Gravity Rules


First Flight


Gravitation is the mutual attraction of all masses in the universe. While its effect decreases in proportion to distance squared, it nonetheless applies, to some extent, regardless of the sizes of the masses or their distance apart. The concepts associated with planetary motions developed by Johannes Kepler (1571-1630) describe the positions and motions of objects in our solar system. Isaac Newton (1643-1727) later explained why Keplers laws worked, by showing they depend on gravitation. Albert Einstein (1879-1955) posed an explanation of how gravitation works in his general theory of relativity, which equates gravity with acceleration.

Isaac Newton realized that the force that makes apples fall to the ground is the same force that makes the planets "fall" around the Sun. Newton had been asked to address the question of why planets move as they do. He established that a force of attraction toward the sun becomes weaker in proportion to the square of the distance from the Sun. Newton postulated that the shape of an orbit should be an ellipse. Circular orbits are merely a special case of an ellipse where the foci are coincident. Newton described his work in the Mathematical Principles of Natural Philosophy, or the Principia, which he published in 1685. Newtons Laws describe the behavior of inertia, they do not explain what the nature of inertia is. This is still a valid question, as is the nature of mass.

Newton gave his Laws of Motion as follows:


1. Every body continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

2. The change of motion (linear momentum) is proportional to the force impressed and is made in the direction of the straight line in which that force is impressed.

3. To every action there is always an equal and opposite reaction, or, the mutual actions of two bodies upon each other are always equal, and act in opposite directions.


There are three ways to modify the momentum of a body. The mass can be changed, the velocity can be changed, (acceleration), or both. Force (F) equals change in velocity (acceleration, A) times mass (M): F = MA. Acceleration may be produced by applying a force to a mass (such as a spacecraft). If applied in the same direction as an object's velocity, the objects velocity increases in relation to an unaccelerated observer. If acceleration is produced by applying a force in the opposite direction from the objects original velocity, it will slow down relative to an unaccelerated observer. If the acceleration is produced by a force at some other angle to the velocity, the object will be deflected.

The world standard of mass is the kilogram, whose definition is based on the mass of a metal cylinder kept in France. Previously, the standard was based upon the mass of one cubic centimeter of water being one gram, which is approximately correct. The standard unit of force is the Newton, which is the force required to accelerate a 1-kg mass 1 m/sec2 (one meter per second per second). A Newton is equal to the force from the weight of about 100 g of water in Earths gravity. That's about half a cup. A dyne is the force required to accelerate a 1-g mass 1 cm/s2.



We know from Einsteins Special Theory of Relativity that mass, time, and length are relative and the speed of light is constant. And from General Relativity, we know that gravitation and acceleration are equivalent, that light bends in the presence of mass, and that accelerating mass radiates gravitational waves at the speed of light.

Spacecraft operate at very high velocities compared to speeds we are familiar with in transportation and ballistics here on our planet. Nonetheless, spacecraft velocities do not approach a significant fraction of the speed of light, and so Newtonian physics work well for operating and navigating throughout the solar system. Once we begin to travel between the stars, much higher velocities will be necessary, and those may well be large enough fractions of light speed that Einsteinian physics will describe their operation more precisely than Newtonian physics can. Spacecraft sometimes carry out experiments to test Special Relativity effects on moving clocks, and experiments to test General Relativity effects such as the space-time warp caused by the sun, the equivalence of gravitation and acceleration, and the search for direct evidence of gravitational waves.

Newtons First Law describes how, once in motion, planets remain in motion. What it does not do is explain how the planets are observed to move in nearly circular orbits rather than straight lines. Enter the Second Law. To move in a curved path, a planet must have an acceleration toward the center of the circle. This is called centripetal acceleration and is supplied by the mutual gravitational attraction between the Sun and the planet.

Keplers Laws, as expressed by Newton, are:


1. If two bodies interact gravitationally, each will describe an orbit that is a conic section about the common mass of the pair. If the bodies are permanently associated, their orbits will be ellipses. If they are not permanently associated with each other, their orbits will be hyperbolas (open curves).

2. If two bodies revolve around each other under the influence of a central force (whether or not in a closed elliptical orbit), a line joining them sweeps out equal areas in the orbital plane in equal intervals of time.

3. If two bodies revolve mutually about each other, the sum of their masses times the square of their period of mutual revolution is in proportion to the cube of the semi-major axis of the relative orbit of one about the other.


