ASTR 1210 (O'Connell) Study Guide
8. GRAVITATIONAL ORBITS AND SPACE FLIGHT
Space Shuttle Discovery launches on
to the Space Station, 2001
"There will certainly be no lack of human pioneers when we have
mastered the art of [space] flight....Let us create vessels and sails adjusted
to the heavenly ether, and there will be plenty of people unafraid of
the empty wastes. In the meantime we shall prepare, for the brave
sky-travelers, maps of the celestial bodies. " |
---- Johannes Kepler (1610)
Newton's theories of dynamics and gravity provided a complete
understanding of the interaction between gravitating bodies and the
resulting orbits for planets and satellites. This guide first
describes the nature of possible gravitational orbits
and some implications of those.
In the mid-twentieth century, Newton's work became the key conceptual
element in space technology, which is introduced in the second
part of the guide. Space technology---rockets, the Space
Shuttle, dozens of robot spacecraft, the human space program---has
provided most of our present knowledge of the Solar System and most of
the material we will discuss in the rest of this course.
A. Newtonian Orbit Theory
Newton's theory can accurately predict gravitational orbits
because it allows us to determine the acceleration of an object
in a gravitational field. Acceleration is the
rate of change of an object's velocity.
If we know the
initial position and velocity of an object, knowing its acceleration
at all later times is enough to completely determine its later path of
To predict the path, we simply substitute Newton's expression
for Fgrav for the force term in his Second
Law and solve for acceleration.
But there is a major complication. The Second Law is not a simple
algebraic expression. Both velocity and acceleration are rates of
change (of position and velocity, respectively). Mathematically,
they are derivatives. The gravitational force changes with
position. Finally, velocity, acceleration and the gravitational force
all have a
directionality as well as a magnitude associated with
them. That is, they are "vectors".
So the Second Law is really a differential vector
equation. To solve it, Newton had to
We don't need to know the mathematical details in order to
understand the basic interaction that shapes Newtonian orbits. Take
as an example the orbit of the Earth around the Sun.
Pick any location on the Earth's orbit. Represent its velocity at that
location as an arrow (a vector) showing the direction and magnitude
of its motion.
An essential element of Newtonian theory is that changes in
the magnitude of the velocity vector (the speed) or in
the direction of motion are both considered to be
"accelerations." In the following drawings, the red arrows represent
the Earth's velocity vector and the blue arrows represent the applied
gravitational force. According to Newton's Second Law, the change
in the velocity vector (a speed-up in the first case or a
deflection of the direction of motion in the second) is in the
direction of the applied force.
Starting from any location, the instantaneous velocity vector and the
rate of change of that vector (the acceleration) combine to determine
where the Earth will be at the next moment of time. Adding up the
motion from one moment to the next traces out the orbital path.
Newton discovered that, for his gravitational force law, the paths
would always be one of the four kinds of "conic sections" described
In Newtonian gravity, the gravitational force acts
radially --- i.e. along the line connecting the Earth and the
Sun. Accordingly, both the
acceleration and the change in the Earth's velocity
vector from one moment in time to the next will also always be in
the radial direction.
You might think that if the acceleration is always toward the Sun,
then the Earth should fall faster and faster on a radial trajectory
until it crashes into the Sun. That's exactly what would
happen if the Earth were ever stationary in its orbit.
In that case, the situation in the left hand drawing above (straight-line
acceleration toward the Sun) would prevail.
But if the Earth's velocity vector has a component which is perpendicular
to the radial direction, then in any interval in time, it will move
"sideways" at the same time as it accelerates toward the Sun.
If the combination of sideways motion and distance from the Sun is
correct, the Earth will avoid collision with the Sun, and it will
stay in permanent orbit. The animation at the right shows the
situation for the Earth's (exaggerated) elliptical orbit around the Sun (the blue line is the
velocity vector, the green line is the acceleration). Note that where
the Earth is nearest the Sun, the gravitational force and inward acceleration
are greatest, but the sideways motion is also greatest, which prevents us
from colliding with the Sun. That motion is in accordance with Kepler's
Therefore, all permanently orbiting bodies are perpetually
falling toward the source of gravity but have enough sideways
motion to avoid a collision.
