ASTR 121 (O'Connell) Study Guide
8. GRAVITATIONAL ORBITS AND SPACE FLIGHT
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. The various kinds of
possible orbits are described further in this lecture.
In the mid-twentieth century, Newton's work became the key element in
space technology. In turn, 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 the
material we will discuss in the rest of this course.
A. NEWTONIAN ORBIT THEORY
"Newton's Mountain": Orbital type depends
on initial velocity.
5) Newton showed that objects in bound elliptical orbits obey Kepler's
Second and Third Laws.
He found that the constant K in the formulation of Kepler's Third
Law in Guide 7 depends on
the mass of the primary body (i.e. the Sun in the case of the
planetary orbits and the Earth in the case of orbiting spacecraft).
The larger the mass of the primary, the shorter the
period for a given orbital size.
- The Third Law therefore has an invaluable
application: the motions of orbiting objects can be used to
determine the mass of the primary.
This is true no matter
how far away the objects are (as long as the orbital motion
can be detected).
- For example, the periods and sizes of the orbits
of the Galilean satellites of Jupiter can be used to
determine Jupiter's mass (as in
Optional Lab 3).
- The Third Law has been critical to such diverse
astronomical problems as measuring the mass of "exoplanets" around
other stars (see Study Guide
11) and establishing the existence of "Dark
Matter" in the distant universe.
6) Orbits are independent of the mass of the orbiting object
Free motion in response to gravity (in the absence of other forces) is
called "free-fall" motion. All four types of conic section
orbits are free-fall.
The mass of the orbiting body always
cancels out of the expression for acceleration under gravity.
E.g. 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.
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). This was first demonstrated experimentally by Galileo and was the subject of Puzzlah#7 (see also Study Guide 7)
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 one
expression of this fact:
- 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, regardless of their mass.
- According to Kepler's Third, 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. 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
geostationary). See the animation above. Most
communications satellites are in such orbits.
B. SPACE FLIGHT
- In a rocket
engine, fuel is burned rapidly 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 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.
- You can think of a rocket in the abstract as a device for
changing from one free-fall orbit to another using the reaction
force of expelled hot gas.
- With its engine turned off, the motion of any
spacecraft is a free-fall orbit.
Likewise, any object (e.g. an astronaut) inside the
rocket is also in a free-fall orbit. A "floating" astronaut is
simply moving in a free-fall orbit that parallels that of the
spacecraft. Both the spacecraft and astronaut have identical
accelerations under the external gravitational fields.
- If the engine is on, the craft is
not in free fall and its orbit will depart from a conic
section.
- An example is shown here.
- The main challenge to spaceflight: obtaining the power needed to
reach escape velocity. For Earth, this is 11 km/sec or 25,000 MPH.
Most scientific spacecraft for planetary missions are relatively small
(i.e. low mass) in order that standard rocket engines can propel them
past Earth escape velocity. This means that many clever
strategies are needed to pack high performance into light packages.
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.
The Apollo program used the extremely powerful Saturn V rockets to launch
large payloads (including 3 crew members) to the Moon. This
technology was, however, retired in the mid-1970's.
The Space Shuttle (at right) is fueled by high energy
liquid oxygen and liquid hydrogen plus solid-rocket boosters. But it
is so massive compared to the power of its engines that it
cannot reach escape velocity. Its maximum altitude is only
about 300 miles. That is why NASA is developing new rocket technologies in its initiative to return
to the Moon.
C. INTERPLANETARY SPACE MISSIONS
Beginning in the early 1960's, NASA (and. to a lesser extent,
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,
and landers.
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 never 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 2006, we had flown at close range past every planet except Pluto;
had placed satellite observatories into orbit around the Moon, Venus,
Mars, Jupiter, Saturn, and the asteroid Eros; had soft-landed on
the Moon, Venus, Mars, and Saturn's moon Titan; and had sent probes
into a comet nucleus and the atmosphere of Jupiter.
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
Solar System has come from the powerful robot observatories.
For a list of these and additional links, click here.
Reading for this lecture:
Seeds text: 5.1 and 5.2 (Newtonian dynamics & gravitational orbits)
Study Guide 8
Reading for next lecture:
Web links:
Last modified
February 2008 by rwo
Text copyright © 1998-2008 Robert W. O'Connell. All
rights reserved. Conic section drawings from ASTR 161 UTenn at
Knoxville. Orbital types drawing copyright ©
Brooks/Cole-Thomson. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 121 at the
University of Virginia.
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