The major application of Keplers First Law is to precisely describe the geometric shape of an orbit: an ellipse, unless perturbed by other objects. Keplers First Law also informs us that if a comet, or other object, is observed to have a hyperbolic path, it will visit the sun only once, unless its encounter with a planet alters its trajectory again.

Keplers Second Law addresses the velocity of an object in orbit. Conforming to this law, a comet with a highly elliptical orbit has a velocity at closest approach to the sun that is many times its velocity when farthest from the Sun. Even so, the area of the orbital plane swept is still constant for any given period of time.

Keplers Third Law describes the relationship between the masses of two objects mutually revolving around each other and the determination of orbital parameters. Consider a small star in orbit about a more massive one. Both stars actually revolve about a common center of mass, which is called the barycenter. This is true no matter what the size or mass of each of the objects involved. Measuring a stars motion about its barycenter with a massive planet is one method that has been used to discover planetary systems associate with distant stars.

These statements apply to a two-dimensional picture of planetary motion. A three-dimensional picture of motion would describe the orbital path through space taking into account the interaction of the gravitational fields of the sun and planets.



Promise


Gravitys strength is inversely proportional to the square of the objects distance from each other. For an object in orbit about a planet, the parts of the object closer to the planet feel a slightly stronger gravitational attraction than do parts on the other side of the object. This is known as gravity gradient. It causes torque to be applied to any mass which is non-spherical and non-symmetrical in orbit, until it assumes a stable attitude with the more massive parts pointing toward the planet. An object whose mass is distributed like a bowling pin would end up in an attitude with its more massive end pointing toward the planet, if all other forces were equal. In the case of a fairly massive body such as our moon in Earth orbit, the gravity gradient effect has caused the moon, whose mass is unevenly distributed, to assume a stable rotational rate which keeps one face towards Earth at all times.

The moon acts upon the Earths oceans and atmosphere, causing two bulges to form. The bulge on the side of Earth that faces the moon is caused by the proximity of the moon and its relatively stronger gravitational pull on that side. The bulge on the opposite side of Earth results from that side being attracted toward the moon less strongly than is the central part of Earth. Earths crust is also affected to a small degree. Other factors, including Earths rotation and surface roughness, complicate the tidal effect. On planets or satellites without oceans, the same forces apply, but they cause slight deformations in the body rather than oceanic tides. This mechanical stress can translate into heat as in the case of Jupiter's volcanic moon Io.

Newtons analogy of a tall mountian with a cannon on top of it describes the physics of Orbital Mechanics. When the cannon is fired, the cannonball follows its ballistic arc, falling as a result of Earths gravity, and of course it hits Earth some distance away from the mountain. If we put more gunpowder in the cannon, the next time it's fired, the cannonball goes faster and farther away from the mountain, meanwhile falling to Earth at the same rate as it did before. Packing still more gunpowder into the cannon, the cannonball goes much faster, and so much farther that it just never has a chance to touch down. All the while it would be falling to Earth at the same rate as it did previously. This time it falls completely around Earth. It has achieved orbit. That cannonball would skim past the south pole, and climb right back up to the same altitude from which it was fired. Its orbit is an ellipse.

This is basically how a spacecraft achieves orbit. It gets an initial boost from a rocket, and then simply falls for the rest of its orbital life. Modern spacecraft are more capable than cannonballs, and they have rocket thrusters that permit the occasional adjustment in its orbit. But it's usually just falling. Part of the orbit comes closer to Earths surface that the rest of it does. This is called the periapsis of the orbit. The higest position of the orbit is called the apoapsis. The altitude affects the time an orbit takes, called the orbit period. The period of the space shuttle's orbit, at say 200 kilometers, is about 90 minutes.

By applying more energy at apoapsis, you have raised the periapsis altitude. A spacecrafts periapsis altitude can be raised by increasing the spacecrafts energy at apoapsis. This can be accomplished by firing on-board rocket thrusters when at apoapsis. If you decrease energy when you're at apoapsis, you'll lower the periapsis altitude. If you increase speed when you're at periapsis it will cause the apoapsis altitude to increase. A spacecraft's apoapsis altitude can be raised by increasing the spacecrafts energy at periapsis. Decreasing energy at periapsis will lower the apoapsis altitude.

In practice, you can remove energy from a spacecrafts orbit at periapsis by firing the onboard rocket thrusters there and using up more propellant, or by intentionally and carefully dipping into the planet's atmosphere to use frictional drag. That's called aerobraking, and it uses much less propellant.