Kinds of Gravitational Orbits
In the case of two gravitating objects (for example, the Earth and the
Moon, the Sun and a planet, or the Earth and an artificial satellite),
the full solutions of Newton's equations give the following
You can interactively explore the relation between the orbit and the
initial velocity vector using the Flash animation
- The relative orbit is confined to a geometric plane.
- The shape of the orbit within the plane is a "conic section",
of which there are only four types.
- A circle
- An ellipse
- A parabola
- A hyperbola
See the illustration at the right.
- The orbital type is determined by the
initial distance, speed and
direction of motion of the orbiting object, as follows:
- Define the "escape velocity" at a given
distance: Vesc(R) = √(2GM/R), where R
is the separation between the two objects and M is
the mass of the primary object.
Vesc for the Earth at the Earth's surface is 25,000 mph (or 11 km/s).
Vesc for the Sun at the Earth's 1 AU distance from the Sun
is 94,000 mph (42 km/s).
- If V < Vesc, the orbit is an ellipse
or circle. It is said to be "closed" or
"bound". The smaller object will permanently
repeat its orbital motion.
- If V ≥ Vesc, the orbit is a parabola or
hyperbola. It is said to be "open" or "unbound".
The smaller object escapes and does not return.
- Only specific values of velocity will yield circular or parabolic
orbits. An object moving exactly at escape velocity will move on a
parabola. To achieve a circular orbit an object must move at 71% of
the escape velocity, and its velocity must be exactly
perpendicular to the radial direction.
- As noted earlier, shapes and motions within the "closed" orbits
for the planets satisfy all three of Kepler's Laws of planetary
Newton illustrated orbital behavior for a simple idealized situation
where a powerful cannon is placed on top of a high mountain on the
Earth. Since both the distance from Earth's center and the direction
of initial flight is fixed, the cannonball follows an orbit that
depends only on the muzzle velocity of the cannon as shown below:
The gravitational force of a spherical body like the Earth acts as
though it originates from the center of the sphere, so elliptical
orbits have the center of the Earth at one focus.
"Newton's Mountain": orbit type depends on
From lower to higher velocities, orbit shapes are:
ellipse, circle, ellipse, parabola, hyperbola.
"Escape velocity" (which
is 25,000 mph at Earth's surface) produces
a parabolic orbit.
B. Important Implications of Newtonian Orbits
Free motion in response to gravity (in the absence of other forces) is
called "free-fall" motion. Conic section
orbits are all "free-fall orbits."
Remember that motion is normal in free-fall. For instance, the
engines of spacecraft do not have to be on in order for them to
move through space on a free-fall orbit. They will "coast" forever on
such an orbit.
Note also that free-fall orbits will depart from simple conic
sections if an object is under the influence of
more than one gravitating body. For instance, comets
are often deflected from their Sun-dominated simple conic orbits by
Jupiter's gravity (see Guide 21).
Free-fall orbits are independent of the mass of the orbiting object.
The mass of the orbiting body always
cancels out of the expression for acceleration under gravity.
For instance, in the case of a planet orbiting the Sun, the
gravitational force on the planet is directly proportional to
the planet's mass but, according to Newton's Second Law, the resulting
acceleration is inversely proportional to its mass. Hence,
mass drops out from the expression for acceleration.
This is true for all orbits under gravity. Hence, a tennis ball in
space, if it were moving with the same speed and direction as the
Earth at any point, would follow the same orbital path as the
Kepler's Third Law (that the orbital period of a planet around the Sun
depends only on orbital size, not on the mass of the planet) is another
manifestation of this fact.
