Abstract
Modeling the path of a rocket-propelled body that is launched from Earth into a stable
orbit requires the derivation of several dynamical systems, as well as Newton’s second law of
motion. The result is several systems of non-linear differential equations that require a numerical
solver such as MATLAB’s ode45 to generate data that can be quantitatively examined and
graphically displayed.
Intro
An accurate workable model for launching a rocket from any point on Earth’s surface
into orbit around the Earth is dependent on hundreds of different variables. In order to simplify
the situation to fit within the scope of Ordinary Differential Equations (ODE’s) and MATLAB’s
solving abilities, several assumptions must be made. The model will not, for example, take into
account variations in humidity or air temperature during the initial phase of the rocket’s flight.
The minor variations in gravitational pull do to other celestial bodies such as the moon will be
neglected, as well as minor variations in Earth’s gravitational field. A complete list of
assumptions is given in section 1.1.
To accurately model the path of a rocket from the surface of the Earth it is best to use a
polar coordinate system, where R is the distance radially outward from the center of the Earth,
and (theta) is the angle measured clockwise from the origin starting from the position of the
rocket.
To find the acceleration of an object the force and mass must be known. Because rockets
use a mass-propellant system, their mass will decrease over the duration of the flight. For the
purposes of this model the rocket engine will be simplified such that it operates at a constant rate,
expelling a constant amount of mass every second at a constant velocity until it is shut off. The
function for the rockets’ mass can then be written as ().
There are three main forces that act on a rocket in flight. The first is thrust. For this
assessment thrust is a constant force that acts in the direction of the rocket’s path. It can be
calculated using Newton’s second law, F=ma, by multiplying the velocity of the gas exiting the
rocket by the rate of mass leaving. The rate at which gas leaves the rocket is equal to the rate of
change of the rocket’s mass, which is a function of time. The exit velocity of the gas is a constant
that is determined by engine design and fuel type, so the equation for the force acting on the
rocket due to thrust can be written as shown in equation ().
The second main force acting on a rocket is atmospheric drag. Drag acts in the opposite
direction of the rocket’s motion, and is proportional to the square of its velocity. Drag can be
modeled as shown in equation (), where RHO represents a simplified model of air density as a
function of the rocket’s altitude, and is given by equation (). The velocity of an object in polar
coordinates can be written as (). To use this in the drag equation, it must be modified to be the
equation of velocity relative to the air in Earth’s atmosphere, as shown in equation ().The ‘A’
term in equation (drag) is a constant that is defined as the area that is acted upon by the force of
air resistance, and Cd is the drag coefficient, which depends on the shape of the rocket and the
properties of the materials on the outside of it.
The last major force that affects a rocket’s motion is the force of gravity. For the
purposes of this paper gravity will always act in the negative R-direction. Newton defined the
force of gravitational pull to be proportional to the product of the masses of two objects, and
inversely proportional to the square of the distance between their respective centers of gravity.
Factoring in the gravitational constant (G), the force acting on a rocket near Earth can be
modeled using equation ().
Because force and drag act in the direction of the path of a rocket, it is important to factor
in the rockets’ angle of incline (alpha). If the rocket starts in the vertical position at 90 degrees
and then slowly leans over during its flight until it is moving parallel to the surface of the Earth,
alpha can be given as().
An objects’ acceleration in polar coordinates can be written as (), where () and () are unit
vectors in the R- and (theta-) directions. Combining this with a=F/m, the acceleration of a rocket
in R and (theta) and be written as equations () and ().
At t=0, R will equal the radius of the earth. The initial velocity in the R direction will be
zero, and the rocket will be launched from theta=0. Due to the rotation of the Earth, the initial
velocity in the theta direction can be found by converting one revolution per day into radians per
second.
After the rocket has expelled all of its fuel and reached its orbiting altitude, a second
model must be created to show the unpowered orbital path. Using the final values of (R), (theta),
and the rocket’s velocity from the launch as new initial conditions, a second model can be
derived as (), which is much the same as the first one, without the forces of thrust and drag. Drag
is excluded because at the rockets’ final orbital altitude, the air is so thin that the force of drag
can be neglected altogether. To maintain a steady orbit, the force of gravity pulling on the rocket
must be equal to the centrifugal force pushing it outward due to the speed of its rotation. This
equality can be shown as (), where (Vf) is the final velocity of the rocket.
Assumptions
1) Earth is a perfect sphere
2) There are no outside bodies (e.g. the moon) whose gravity could have an effect on the
rocket’s trajectory
3) There is no wind
4) The final orbiting altitude of the rocket will be such that air resistance can be neglected
5) The mass of the rocket does not affect Earth’s position
6) The rocket engine burns at a constant rate the entire time that it is on
7) The engine stops burning at the instant that the rocket reaches the desired altitude at
the correct orbital velocity
8) The rocket is launched from the equator
9) Small variations in humidity and things like clouds do not interfere with the rocket’s
path