In practical terms, you don't generally want to be less than about 150 kilometers above surface of Earth. At that altitude, the atmosphere is so thin that it doesn't present much frictional drag to slow you down. You need your rocket to speed the spacecraft to the neighborhood of 30,000 km/hr (about 19,000 mph). Once you've done that, your spacecraft will continue falling around Earth. No more propulsion is necessary, except for occasional minor adjustments. It can remain in orbit for months or years before the presence of the thin upper atmosphere causes the orbit to degrade. These same mechanical concepts, but different numbers for altitude and speed, apply whether you're talking about orbiting Earth, Venus, Mars, the Moon, the Sun, or anything.



Monuments of Mars


If you ride along with an orbiting spacecraft, you feel as if you are falling, as in fact you are. The condition is properly called free fall. You find yourself falling at the same rate as the spacecraft, which would appear to be floating there (falling) beside you, or around you if you're aboard the International Space Station. You'd just never hit the ground. An orbiting spacecraft has not escaped Earths gravity, it is giving the mass the centripetal acceleration it needs to stay in orbit. It just happens to be balanced out by the speed that the rocket provided when it placed the spacecraft in orbit. Gravity is a little weaker on orbit, simply because you're farther from Earths center, but it's mostly there. Gravity is still dominant, but some of its familiar effects are not apparent on orbit.

To launch a spacecraft from Earth to an outer planet such as Mars using the least propellant possible, first consider that the spacecraft is already in solar orbit as it sits on the launch pad. This existing solar orbit must be adjusted to cause it to take the spacecraft to Mars: The desired orbit's perihelion (closest approach to the Sun) will be at the distance of Earths orbit, and the aphelion (farthest distance from the Sun) will be at the distance of Mars' orbit. This is called a Hohmann Transfer orbit. The portion of the solar orbit that takes the spacecraft from Earth to Mars is called its trajectory.

We know that the task is to increase the apoapsis (aphelion) of the spacecrafts present solar orbit. A spacecrafts apoapsis altitude can be raised by increasing the spacecrafts energy at periapsis. Well, the spacecraft is already at periapsis. So the spacecraft lifts off the launch pad, rises above Earths atmosphere, and uses its rocket to accelerate in the direction of Earths revolution around the sun to the extent that the energy added here at periapsis (perihelion) will cause its new orbit to have an aphelion equal to Mars' orbit. After this brief acceleration away from Earth, the spacecraft has achieved its new orbit, and it simply coasts the rest of the way.

Getting to the planet Mars, rather than just to its orbit, requires that the spacecraft be inserted into its interplanetary trajectory at the correct time so it will arrive at the Martian orbit when Mars will be there. This task might be compared to throwing a dart at a moving target. You have to lead the aim point by just the right amount to hit the target. The opportunity to launch a spacecraft on a transfer orbit to Mars occurs about every 25 months.

To be captured into a Martian orbit, the spacecraft must then decelerate relative to Mars using a retrograde rocket burn or some other means. To land on Mars, the spacecraft would have to decelerate even further using a retrograde burn to the extent that the lowest point of its Martian orbit will intercept the surface of Mars. Since Mars has an atmosphere, final deceleration may also be performed by aerodynamic braking direct from the interplanetary trajectory, and/or a parachute, and/or further retrograde burns.

To launch a spacecraft from Earth to an inner planet such as Venus using least propellant, its existing solar orbit (as it sits on the launch pad) must be adjusted so that it will take it to Venus. In other words, the spacecraft's aphelion is already the distance of Earths orbit, and the perihelion will be on the orbit of Venus. This time, the task is to decrease the periapsis (perihelion) of the spacecrafts present solar orbit. A spacecrafts periapsis altitude can be lowered by decreasing the spacecraft's energy at apoapsis. To achieve this, the spacecraft lifts off of the launch pad, rises above Earths atmosphere, and uses its rocket to accelerate opposite the direction of Earth's revolution around the sun, thereby decreasing its orbital energy while here at apoapsis (aphelion) to the extent that its new orbit will have a perihelion equal to the distance of Venus's orbit.

Of course the spacecraft will continue going in the same direction as Earth orbits the Sun, but a little slower now. To get to Venus, rather than just to its orbit, again requires that the spacecraft be inserted into its interplanetary trajectory at the correct time so it will arrive at the Venusian orbit when Venus is there. Venus launch opportunities occur about every 19 months.

