotation and decrease its orbital momentum.

Almost all moons in the solar system are tidally locked with their primaries, since they orbit closely and tidal force strengthens rapidly with decreasing distance. Mercury is tidally locked with the Sun in a 3:2 resonance (when two orbital periods make an integer ratio). Mercury is the only solar system body in a 3:2 resonance with the Sun. For every two times Mercury revolves around the Sun, it rotates on its own axis three times. More subtly, the planet Venus is tidally locked with the planet Earth, so that whenever the two are at their closest approach to each other in their orbits Venus always has the same face toward Earth (the tidal forces involved in this lock are extremely small). In general, any object that orbits another massive object closely for long periods is likely to be tidally locked to it.

Just as planets are not truly spheres, the orbits of solar system objects are not circular. Johannes Kepler, the prominent 17th-century

German mathematician and astronomer, first realized that the orbits of planets are ellipses after analyzing a series of precise observations of the location of Mars that had been taken by his colleague, the distinguished Danish astronomer Tycho Brahe. Kepler drew rays from the Sun's center to the orbit of Mars and noted the date and time that Mars arrived on each of these rays. He noted that Mars swept out equal areas between itself and the Sun in equal times and that Mars moved much faster when it was near the Sun than when it was farther from the Sun.Together these observations convinced Kepler that the orbit was shaped as an ellipse and not as a circle, as had been previously assumed. Kepler defined three laws of orbital motion (listed in the table on page 73), which he published in 1609 and 1619 in his books New Astronomy and The Harmony of theWorld. These three laws are still used as the basis for understanding orbits.

As Kepler observed, all orbits are ellipses, not circles. An ellipse can be thought of simply as a squashed circle, resembling an oval.The proper definition of an ellipse is the set of all points that have the same sum of distances to two given fixed points, called foci.To demonstrate this definition, take two pins, push them into a piece of stiff cardboard, and loop a string around the pins (see figure on page 74).The two pins are the foci of the ellipse. Pull the string away from the pins with a pencil and draw the ellipse, keeping the string taut around the pins and the pencil all the way around. Adding the distance along the two string segments from the pencil to each of the pins will give the same answer each time: The ellipse is the set of all points that have the same sum of distances from the two foci.

The mathematical equation for an ellipse is x2 y2

where x and y are the coordinates of all the points on the ellipse, and a and b are the semimajor and semiminor axes, respectively.The semimajor axis and semiminor axis would both be the radius if the shape was a circle, but two radii are needed for an ellipse. If a and b are equal, then the equation for the ellipse becomes the equation for a circle:

KEPLER'S LAWS | |

Kepler's first law: |
A planet orbits the Sun following the path of an ellipse with the Sun at one focus. |

Kepler's second law: |
A line joining a planet to the Sun sweeps out equal areas in equal times (see figure below). |

Kepler's third law: |
The closer a planet is to the Sun, the greater its speed. This is stated as: The square of the period of a planet T is proportional to the cube of its semimajor axis R, or T a R2 as long as T is in years and R in AU. |

Sweeping Equal Areas in Equal Times: Kepler's Second Law

Sweeping Equal Areas in Equal Times: Kepler's Second Law

Kepler's second law shows that the varying speed of a planet in its orbit requires that a line between the planet and the Sun sweep out equal areas in equal times.

Making an ellipse with string and two pins: Adding the distance along the two string segments from the pencil to each of the pins will give the same sum at every point around the ellipse. This method creates an ellipse with the pins at its foci.

When drawing an ellipse with string and pins, it is obvious where the foci are (they are the pins). In the abstract, the foci can be calculated according to the following equations: Coordinates of the first focus

In the case of an orbit the object being orbited (for example, the Sun) is located at one of the foci.

An important characteristic of an ellipse—perhaps the most important for orbital physics—is its eccentricity: a measure of how different the semimajor and semiminor axes of the ellipse are. Eccentricity is dimensionless and ranges from 0 to 1, where an eccentricity of zero means that the figure is a circle, and an eccentricity of 1 means that the ellipse has gone to its other extreme, a parabola (the reason an extreme ellipse becomes a parabola results from its definition as a conic section). One equation for eccentricity is where a and b are the semimajor and semiminor axes, respectively (see figure on page 75, top). Another equation for eccentricity is

Coordinates of the second focus

Semimajor and Semiminor Axes, Foci

The semimajor and semiminor axes of an ellipse (or an orbit) are the elements used to calculate its eccentricity, and the body being orbited always lies at one of the foci.

a where c is the distance between the center of the ellipse and one focus. The eccentricities of the orbits of the planets vary widely, though most are very close to circles, as shown in the figure below. Pluto has the most eccentric orbit at 0.244, and Mercury's orbit is also very eccentric, but the rest have eccentricities below 0.09.

While the characteristics of an ellipse drawn on a sheet of paper can be measured, orbits in space are more difficult to characterize. The ellipse itself has to be described, and then the ellipse's position in

Eccentricity of Pluto's Orbit Compared to a Circle

A circle

Though the orbits of planets are measurably eccentric, they deviate from circularity very little. This figure shows the eccentricity of Pluto's orbit in comparison with a circle.

space, and then the motion of the body as it travels around the ellipse. Six parameters are needed to specify the motion of a body in its orbit and the position of the orbit. These are called the orbital elements (see the figure below).The first three elements are used to determine where a body is in its orbit.

a semimajor axis The semimajor axis is half the width of the widest part of the orbit ellipse. For solar system bodies, the value of the semimajor axis is typically expressed in units of AU. Mercury's semimajor axis is 0.387 AU. e eccentricity Eccentricity measures the amount by which an ellipse differs from a circle, as described above. An orbit with e of A series of parameters called 0 is circular, and an orbit with e of one stretches into infinity and orbital elements are used to becomes a parabola. In between, the orbits are ellipses. The orbits describe exactly the orbit of of all large planets are almost circles:The Earth, for instance, has a body. an eccentricity of 0.0068, and Mercury's eccentricity is 0.2056.

Orbital Elements

Orbital Elements

a semimajor axis i inclination

Q longitude of the ascending node to argument of the perigee

*"£r foci p center of ellipse perihelion aphelion vernal equinox ascending node semiminor axis distance from center to one focus a semimajor axis i inclination

Q longitude of the ascending node to argument of the perigee

*"£r foci p center of ellipse perihelion aphelion vernal equinox ascending node semiminor axis distance from center to one focus

M mean anomaly Mean anomaly is an angle that moves in time from 0° to 360° during one revolution, as if the planet were at the end of a hand of a clock and the Sun were at its center. This angle determines where in its orbit a planet is at a given time, and is defined to be 0° at perigee (when the planet is closest to the Sun) and 180° at apogee (when the planet is farthest from the Sun).The equation for mean motion M is given as

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