[The following is an excerpt from a letter to Allen Meece.]

[Updated 16 September 2002]

Regarding vertical velocity for orbital launches, the space shuttle does indeed roll over horizontally shortly after launch in order to put on most of its velocity horizontally. Anything we launch to orbit should do the same.

An orbital launch from the platform will need some vertical velocity in order to rise to its orbital altitude, just like a thrown ball needs some vertical velocity to rise to the apex of its trajectory. The approximate minimum delta-v required is

Dv.vertical = SQRT( 2 G M.earth ( 1/R.earth - 1/(R.earth + h.orbit) ) ) + g t.vert

where G is the universal gravitational constant

M.earth is the mass of the earth

R.earth is the radius of the earth

h.orbit is the orbital altitude

g is the orbital acceleration due to gravity during vertical acceleration (assumed constant)

t.vert is the time required for vertical acceleration (assumed short)

The whole business of rolling the rocket over to horizontal flight is to minimize t.vert. Every second of vertical acceleration is one more bit of final velocity gravity saps away from you. After the rocket rolls over, gravity is still reducing its vertical velocity. It just has enough already to lose some.

Note that

U = m G M.earth ( 1/R.earth - 1/(R.earth + h.orbit) )

is just the potential energy associated with the orbital altitude, much as

U = m g h.orbit

would be the potential energy of the orbit if the acceleration due to gravity were the same all the way up.

As it happens, for orbits close to the earth's surface, U is relatively small compared to the final kinetic energy of the rocket. Small enough that a rocket launched from the ground at a high enough acceleration can put on almost all the vertical delta-v it needs during the first few dozen kilometers of its flight. A rocket launched from the VBP can do it in an even shorter time, because it doesn't have to contend with air resistance. In fact, with no air to punch through, we can play with the orbital trajectory, launching at angles that will put on vertical velocity at any rate we please. The only requirement is

a.rocket sin(Q.trajectory) > g

where a.rocket is the acceleration of the rockey and Q.trajectory is the angle of the trajectory from horizontal at launch. Anything that violates this equation can't hold itself up against gravity at the platform and just falls back toward the surface of the earth.

The overall delta-v of an orbital rocket is at least

Dv^2 = Dv.orbit^2 + Dv.vertical^2

Saddly, the VBP can't reduce the vertical delta-v to zero. But we can come 600m/s closer than the space shuttle or any other surface launched rocket, and we can launch at angles (like 45 degrees) that will put on vertical and horizontal velocity at the same time.

We'd launch already halfway through our roll maneuver.

Another special advantage of launching off of the VBP is that there is no lower limit to the size of the rocket you can launch into orbit.

Rockets with 1kg payloads are quite impractical from the surface of the earth due to drag losses. A rocket that small has an unfavorable cube:square ratio. The forces on it due to drag are so huge compared to its weight that it's like trying to accelerate two or three other small rockets along with it. A more massive rocket requires more thrust, but the drag on it is smaller compared to the required thrust. It's simply easier to move a big mass with a small surface area through the air than it is to move a tiny mass with the same surface area. This is one reason the "Big Dumb Booster" method is so attractive when launching from the earth's surrface. Rockets launched from the VBP won't have as much of a problem with this. We'll be able to put up 10kg microsatellites without having to piggyback them on larger payloads.

CME