Rocket Propulsion:Boost From A Planet

Boost From Nonrotating Planet

To go from a nonrotating planetary surface orbit requires that a rocket change its velocity from a rest velocity (zero) to a velocity that will keep the payload in orbit. If our rocket maintains a constant thrust during its ascent, then the total velocity change is

$\underset{0}{\overset{t_{orbit}}{\int }}adt=\underset{0}{\overset{t_{orbit}}{\int }}\frac{T}{m}-\frac{D}{m}-gdt$

where $a$ is the acceleration, $D$ is the drag, and $g$ is the planets gravitational pull.

Staging

Many rockets do not have the capability to reach the required orbital trajectory using a single stage. Also, the mass efficiency (ratio of useful payload to total mass) increases with staging. In the end, we desire a rocket with a number of stages that optimizes the economic efficiency (cost per payload unit mass). The economic efficiency depends on a number of factors, the mass efficiency being only one factor.

Let us assume that we desire to launch a payload of weight P. The weight of each stage in the stack is

$W_i=P{w}_i$

where $w_i$ is a normalized weight for the stage. The total stack weight is thus

$W=P(1+\sum _{n=1}^N{w}_i)$

The change of velocity per unit mass for each stage is

$\mathrm{\Delta }{v}_i=I_{s{p}_i}\ln {\mu }_i$

where ${\mu }_i$ is the ratio of the weight before the burn of the ith stage to the weight after the burn of that stage. Thus, ${\mu }_i$ will always have a value greater than 1. The total change in velocity per unit mass for each stage is

$\mathrm{\Delta }v=\sum _{n=1}^N{I}_{s{p}_i}\ln {\mu }_i$