14. SPACE FLIGHT

Lunar Excursion Module. From the Boeing Web Page, Gallery, History Section. See www.boeing.com

Using conventional methods of rocket propulsion, all of the propellant (and the working fluid) has to be stored on board the space vehicle, and gets used up as the vehicle produces thrust to accelerate. This is because the density of gases in outer space is extremely small: you might see a molecule or two every few miles. Consider a rocket with effective exhaust velocity ce. The effective exhaust velocity is a way of expressing the thrust in a simple manner by adding up the momentum thrust and the pressure thrust and dividing it by the mass flow rate. Also, as the propellant is blasted out the exhaust nozzle, the mass of the vehicle decreases. This is substantial in the case of the rocket as compared to air-breathing engines, because all the propellant comes from inside the vehicle. From Newton's Second Law,  .

Hence,  . Integrating,  . or,  where M₁ is the initial mass, which includes the propellant, and M₂ is the mass after the propellant has been used up to achieve the velocity increment DV.

Define the Specific Impulse of the propellant  where g is the standard value of acceleration due to gravity at sea-level (9.8m/s²). Note that the unit of Specific Impulse is seconds. Using this definition,  . Thus, the Mass Ratio of a rocket is . Note that for missions such as a launch from Earth's surface to a trajectory which will escape from earth's graviational field, this Mass Ratio is large number.

Example:

Newton's Law of Gravitation

To find the velocity increment required for various missions, we must calculate trajectories and orbits. This is done using Newton's Law of Gravitation:

 Here the lhs is the "radial force" of attraction due to gravitation, between two bodies; the big one of mass M, and the little one of mass m. The universal gravitational constant G is 6.670 * 10-11 Nm²/kg².

Kepler's Laws

(applied to satellite of mass m, orbiting a much bigger object of mass M. i.e, m << M):

2. The radius vector from the center of attraction sweeps equal areas of the orbit per unit time. As the satellite moves away, its speed decreases. As it nears the center of attraction, its speed increases.
 

3. The ratio of the squares of the orbital periods of any two satellites about the same body equals the ratio of the cubes of the semi-major axes of the respective orbits.

.
 

Speed at any point in an elliptical orbit:

Time for one orbit is:

Now this is similar to what happens when, for example, your professor jumps up. He becomes a vehicle in an elliptic trajectory around the center of attraction, which is the center of the earth. He has maximum velocity at launch (perigee) and reaches minimum velocity at the apogee of the orbit. Unfortunately, this orbit intersects the surface of the earth, and is brought to an abrupt halt.

A stone thrown from the surface goes up a little more. A ballistic missile launched with sufficient velocity increment to  reach an altitude which is above the atmosphere, then re-enters the atmosphere, reaching just about anywhere on the surface. With a little more energy, the vehicle can go into an orbit which does not intersect the surface, and stays outside the atmosphere, so that there is no air drag to drain away the energy of the vehicle.

This was one of the concepts which was considered for the X-33, reusable launch vehicle program. This is the McDonnell-Douglas Delta Clipper, which was to be a single-stage-to-orbit vehicle, which would return and land vertically as well.