ASTR 1210 (O'Connell) Study Guide




8. GRAVITATIONAL ORBITS AND SPACE FLIGHT


STS-105 Launch

Space Shuttle Discovery launches on
a mission 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)


Kepler was right about the multitudes of people eager to travel into space, but it took another 350 years of technological development to build the "vessels" needed to carry them. Space travel is difficult. However, the theoretical key to space flight was discovered by Newton only 80 years after Kepler's work.

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.

Two hundred and fifty years after he died, Newton's work became the foundation of space technology, which is introduced in the second part of the guide. Space technology---rockets, the Space Shuttle, scores 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. Commercial space technology (e.g. GPS, communications, and remote observing satellites) is already an integral part of modern life.

The mid-20th century was the first time humans had ever sent machines beyond the Earth's atmosphere. By 2015, we had explored every large body in the Solar System out to the orbit of Pluto. Even such far-sighted thinkers as Galileo and Newton himself would never have thought that possible in the mere 400 years that had elapsed since Kepler's Laws were formulated. This was an amazing accomplishment, the greatest exploratory feat of humanity to date.


A. Newtonian Orbit Theory

Orbital Dynamics

Newton's theory can accurately predict gravitational orbits because it allows us to determine the acceleration of an object in a gravitational field.

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), Newton found that the full solutions of his equations give the following results:

You can interactively explore the relation between the orbit and the initial velocity vector using the Flash animation Gravity Chaos.


Newton's Mountain

Newton illustrated orbital behavior for a simple idealized situation where a powerful cannon is fixed in position on top of a high mountain on the Earth's equator. It is allowed to fire only with its barrel parallel to the Earth's surface (see the illustration below). Since both the distance from Earth's center and the direction of initial flight are fixed, the cannonball follows an orbit that depends only on the muzzle velocity of the cannon as shown below.

Orb-types

"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.


General Relativity

Much later (1915), Newton's theory was shown by Albert Einstein to be inadequate in the presence of large masses or over large distances and has been replaced by the General Theory of Relativity in such situations. Relativity theory profoundly changed our understanding of space and time, for example by demonstrating that mass and energy can affect the structure of space and time, something that Newton never contemplated. It is much more complicated mathematically than Newton's formulation. But as a practical matter, Newton's theory is an entirely satisfactory description of "everyday" gravity. Only very minor corrections to the Newtonian predictions are necessary, for example, to send spacecraft with high accuracy throughout the solar system.


B. Important Implications of Newtonian Orbits

"Free-Fall" Orbits

The Russian "Mir" space station (1986-2001) orbiting Earth in free-fall at an altitude of 200 miles with a velocity of 17,000 mph

Geosynchronous Orbits

Applications of Kepler's Third Law


Liquid Rocket Engine

Schematic diagram of a liquid-fueled rocket engine. Rockets carry both fuel and an oxidizer,
which allows the fuel to burn even in the absence of an oxygen-rich atmosphere.
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 rocket engine.


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.

We also put a number of highly capable remote-controlled observatories for studying the Solar System and the distant universe (such as the Hubble Space Telescope and the Chandra X-Ray Observatory) into orbit around the Earth and the Sun.

Of course, the Apollo program in the 1960's also sent human beings to the Moon. But, by far, most of what we know about the denizens of the Solar System has come from our powerful robot missions and observatories, most of which were not even in development by the end of the Apollo program in 1972.

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

For a nice, prospective view of what human expansion over the next couple centuries into the deep space of our Solar System might look like, see this video, based on Carl Sagan's writings.


E. The Cost of Space Missions

Ever wonder why space flight is so expensive? It's because thousands of people are involved in almost any space endeavor, and it takes anywhere from 5 to 30 years to prepare a space mission.

Spacecraft require not only their own launching and maneuvering engines but also command and control systems, power supplies, thermal control, pointing and stabilization systems, protection from a harsh environment, sensors and scientific instruments, data taking and storage systems, communications systems, and more -- all of these subject to stringent weight and volume constraints -- plus extensive Earth-based infrastructure to monitor and operate them. All those people are required in order to design and build these myriad components and then certify that they will be at least 95% reliable after launch --- because, with very few exceptions, no repair is possible. They must identify and mitigate all possible failure modes. Reliability is the principal cost driver of space missions. Needless to say, because of the demands of human safety, the cost of crewed missions greatly exceeds that of comparably-scaled robot missions.

Two examples: The overall unique scientific, commercial or exploratory value of a space mission must be weighed carefully against its projected cost.

To get a feel for the scale of effort involved in preparing and launching major spacecraft, watch this:


HST on orbit

The Hubble Space Telescope on orbit 300 miles above Earth


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Last modified December 2020 by rwo

Text copyright © 1998-2020 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.