91影视

The orbit radius of Earth is,$r_E=1.50脳{10}^{11}m$ .

The orbit radius of Mars is,$r_M=2.28脳{10}^{11}\mathrm{m}$.

The semi major axis radius is, $a=\frac{1}{2}\left(r_E+r_M\right)$.

As stated in Kepler's third law, the period's square is directly proportional to semi-major axis of orbit's sphere of radius.

The law can be given as,

$T=\frac{2{\mathrm{蟺补}}^{{\displaystyle \frac{3}{2}}}}{\sqrt{G{m}_s}}$鈥� (i)

The arithmetic average of orbit can be evaluated as,

$E=\frac{-G{m}_{s}m}{2r}$

The rockets are launched in the opposite direction of movement, that is, in the direction which increases the speed, in order to go from the Earth's circular orbit to the transfer orbit. Once in Mars orbit, the rockets must be launched in the opposite direction of motion in order to boost the energy.

Intervals between circular orbits may be found in the spacecraft's energy when it moves from one to another in the transfer zone.

Thus, the rockets are launched in the opposite direction of speed while returning from Mars to the Earth.

The semi-axis radius can be given by,

$$\begin{gathered} a=\frac{1}{2}\left(r_E+r_M\right) \\ a=\frac{1}{2}\left[\left(1.50脳{10}^{11}m\right)+\left(2.28脳{10}^{11}m\right)\right] \\ a=1.89脳{10}^{11}m \end{gathered}$$

Substitute above value in equation (i),

$$\begin{gathered} t=\frac{T}{2}=\frac{\mathrm{蟺}{\left(1.89脳{10}^{11}\mathrm{m}\right)}^{{\displaystyle \frac{3}{2}}}}{\sqrt{\left(6.673脳{10}^{-11}N{m}^2/k{g}^2\right)\left(1.99脳{10}^{30}kg\right)}} \\ t=2.24脳{10}^{7}s=259days \end{gathered}$$

Thus, the time taken for one way trip from Earth to Mars between the firings of rocket

is, 259 days which is more than $8\frac{1}{2}\mathrm{months}$.

The Mars will be in a 360-degree orbit at this period.

The angle between a Sun-Mars and Sun-Earth can be evaluated by,

$$\begin{gathered} 胃=\left({360}^{鈭�}\right)\frac{\left(2.24脳{10}^{7}s\right)}{\left(687d\right)\left(86,400s/d\right)} \\ 胃=135.9^{鈭�} \end{gathered}$$

And the spaceship does a 180-degree turn. So, the angle between a Sun-Mrs like and Sun-Earth line is,$44.1^{鈭�}$.