ORBITAL MECHANICS

•Conic Sections

•Orbital Elements

•Types of Orbits

•Newton's Laws of Motion and Universal Gravitation

•Uniform Circular Motion

•Motions of Planets and Satellites

•Launch of a Space Vehicle

•Position in an Elliptical Orbit

•Orbit Perturbations

•Orbit Maneuvers

•Escape Velocity

Orbital mechanics, also called ﬂight mechanics, is the study of the motions of artiﬁcial satellites and space

vehicles moving under the inﬂuence of forces such as gravity, atmospheric drag, thrust, etc. Orbital

mechanics is a modern oﬀshoot of celestial mechanics which is the study of the motions of natural

celestial bodies such as the moon and planets. The root of orbital mechanics can be traced back to the

17th century when mathematician Isaac Newton (1642-1727) put forward his laws of motion and

formulated his law of universal gravitation. The engineering applications of orbital mechanics include

ascent trajectories, reentry and landing, rendezvous computations, and lunar and interplanetary

trajectories.

Conic Sections

A conic section, or just conic, is a curve

formed by passing a plane through a right

circular cone. As shown in the ﬁgure to the

right, the angular orientation of the

plane relative to the cone determines whether

the

conic section is a circle, ellipse, parabola, or

hyerbola. The circle and the ellipse arise

when the intersection of cone and plane is a

bounded curve. The circle is a special case of

the ellipse in which the plane is

perpendicular to the axis of the cone. If the

plane is parallel to a generator line of

the cone, the conic is called a parabola.

Finally, if the intersection is an unbounded

curve and the plane is not parallel to a

generator line of the cone, the ﬁgure is a

hyperbola. In the latter case the plane will

intersect both halves of the cone, producing two

separate curves.

We can deﬁne all conic sections in terms of the eccentricity. The type of conic section is also related to the

semi-major axis and the energy. The table below shows the relationships between eccentricity, semi-major

axis, and energy and the type of conic section.

Conic Section Eccentricity, e Semi-major axis Energy

Circle 0 = radius < 0

Ellipse 0 < e < 1 > 0 < 0

Parabola 1 infinity 0

Hyperbola > 1 < 0 > 0

Satellite orbits can be any of the four conic sections. In this section we will discuss bounded conic orbits,

i.e. circles and ellipses.

Orbital Elements

To mathematically describe an orbit one must deﬁne six quantities, called orbital elements. They are

•Semi-Major Axis, a

•Eccentricity, e

•Inclination, i

•Argument of Periapsis,

•Time of Periapsis Passage, T

•Longitude of Ascending Node,

An orbiting satellite follows an oval shaped path known as an ellipse with the body being orbited, called

the primary, located at one of two points called foci. An ellipse is deﬁned to be a curve with the following

property: for each point on an ellipse, the sum of its distances from two ﬁxed points, called foci, is

constant (see ﬁgure to right). The longest and shortest lines that can be drawn through the center of an

ellipse are called the major axis and minor axis, respectively. The semi-major axis is one-half of the major

axis and represents a satellite's mean distance from its primary. Eccentricity is the distance between the

foci divided by the length of the major axis and is a number between zero and one. An eccentricity of zero

indicates a circle.

Inclination is the angular distance between a satellite's orbital plane and the equator of its primary (or the

ecliptic plane in the case of heliocentric, or sun centered, orbits). An inclination of zero degrees indicates

an orbit about the primary's equator in the same direction as the primary's rotation, a direction called

prograde (or direct). An inclination of 90 degrees indicates a polar orbit. An inclination of 180 degrees

indicates a retrograde equatorial orbit. A retrograde orbit is one in which a satellite moves in a direction

opposite to the rotation of its primary.

Periapsis is the point in an orbit closest to the primary. The opposite of periapsis, the farthest point in an

orbit, is called apoapsis. Periapsis and apoapsis are usually modiﬁed to apply to the body being orbited,

such as perihelion and aphelion for the Sun, perigee and apogee for Earth, perijove and apojove for

Jupiter, perilune and apolune for the Moon, etc. The argument of periapsis is the angular distance between

the ascending node and the point of periapsis (see ﬁgure below). The time of periapsis passage is the time

in which a satellite moves through its point of periapsis.