Therefore, the acceleration of all objects is the same in a
given gravity field (e.g. at a given distance from the Sun or near the
Earth's surface), regardless of their masses. This was first
and was the subject of a Puzzlah (see also Study Guide 7)
A more familiar manifestation is the phenomenon of "floating"
astronauts on space missions. Even though the spacecraft is much
more massive, both the astronauts and the
spacecraft have identical accelerations under the external gravitational
fields. They are moving on parallel free-fall orbits, so the
astronauts appear to be floating and stationary with respect to
the spacecraft, even though (in near-Earth orbit) they might be
moving at tens of thousands of miles per hour.
Rocket engines are described under (C) below. You can think of a
rocket engine in the abstract as a device for
changing from one free-fall orbit to another by applying a
With its engine turned off, the motion of any
spacecraft is a free-fall orbit.
If the engine is on, the craft is
not in free fall. For instance, the orbit of the Space Shuttle
launching from the Earth will depart from a conic section until its engines
An example of using a rocket engine to change from one
free-fall orbit to another is shown here.
The Russian "Mir" space station (1986-2001) orbiting
Earth at an altitude of 200 miles with a velocity of 17,000
According to Kepler's Third Law, the orbital period of a satellite will
increase as its orbital size increases. We have exploited that
fact in developing one of the most important practical
applications of space technology: geosynchronous satellites.
- Spacecraft in "low" Earth orbits (less than about 500 mi),
like the Mir space station (seen above) or the Space Shuttle, all
orbit Earth in about 90 minutes, at 17,000 miles per hour,
regardless of their mass.
- The orbital period of a spacecraft in a
larger orbit will be longer. For an orbit of radius
about 26,000 mi, the period will be 24 hours---the same
as the rotation period of the Earth. Spacecraft here, if they are
moving in the right direction, will appear to "hover" over a
given point on the Earth's surface. These orbits are therefore called
geosynchronous or "geostationary." See the animation
above. This is the ideal location for placing communications satellites.
[The concept of geosynchronous communications satellites was first
proposed by science fiction
C. Clarke. He deliberately did not patent his idea, which became
the basis of a trillion-dollar industry.]
Applications of Kepler's Third Law
Newton's theory provided a physical interpretation of Kepler's Third
Law. According to his formulation of gravitational force, he found
that the value of the constant K in the formulation of Kepler's Third
Law in Guide 7 is K =
4π2/GM, where M is the mass of the Sun.
More generally, K depends on the mass of the primary body
(i.e. the Sun in the case of the planetary orbits but the Earth in the
case of orbiting spacecraft). The larger the mass of the
primary, the shorter the period for a given orbital
- The Third Law therefore has an invaluable astrophysical
application: once the value of the "G" constant has been determined
(in the laboratory), the motions of orbiting objects can be used to
determine the mass of the primary. This is true no matter how far
from us the objects are (as long as the orbital motion and size can be
- In the Solar System, the Third Law allows us to determine the
mass of the Sun from the size and periods of the planetary orbits. In
the case of Jupiter, for example, the periods and sizes of the orbits
of the Galilean satellites can be used to determine Jupiter's mass (as
Optional Lab 3).
- The Third Law has been critical to such diverse
astronomical problems as measuring the masses of "exoplanets" around
other stars (see Study Guide
11) and establishing the existence of "Dark
Matter" in distant galaxies.
Schematic diagram of a liquid-fueled rocket
engine. The thrust of the engine is
proportional to the velocity
of the exhaust gases (Ve).
C. Space Flight
If the primary technology enabling space flight is Newtonian
orbit theory, the second most important technology is the
- In a rocket engine such as that shown in the diagram above, fuel
is burned rapidly in a combustion chamber and converted into a large
quantity of hot gas. The gas creates high pressure, which causes it
to be expelled out a nozzle at very high velocity.
The exhaust pressure
simultaneously forces the body of the rocket forward. You can think
of the rocket as "pushing off" from the moving molecules of exhaust
gas. The higher the exhaust velocity, the higher the thrust.
Note: rockets do not "push off" against the air or against the
Earth's surface. Rather, it is the "reaction force" between the
expelled exhaust and the rocket that impells the rocket forward.