Planets retain most of the solar systems angular momentum. This momentum can be tapped to accelerate spacecraft on so-called "gravity-assist" trajectories. It is commonly stated in the news media that spacecraft such as Voyager, Galileo, and Cassini use a planets gravity during a flyby to slingshot it farther into space. By using gravity to tap into the planets tremendous angular momentum. In a gravity-assist trajectory, angular momentum is transferred from the orbiting planet to a spacecraft approaching from behind the planet in its progress about the Sun.



If the interplanetary trajectory carries the spacecraft less than 180 degrees around the Sun, it's called a Type-I Trajectory. If the trajectory carries it 180 degrees or more around the Sun, it's called a Type-II.

Voyager 2 toured the Jovian planets. The spacecraft was launched on a Type-II Hohmann transfer orbit to Jupiter. Had Jupiter not been there at the time of the spacecrafts arrival, the spacecraft would have fallen back toward the Sun, and would have remained in elliptical orbit as long as no other forces acted upon it. Perihelion would have been at 1 AU, and aphelion at Jupiters distance of about 5 AU.

However, Voyagers arrival at Jupiter was carefully timed so that it would pass behind Jupiter in its orbit around the Sun. As the spacecraft came into Jupiters gravitational influence, it fell toward Jupiter, increasing its speed toward maximum at closest approach to Jupiter. Since all masses in the universe attract each other, Jupiter sped up the spacecraft substantially, and the spacecraft tugged on Jupiter, causing the massive planet to actually lose some of its orbital energy.

The spacecraft passed on by Jupiter since Voyagers speed was greater than Jupiters escape velocity, and of course it slowed down again relative to Jupiter as it climbed out of the huge gravitational field. Its Jupiter-relative velocity outbound dropped to same as its velocity inbound. But relative to the Sun, it never slowed all the way to its initial Jupiter approach speed. It left the Jovian environment carrying an increase in angular momentum stolen from Jupiter. Jupiters gravity served to connect the spacecraft with the planets ample reserve of angular momentum. This technique was repeated at Saturn and Uranus.

Gravity assists can be also used to decelerate a spacecraft, by flying in front of a body in its orbit, donating some of the spacecrafts angular momentum to the body. When the Galileo spacecraft arrived at Jupiter, passing close in front of Jupiter's moon Io in its orbit, Galileo experienced deceleration, helping it achieve Jupiter orbit insertion and saving propellant.

The gravity assist technique was championed by Michael Minovitch in the early 1960s, while he was a UCLA graduate student working during the summers at JPL. Prior to the adoption of the gravity assist technique, it was believed that travel to the outer solar system would only be possible by developing extremely powerful launch vehicles using nuclear reactors to create tremendous thrust, and basically flying larger and larger Hohmann transfers.

An interesting fact to consider is that even though a spacecraft may double its speed as the result of a gravity assist, it feels no acceleration at all. If you were aboard Voyager 2 when it more than doubled its speed with gravity assists in the outer solar system, you would feel only a continuous sense of falling. No acceleration.

General Principal states "There is no way to distinguish the effects produced by the inertial force of acceleration from the effects produced by gravitational force". General Theory Of Relativity, also known as the evevator analogy, states "There is no way for the observer inside the elevator to determine whether they are falling in a gravitational field or simply free floating in the void of deep space".

The effect of the Earths force of gravity extends for an infinite distance, becoming weaker and weaker. We can consequently never completely escape the gravitational field of the Earth, never reaching the actual gravitational boundary of the Earth. It can, nevertheless, be calculated what amount of work would theoretically be required in order to overcome the Earths total gravitational field.

To this end, an energy not less than 6,380 meter-tons would have to be used for every kilogram of load. Furthermore, it can be determined at what velocity an object would have to be launched from the Earth, so that it no longer returns to Earth. The velocity is 11,180 meters per second. This is the same velocity at which an object would strike the Earths surface if it fell freely from an infinite distance onto the Earth. In order to impart this velocity to a kilogram of mass, the same amount of work of 6,380 meter-tons is required that would have to be expended to overcome the total Earths gravitational field per kilogram of load. The radius of the Earth happens to be 6,380 km.



Images and data courtesy NASA and the Jet Propulsion Laboratory



Back To Top


Home


Valid XHTML 1.0 Transitional Valid CSS level 3!