Nodes are the points where an orbit crosses a plane, such as a satellite crossing the Earth's equatorial

plane. If the satellite crosses the plane going from south to north, the node is the ascending node; if

moving from north to south, it is the descending node. The longitude of the ascending node is the node's

celestial longitude. Celestial longitude is analogous to longitude on Earth and is measured in degrees

counter-clockwise from zero with zero longitude being in the direction of the vernal equinox.

In general, three observations of an object in orbit are required to calculate the six orbital elements. Two

other quantities often used to describe orbits are period and true anomaly. Period, P, is the length of time

required for a satellite to complete one orbit. True anomaly, v, is the angular distance of a point in an orbit

past the point of periapsis, measured in degrees.

Types Of Orbits

For a spacecraft to achieve Earth orbit, it must be launched to an elevation above the Earth's atmosphere

and accelerated to orbital velocity. The most energy eﬃcient orbit, that is one that requires the least

amount of propellant, is a direct low inclination orbit. To achieve such an orbit, a spacecraft is launched in

an eastward direction from a site near the Earth's equator. The advantage being that the rotational speed

of the Earth contributes to the spacecraft's ﬁnal orbital speed. At the United States' launch site in Cape

Canaveral (28.5 degrees north latitude) a due east launch results in a "free ride" of 1,471 km/h (914 mph).

Launching a spacecraft in a direction other than east, or from a site far from the equator, results in an

orbit of higher inclination. High inclination orbits are less able to take advantage of the initial speed

provided by the Earth's rotation, thus the launch vehicle must provide a greater part, or all, of the energy

required to attain orbital velocity. Although high inclination orbits are less energy eﬃcient, they do have

advantages over equatorial orbits for certain applications. Below we describe several types of orbits and

the advantages of each:

Geosynchronous orbits (GEO) are circular orbits around the Earth having a period of 24 hours. A

geosynchronous orbit with an inclination of zero degrees is called a geostationary orbit. A spacecraft in a

geostationary orbit appears to hang motionless above one position on the Earth's equator. For this reason,

they are ideal for some types of communication and meteorological satellites. A spacecraft in an inclined

geosynchronous orbit will appear to follow a regular ﬁgure-8 pattern in the sky once every orbit. To attain

geosynchronous orbit, a spacecraft is ﬁrst launched into an elliptical orbit with an apogee of 35,786 km

(22,236 miles) called a geosynchronous transfer orbit (GTO). The orbit is then circularized by ﬁring the

spacecraft's engine at apogee.

Polar orbits (PO) are orbits with an inclination of 90 degrees. Polar orbits are useful for satellites that

carry out mapping and/or surveillance operations because as the planet rotates the spacecraft has access

to virtually every point on the planet's surface.

Walking orbits: An orbiting satellite is subjected to a great many gravitational inﬂuences. First, planets

are not perfectly spherical and they have slightly uneven mass distribution. These ﬂuctuations have an

eﬀect on a spacecraft's trajectory. Also, the sun, moon, and planets contribute a gravitational inﬂuence on

an orbiting satellite. With proper planning it is possible to design an orbit which takes advantage of these

inﬂuences to induce a precession in the satellite's orbital plane. The resulting orbit is called a walking

orbit, or precessing orbit.

Sun synchronous orbits (SSO) are walking orbits whose orbital plane precesses with the same period as

the planet's solar orbit period. In such an orbit, a satellite crosses periapsis at about the same local time

every orbit. This is useful if a satellite is carrying instruments which depend on a certain angle of solar

illumination on the planet's surface. In order to maintain an exact synchronous timing, it may be

necessary to conduct occasional propulsive maneuvers to adjust the orbit.