Designers work to achieve the highest possible exhaust velocity per
gram of fuel. Newton's second law of motion and various elaborations
of it are essential for understanding and designing rocket motors.
- The main challenge to spaceflight is obtaining the power needed to
reach escape velocity. For Earth, this is 11 km/sec or 25,000 mph.
"Standard" rocket engines are designed for launching commercial
payloads to synchronous orbit or delivering intercontinental ballistic
missles---neither of which involve reaching escape velocity from
Earth. Therefore, most scientific spacecraft for planetary missions
are relatively small (i.e. low mass) in order that standard engines
can propel them past Earth escape velocity. This means that many
clever strategies are needed to pack high performance into light
Example: The New Horizons
spacecraft, launched on a super-high velocity trajectory to Pluto in
2006, has a mass of only 1050 lbs; its launching rocket weighed
1,260,000 lbs, over 1000 times more! New Horizons is currently
beyond the orbit of Uranus and is traveling at over 34,000
A rocket launched at exactly escape velocity from a given parent
body will, at very large distances, slow to exactly zero velocity
with respect to that body (ignoring the effect of other gravitating
The Apollo program used the extremely
powerful Saturn V rockets to
launch payloads with masses up to 100,000 pounds (including 3 crew
members) to the Moon. This technology was, however, retired in the
mid-1970's because it was thought, erroneously, that the reusable
Space Shuttle vehicles would be cheaper to operate. A big
The Space Shuttle (shown above) was fueled by high energy
liquid oxygen and liquid hydrogen plus solid-rocket boosters. But it
was so massive compared to the power of its engines that it
could not reach escape velocity from Earth. Its maximum altitude
is only about 300 miles. That is why NASA is
"heavy lift" rocket technologies to replace the Shuttle.
D. Interplanetary Space Missions
Beginning in the early 1960's, NASA and
foreign space agencies developed a series of ever-more sophisticated
robot probes to study the Sun, Moon, planets, and the
interplanetary medium. These included flyby spacecraft,
orbiters, landers, rovers, and sample-return vehicles.
The mid-20th century was the first time humans had ever sent
machines beyond the Earth's atmosphere. Even such far-sighted
thinkers as Galileo and Newton had not envisioned that to be
possible in the mere 350 years that elapsed between Kepler's Laws and
the first landings on the Moon. This was an amazing accomplishment.
By 2012, we had flown at close range past every planet except Pluto;
had placed robotic observatories into orbit around the Moon, Mercury,
Venus, Mars, Jupiter, Saturn, and the asteroids Eros and Vesta; had
soft-landed on the Moon, Venus, Mars, and Saturn's moon Titan; had
returned to Earth samples obtained from the coma of the comet Wild 2
and from a soft-landing on the asteroid Itokawa; and had sent probes
into a comet nucleus and the atmosphere of Jupiter. At right is an
artist's concept painting of the Cassini mission in orbit around
Saturn. We also put a number of highly capable observatories for
studying the distant universe (such as the Hubble Space Telescope and
the Chandra X-Ray Observatory) into orbit around the Earth and the
Of course, the Apollo program in the 1960's also sent human beings to
the Moon. This was very fruitful in learning about lunar geology and
surface history. But, by far, most of what we know about the
denizens of the Solar System has come from the powerful robot
For a list of these missions and additional
links, click here.
Reading for this lecture:
Bennett textbook: Ch. 4.1-4.4 (Newtonian dynamics &
Study Guide 8
Reading for next lecture:
September 2013 by rwo
Text copyright © 1998-2013 Robert W. O'Connell. All
rights reserved. Orbital animation copyright © Jim Swift,
Northern Arizona University. Conic section drawings from ASTR
161, University of Tennessee at Knoxville. Newton's Mountain
drawing copyright © Brooks/Cole-Thomson. These notes are
intended for the private, noncommercial use of students enrolled
in Astronomy 1210 at the University of Virginia.