Question

# The nearest star, Proxima Centauri is 4.2 light years away and was recently found to have...

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!

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)

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

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