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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

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.

Semimajor and Semiminor Axes, Foci

Eccentricity of Pluto's Orbit Compared to a Circle

A circle

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

dimensionless and ranges from 200 to one, where an eccentricity of zero means that the figure is a circle, and an eccentricity of one 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 66). Another equation for eccentricity is c 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 here. Pluto has the most eccentric orbit, and Mercury's orbit is also very eccentric, but the rest have eccentricities below 0.09.

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

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 R 2, 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

Elliptical planetary orbit

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.

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, not as a circle, as had been previously assumed. Kepler defined three laws of orbital motion (listed in the table on page 70), which he published in 1609 and 1619 in his books New Astronomy and The Harmony of the World. These three laws, listed in the following table, are still used as the basis for understanding orbits.

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 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 on page 73).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 semimajor axis is typically expressed in units of AU. Neptune's semimajor axis is 30.069 AU. e eccentricity Eccentricity measures the amount by which an ellipse differs from a circle, as described above. An orbit with e = 0 is circular, and an orbit with e = 1 stretches into infinity and becomes a parabola. In between, the orbits are ellipses.The orbits of all large planets are almost circles:The Earth, for instance, has an eccentricity of 0.0068, and Neptune's eccentricity is 0.0113. M the mean anomaly Mean anomaly is an angle that moves in time from zero to 360 degrees 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 zero degrees at perigee (when the planet is closest to the Sun), and 180 degrees at apogee (when the planet is farthest from the Sun). The equation for mean anomaly M is given as

where M is the value of M at time zero, T is the orbital period, and t is the time in question.

The next three Keplerian elements determine where the orbit is in space.

i inclination In the case of a body orbiting the Sun, the inclination is the angle between the plane of the orbit of the body and the plane of the ecliptic (the plane in which the Earth's orbit lies). In the case of a body orbiting the Earth, the inclination is the angle between the plane of the body's orbit and the plane of the Earth's equator, such that an inclination of zero indicates that the body orbits directly over the equator, and an inclination of 90 indicates that the body orbits over the poles. If there is an orbital inclination greater than zero, then there is a line of intersection between the ecliptic plane

OBLIQUITY, ORBITAL INCLINATION, AND

ROTATION DIRECTION FOR ALL THE PLANETS

Orbital inclination to the

Obliquity (inclination of ecliptic (angle between the

the planet's equator to planet's orbital plane and

its orbit; tilt); remarkable the Earth's orbital plane);

Planet

values are in italic remarkable values are in italic Rotation direction

Mercury

0° (though some scientists 7.01° believe the planet is flipped over, so this value may be 180°)

prograde

Venus

177.3° 3.39°

retrograde

Earth

23.45° 0° (by definition)

prograde

Mars

25.2° 1.85°

prograde

Jupiter

3.12° 1.30°

prograde

Saturn

26.73° 2.48°

prograde

Uranus

97.6° 0.77°

retrograde

Neptune

29.56° 1.77°

prograde

Pluto

122.5° 17.16°

retrograde

Orbital Elements

Orbital Elements

A series of parameters called orbital elements is used to describe exactly the orbit of a and the orbital plane.This line is called the line of nodes. Neptune's orbital inclination is 1.769 degrees (see table on page 72). ^ longitude of the ascending node After inclination is specified, an infinite number of orbital planes is still possible: The line of nodes could cut through the Sun at any longitude around the Sun. Notice that the line of nodes emerges from the Sun in two places. One is called the ascending node (where the orbiting planet crosses the Sun's equator going from south to north).The other is called the descending node (where the orbiting planet crosses the Sun's equator going from north to south). Only one node needs to be specified, and by convention the ascending node is used. A second point in a planet's orbit is the vernal equinox, the spring day in which day and night have the same length (equinox means "equal night"), occurring where the plane of the planet's equator intersects its orbital plane.The angle between the vernal

A series of parameters called orbital elements is used to describe exactly the orbit of a equinox y and the ascending node N is called the longitude of the ascending node. Neptune's longitude of the ascending node is 131.722 degrees.

w argument of the perigee The argument of the perigee is the angle (in the body's orbit plane) between the ascending node N and perihelion P measured in the direction of the body's orbit. Neptune's argument of the perigee is 273.249 degrees.

The complexity of the six measurements shown above demonstrates the extreme attention to detail that is necessary when moving from simple theory ("every orbit is an ellipse") to measuring the movements of actual orbiting planets. Because of the gradual changes in orbits over time caused by gravitational interactions of many bodies and by changes within each planet, natural orbits are complex, evolving motions. To plan with such accuracy space missions such as the recent Mars Exploration Rovers, each of which landed perfectly in their targets, just kilometers long on the surface of another planet, the mission planners must be masters of orbital parameters. The complexity of calculating an actual orbit also explains the difficulty earlier astronomers had in calculating Neptune's, Uranus's, and Saturn's orbits. Their skill is apparent in the fact that only an incorrect value for Neptune's mass stood between them and reproducing the planet's orbits perfectly.The discrepancy stood for more than 100 years, until Voyager 2 visited Neptune and other planets and took more precise measurements.

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