2016年IOAA理论第9题-火星环绕任务

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英文题目

(T9) Mars Orbiter Mission

India's Mars Orbiter Mission (MOM) was launched using the Polar Satellite Launch Vehicle (PSLV) on 5 November 2013. The dry mass of MOM (body + instruments) was 500 kg and it carried fuel of mass 852 kg. It was initially placed in an elliptical orbit around the Earth with perigee at a height of 264.1 km and apogee at a height of 23903.6 km, above the surface of the Earth. After raising the orbit six times, MOM was transferred to a trans-Mars injection orbit (Hohmann orbit).

The first such orbit-raising was performed by firing the engines for a very short time near the perigee. The engines were fired to change the orbit without changing the plane of the orbit and without changing its perigee. This gave a net impulse of 1.73 × 10⁵kg m s ⁻¹ to the satellite. Ignore the change in mass due to burning of fuel.

(T9.1) What is the height of the new apogee, ℎ_a above the surface of the Earth, after this engine burn?

(T9.2) Find the eccentricity (𝑒) of the new orbit after the burn and the new orbital period (𝑃) of MOM in hours.

中文翻译

(T9) 火星环绕任务

火星轨道器任务（MOM）于2013年11月5日使用极轨卫星运载火箭（PSLV）发射。MOM（箭体+仪器）的干质量为$$500\rm{kg}$$，携带的燃料质量为$$852\rm{kg}$$。它最初被发射在环绕地球的椭圆轨道上，近地点高度为距地球表面$$264.1\rm{km}$$，远地点高度为$$23903.6\rm{km}$$。在六次提升轨道后，MOM转移到一个火星转移轨道（霍曼轨道）。

第一次这样的轨道提升是通过在近地点附近很短时间内点燃发动机来完成的。点燃发动机是为了在不改变轨道平面和近地点的情况下改变轨道。这给卫星带来了$$1.73×10^5\rm{kg}\cdot \rm m \cdot s^{-1}$$的净冲量。忽略燃料燃烧引起的质量变化。

(T9.1) 发动机点燃后，新的远地点，距离地球表面的高度$$h_a$$是多少？

(T9.2) 求MOM新轨道的偏心率$$e$$和轨道周期$$P$$（单位为小时）。