The orbit of the spacecraft at perihelion touches the orbit of Venus, and at aphelion it touches the orbit of Mars. After approaching one of these planets as a result of active gravitational maneuver, the orbital period of the spacecraft has decreased by 2 times.
1) Approach to which planet led to a decrease in the period?
2) What will be the new circulation period?
3) Will the spacecraft, moving in a new orbit, cross the Earth’s orbit?
The radius of the orbit of Venus is 0.72 AU. e., Mars – 1.52 AU. e.
Let us first define a new spacecraft period. The semiaxis of the original orbit is (0.72 + 1.52) / 2 = 1.12 AU. e. From Kepler’s III law, we obtain that the period of revolution in such an orbit is 1.12 ^ 3/2 = 1.18 years. Then the new period of the spacecraft is 0.59 years. The semi-major axis of the new orbit is 0.59 ^ 2/3 = 0.70 AU. e. If the spacecraft braked near Mars, then its perihelion decreased. The smallest possible orbit with aphelion of Mars will have a semi-major axis equal to 1.52 / 2 = 0.76 AU. e. This is more than the semi-major axis of the new orbit. This means that the spacecraft braked at Venus. It can be seen that the semi-major axis of the new orbit is less than the radius of the orbit of Venus. This means that the perihelion of the old orbit became the aphelion of the new one. The new orbit lies entirely within the orbit of Venus, that is, the spacecraft will not cross the Earth’s orbit.
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