Consider a circular, perfectly reflecting Solar sail that is initially at rest a

Question

Consider a circular, perfectly reflecting Solar sail that is initially at rest at a distance 1.1 of I AU from the Sun and pointing directly at it. The sail is carrying a payload of 1000 kg (which dominates its mass) and its radius is R 500 m. (a) Derive an expression for the acceleration as a function of distance, a(r), for this Solar sail. Include the Sun's gravity as well as its radiation pressure. (The constants may be evaluated to simplify the expression.) (b) Manipulate and integrate this equation to find an expression for the velocity of the Solar sail as a function of distance, v(r). Evaluate the expression to find the velocity of the Solar sail by the time it reaches the orbit of Mars. (c) Finally, derive an expression for the time it would take for the sail to reach the orbit of Mars. Evaluate it to find the time. Express the time as seconds, months, or years whatever is most appropriate.

Explanation / Answer

1.11 consider a perferctly circular refelcting solar sail

initially at rest at a distance 1 AU fromthe sun

p[oionting dirfectly at it

vo = 0 m/s

m = 1000 kg

Rs = 500 m

a. solar constant Eo = 1388 W/m^2

solar pressure for reflective sail

p = 2E/c

hence

F = dp/dt = 2En/c

where n is number of photons per secon

En = energy per second

hence

En = Eo*piRs^2*(d/r)^2

r is distacne from sun, d = 1 AU

En = 1090132650.795658*(d/r)^2

for

F = 2En/c

for

F = 7.2675510*(d/r)^2 ( r in AU)

F = 7.26755/r^2

now if we consider gravity

F = 7.26755/r^2 - GMs*1000/r^2

r in m

Ms = 2*10^30 kg

G = 6.67*10^-11

hence

F = 1.3069134/r^2

F = ma

mdv/dt = 1.6264*10^23/r^2 = m*v*dv/dr

also

m = mo/sdqrt(1 - v^2/c^2)

hence

1.6264*10^20*dr/r^2 = vc*dv/sqrt(c^2 - v^2)

542133333333.33333*dr/r^2 = vdv/sqrt(c^2 - v^2)

inetgrating

3.6238859180*(1 - 1/r) = c - sqrt(c^2 - v^2)

at mars r = 1.524 AU

hence

v = 27611.389489 m/s

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