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The hyperbolic flight path is defined by the planet's radius R extending nearly to the periapsis (solid dot) at declination [[delta].sub.a] and the distance C from the center of mass at the origin O to the intersection of inbound and outbound asymptotes (dashed lines) with declinations [[delta].sub.i] and [[delta].sub.o].
(i) A numerical integration is performed in backward time, with the particle or spacecraft starting at the periapsis until it reaches point A (Figure 1).
(ii) Then, the numerical integration is made again, with the particle or spacecraft starting at the periapsis again, but this time in forward time.
Several values were considered for the velocity at periapsis [v.sub.p]: 0.3, 0.5, and 0.7 canonical units, where 1.0 canonical unit is equal to 29.78 km/s, that is, the velocity of the Earth in its motion around the Sun.
The inclination of those lines increases with the velocity at the periapsis, because the drag force is proportional to the square of the velocity of the particle or spacecraft.
The velocity of the particle or spacecraft with respect to the Earth points towards the positive direction of the vertical axis when [psi] = 0[degrees], so both velocities have the same sense and their magnitudes added together cause an increase in the energy with the increase of the velocity at the periapsis. In the situation where [psi] = 180[degrees], the velocity of the particle with respect to the Earth is in the negative direction of the vertical axis, so the velocities have opposite senses and their magnitudes have to be subtracted from each other to get its value with respect to the Sun, causing a decrease in the energy with the increase of the velocity at the periapsis.
It is also noted that the situation where [psi] = 0[degrees] shows the presence of hyperbolic and elliptic orbits, depending on the velocity at the periapsis.
Note that this gap increases when the velocity at the periapsis decreases, due to the fact that lower velocities allow longer interactions between the bodies, and so the variation of energy is stronger.
However, given its elliptical orbit, meaningful gravity data could be acquired only when the spacecraft was within about 30 |degrees~ of periapsis (which presently occurs near 10 |degrees~ north latitude).
A series of thruster firings lowered periapsis to near 140 km -- enough to bring the spacecraft into the upper vestiges of the Venusian atmosphere.
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