Molniya orbits are highly eccentric Earth orbits with periods of approximately 12 hours (2 revolutions per

day). The orbital inclination is chosen so the rate of change of perigee is zero, thus both apogee and

perigee can be maintained over ﬁxed latitudes. This condition occurs at inclinations of 63.4 degrees and

116.6 degrees. For these orbits the argument of perigee is typically placed in the southern hemisphere, so

the satellite remains above the northern hemisphere near apogee for approximately 11 hours per orbit.

This orientation can provide good ground coverage at high northern latitudes.

Hohmann transfer orbits are interplanetary trajectories whose advantage is that they consume the

least possible amount of propellant. A Hohmann transfer orbit to an outer planet, such as Mars, is

achieved by launching a spacecraft and accelerating it in the direction of Earth's revolution around the sun

until it breaks free of the Earth's gravity and reaches a velocity which places it in a sun orbit with an

aphelion equal to the orbit of the outer planet. Upon reaching its destination, the spacecraft must

decelerate so that the planet's gravity can capture it into a planetary orbit.

To send a spacecraft to an inner planet, such as Venus, the spacecraft is launched and accelerated in the

direction opposite of Earth's revolution around the sun (i.e. decelerated) until it achieves a sun orbit with

a perihelion equal to the orbit of the inner planet. It should be noted that the spacecraft continues to move

in the same direction as Earth, only more slowly.

To reach a planet requires that the spacecraft be inserted into an interplanetary trajectory at the correct

time so that the spacecraft arrives at the planet's orbit when the planet will be at the point where the

spacecraft will intercept it. This task is comparable to a quarterback "leading" his receiver so that the

football and receiver arrive at the same point at the same time. The interval of time in which a spacecraft

must be launched in order to complete its mission is called a launch window.

Newton's Laws of Motion and Universal Gravitation

Newton's laws of motion describe the relationship between the motion of a particle and the forces acting

on it.

The ﬁrst law states that if no forces are acting, a body at rest will remain at rest, and a body in motion will

remain in motion in a straight line. Thus, if no forces are acting, the velocity (both magnitude and

direction) will remain constant.

The second law tells us that if a force is applied there will be a change in velocity, i.e. an acceleration,

proportional to the magnitude of the force and in the direction in which the force is applied. This law may

be summarized by the equation

where F is the force, m is the mass of the particle, and a is the acceleration.

The third law states that if body 1 exerts a force on body 2, then body 2 will exert a force of equal

strength, but opposite in direction, on body 1. This law is commonly stated, "for every action there is an

equal and opposite reaction".

In his law of universal gravitation, Newton states that two particles having masses m1 and m2 and

separated by a distance r are attracted to each other with equal and opposite forces directed along the

line joining the particles. The common magnitude F of the two forces is

where G is an universal constant, called the constant of gravitation, and has the value 6.67259x10 -11 N-

m2/kg2 (3.4389x10-8 lb-ft2/slug2).

Let's now look at the force that the Earth exerts on an object. If the object has a mass m, and the Earth

has mass M, and the object's distance from the center of the Earth is r, then the force that the Earth exerts

on the object is GmM /r2 . If we drop the object, the Earth's gravity will cause it to accelerate toward the

center of the Earth. By Newton's second law (F = ma), this acceleration g must equal (GmM /r2)/m, or

At the surface of the Earth this acceleration has the valve 9.80665 m/s2 (32.174 ft/s2).

Many of the upcoming computations will be somewhat simpliﬁed if we express the product GM as a

constant, which for Earth has the value 3.986005x1014 m3/s2 (1.408x1016 ft3/s2). The product GM is often

represented by the Greek letter .

For additional useful constants please see the appendix Basic Constants.

For a refresher on SI versus U.S. units see the appendix Weights & Measures.

Uniform Circular Motion

In the simple case of free fall, a particle accelerates toward the center of the Earth while moving in a

straight line. The velocity of the particle changes in magnitude, but not in direction. In the case of uniform

circular motion a particle moves in a circle with constant speed. The velocity of the particle changes

continuously in direction, but not in magnitude. From Newton's laws we see that since the direction of the

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