Question

The nearest star, Proxima Centauri is 4.2 light years away and was recently found to have a planet with at least 1.3 times earth’s mass and having temperatures likely to be in the range where water could exist. Let’s check it out! Assume that we want a rocket that will accelerate at +1 g for half the voyage and decelerate at −1 g for the 2nd half, so that we don’t just whiz on by.

Chemical propellants have an exhaust velocity of order 4000 m/s. What is the numerical fraction left for payload (and structure)? Show that at such a low exhaust velocity, not much is left for the payload! Show that even if vex = c, there would be a problem.

Thanks!

Answer 1

given distance from proxima centaurai, d = 4.2 LY
mass of planet, mp = 1.3me
me = mass of earth
temperatures likely in rage for liquid water to exist

a1 = 1 g ( half voyage)
a2 = -1 g ( other half)

vex = 4000 m/s
so, from rocekt equation

dV = vex*ln(mo/mf)
now,

let tiem of journey = 2t
then

d = 0.5*t*vo + 0.5*t*vo = vo*t
also
vo = a1*t = gt
hence
d = gt^2
t = sqrt(d/g) = sqrt(4.2*c*365.25*24*3600/g) = 4.053269*10^15 s
hence
vo = dV = 39.76257*10^15 m/s
hence

mo/mf = exp(9940644,000,000)

hence
mass of payl;oad and structure = m
m = mf
hence
mf/mo = exp(-9940644,000,000)
this is about 0

if vex = c
then too
mo/mf = exp(132,541,900) still a very large numebr