Accelerating force

$$ F=mv_e $$ thrust of the rocket is expressed in terms of the *mass flow rate * m and the efficient exhaust velocity \(v_e\)

$$ V = V_e\log_e\frac{M_0}{M} $$ $M_0$ - mass of the rocket at ignition $M$ - Current mass of the rocket

mass ratio $$ R = \frac{M_0}{M} $$

AcceleratedForce <- function(m,v.e) { m*v.e }

RocketEquation <- function(mass.ratio, exhaust.velocity)
  exhaust.velocity*log(mass.ratio, base = exp(1))
RocketEquationGraph <- data.frame(Mass.Ratio = c(seq(1:20)))
RocketEquationGraph <- RocketEquationGraph %>% 
  mutate( M4000 = RocketEquation(Mass.Ratio, 4000),M3000 = RocketEquation(Mass.Ratio, 3000),M2000 = RocketEquation(Mass.Ratio, 2000), M1000 = RocketEquation(Mass.Ratio, 1000) )
g <- RocketEquationGraph %>% ggplot() + 
  geom_line(aes(Mass.Ratio, M4000), color = "red")+
  geom_line(aes(Mass.Ratio, M3000)) + 
  geom_line(aes(Mass.Ratio, M2000))+
  geom_line(aes(Mass.Ratio, M1000)) + 
  labs(title = "Rocket Velocity as a function of the mass ratio",
       x = "Mass ratio (M_0/M)",
       y = "Rocket Velocity (m/s)")

The rocket equation shows that the final speed depends upon only two numbers; the final mass ratio, and the exhaust velocity. It does not depend on the thrust rather surprisingly, or the size of the rocket engine, or the time the rocket burns, or any other parameter.

Much of the effort in rocket design goes into increasing the exhaust velocity.

Gunpowder, and the range of propellants of the 19th century produced an exhaust velocity of around 2000 $ms^{-1}$ while today we reach around 4500 $ms^{-1}$

The point at which the rocket’s speed exceeds the exhaust speed is when the mass ratio becomes equal to e, or 2.718, the base of natural logarithms.

1.3.1 Orbits

$$ \frac{1}{r} = \frac{GM_{\bigoplus}M^{2}}{h^{2}}1+\epsilon\cos\theta $$

  • $G$ gravitational constant ($6.670\times10^{-11}Nm^2kg^2$)
  • $M_\bigoplus$ mass of the earth
  • $h$ constant angular momentum
  • $\epsilon$ - eccentricity of the orbit

Eccentricity defines the same of the orbit. For an ellipse $\epsilon$ is the ratio of the distance between the foci, to the length of the major axis. However in order to understand how the orbit varies with the initial velocity of the spacecraft, the angular momentum and the eccentricity have to be expressed in terms of useful parameters.

$$ h = MrV $$ $$ \epsilon = \frac{h^{2}}{GM_{\bigoplus}M^{2}r_{0}}-1 $$

The shape of the orbit depends only on the initial velocity and the distance from the center of the Earth.

A circular orbit with eccentricity of zero An elliptical orbit of eccentricity 0.65 A parabolic orbit of eccentricity 1.0

So to see this in practical terms; we express the eccentricity in erms of the initial velocity and height of the spacecraft

$$ \epsilon = \frac{r_{0}V_{0}^{2}}{GM_{\bigoplus}}-1 $$ We can see that if the ratio when $r_0V_0^2 = GM_{\bigoplus}$ the $\epsilon$ becomes 0 and the orbit is circular.

Or we can substitute zero in the orbit equation: the $\cos\theta$ term goes to zero and the radius is independent of the angle; that is constant. Thus the orbit is circular. Since the condition for a circular orbit is that $r_0V_0^2 = GM_{\bigoplus}$, it is easy to calculate the initial velocity, given the distance from the center of the earth.

$$ V_0 = \sqrt{\frac{GM_{\bigoplus}}{r_0}} $$

The mass of the Earth is $5.975\times10^{24}kg$ and the mean radius is 6,371 km. Therefore an initial radius of 500 km above the Earth’s surface, the initial velocity is 7.6$km s^{-1}$.

Now this doesn’t give us the velocity to get into space, but it is the velocity to stay there.

As the velocity given to the spacecraft increases, the eccentricity of the elliptical orbit becomes greater, and the apogee moves farther out.

  • 6.670 $Nm^{2}kg^{2}$
  • 5.975 kg
  • 63
G <- 6.670*10^-11
eMass <- 5.975*10^24

initialVelocity <-function(r) { sqrt((398600.4415)/(6371+r)) }

marsInitialVelocity <- function(r) { sqrt((42828.37)/(3389.5+r)) }

fx <- data.frame(miles = 1:2000)
fx <- fx %>% mutate( = initialVelocity(miles), v.mars = marsInitialVelocity(miles))

g <- fx %>% ggplot() +
  geom_point(aes(miles,,color = "blue") + 
  geom_point(aes(miles,v.mars)) +
  labs(title = "Velocity Curve Above Planets", 
       x = "Velocity (km/sec)", 
       y = "above the Surface (km)")

Elliptical transfer orbits

$$ V_0=\sqrt{\frac{2GM_{\bigoplus}}{r_0}} $$