ASTR 1210 (O'Connell) Study Guide

8. GRAVITATIONAL ORBITS AND SPACE FLIGHT

Space Shuttle Discovery launches on
a mission to the Space Station, 2001

A. Newtonian Orbit Theory

Orbital Dynamics

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 motion.
To predict the path, we simply substitute Newton's expression for F_grav 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 invent calculus.

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 Moon around the Earth.
Pick any location on the Moon'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 Moon'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 Moon 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 below.
In Newtonian gravity, the gravitational force acts radially --- i.e. along the line connecting the Earth and the Moon. Accordingly, both the acceleration and the change in the Moon'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 Earth, then the Moon should fall faster and faster on a radial trajectory until it crashes into the Earth. That's exactly what would happen if the Moon were ever stationary in its orbit. In that case, the situation in the left hand drawing above (straight-line acceleration toward Earth) would prevail.
But if the Moon'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 Earth. If the combination of sideways motion and distance from the Earth is correct, the Moon will avoid collision with the Earth, and it will stay in permanent orbit. The animation at the right shows the situation for an elliptical orbit around the Sun (the blue line is the velocity vector, the green line is the acceleration). Note that where the planet is nearest the Sun, the gravitational force and inward acceleration are greatest, but the sideways motion is also greatest, which prevents the planet from colliding with the Sun. That motion is in accordance with Kepler's second law.
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

• 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: V_esc(R) = √(2GM/R), where R is the separation between the two objects and M is the mass of the primary object.
    V_esc for the Earth at the Earth's surface is 25,000 mph (or 11 km/s). V_esc for the Sun at the Earth's 1 AU distance from the Sun is 94,000 mph (42 km/s).

  
• If V < V_esc, 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 ≥ V_esc, 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 motion.

Newton's Mountain

"Newton's Mountain": orbit type depends on initial velocity.
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-Fall" 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. 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 Earth.

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 demonstrated experimentally by Galileo and was the subject of a Puzzlah (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 another manifestation of this fact.
A more familiar manifestation is the phenomenon of "floating" astronauts on space missions. 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 non-gravitational force.
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 turn off.
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 mph

Geosynchronous Orbits

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 writer Arthur 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 size.
• 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 measured).

• In the Solar System, the Third Law allows us to determine the mass of the Sun from the size and period 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 in 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 (V_e).

C. Space Flight

• 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.
  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. ("Standard" engines are designed for launching commercial payloads to synchronous orbit or delivering intercontinental ballistic missles---neither of which involve reaching 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, over 1000 times more! New Horizons is currently beyond the orbit of Uranus and is traveling at over 34,000 mph.

  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 mistake.
  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 developing new "heavy lift" rocket technologies to replace the Shuttle.

D. Interplanetary Space Missions

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.

For a list of these missions and additional links, click here.

Bennett textbook: Ch. 4.1-4.4 (Newtonian dynamics & gravitational orbits)
Study Guide 8

Study Guide 9

Recommended interactive sites illustrating orbits:
Newton's Mountain (Java animation), from Mike Fowler's UVa Physics course.
Gravity Chaos (Flash animation)--try to build multi-planet systems that don't self-destruct.

Orbits and Newton's Laws (Wikipedia)
Newton's Laws & Rockets. (Also see other pages on aeronautics at this site.)
Newton's Explanation of Kepler's Laws (U. Tenn)
JPL Notes on Basics of Space Flight
History of Space Flight (James Schombert, Univ. of Oregon)
Wade's Encyclopedia of Astronautics
Retrospective on 50 Years of the Space Race (New York Times)
The Saturn Moon rockets (the most powerful ever)
Complete list of current, planned, and past NASA science missions
NASA/Johnson Space Center Digital Image Collection
Introduction to UV and Space Astronomy (O'Connell)
The Astro/Ultraviolet Imaging Telescope Missions(O'Connell)
Ever wonder why space flight is so expensive? It's because thousands of people are involved in almost any space endeavor. To get a feel for what has to be done, watch this:
Apollo 11 Launch Video (slo-mo, up-close-and-personal view of a Saturn V)
or this:
Shuttle Discovery Prep to Launch (remarkable time-lapse video)

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Last modified February 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.