-aε Lecture 4. Subjects: Hyperbolic orbits. Interplanetary transfer. (1) Hyperbolic orbits

Transcription

1 6.50 Lecture 4 ubjects: Hyerbolic orbits. Interlanetary transfer. () Hyerbolic orbits The trajectory is still described by r, but now we have ε>, so that the +! cos" radius tends to infinity at the asymtotic angle! " # $ cos $ ( / % ). -a -aε r δ θ θ Δ The arameter still has the geometrical significance indicated in the figure, and is therefore a ositive number. It is still related to a and ε through a(! " ), but now a is a negative number, so it is (-a) that has a geometrical significance, as indicated in the figure. Note also tat ε is still defined as the ratio of the distance from eriasis to center to the distance from focus to center. The energy is still given by E v!!, and is now ositive. The angular r a momentum is still given by h r!! a(" # ). There are a few new arameters of interest in this case: The trajectory deflection, #! " # (" # $ % ) " # cos sin # & &

2 The miss distance! "a# sin($ " % & ) "a# sin% & # " The excess hyerbolic velocity, v! E ("a) In the secialized technical literature, the term c 3 is often used, meaning simly v!. ) Interlanetary transfer We assume for the moment that our craft has escaed the field of lanet (meaning it is outside its shere of influence), and so may be considered to be in orbit about the un. In order for it to reach lanet, its orbit about the un must intersect that of lanet. P at launch r P at launch un r P and P at arrival Assume that the lanetary orbits are circular. Then it is clear that the trajectory of least energy which will allow the transfer is that which is just tangent to the orbits of the home and target lanets; this is called the Hohmann transfer orbit, which is the half-ellise that is sketched in the figure. The Heliocentric velocity at the start of this Hohman arc is the eriasis (erihelion in this case) velocity, as described at the end of the last lecture: v r + r

3 so that if we launch the shi in the direction of motion of the lanet, it must have a relative velocity r v rel, (!) + r with resect to lanet after escae from the lanet. By definition, this is the excess hyerbolic velocity relative to the lanet, v, and the total energy relative to lanet at he edge of the shere of influence is simly ½( v uose the launch was for the surface of lanet (radius R ), and ignore its rotation. Just after launch, during which the rocket has imarted an ins tantaneous velocity increment ΔV, the energy er unit mass (relative to the lanet) is (!V ) ", and this R must be the same as ½( v ), by energy conservation with resect to lanet inside the shere of influence. We then have r (!V ) " v ( " ) R # r + r from which the first delta-v delivered by the rockets must be ). r!v + ( " ) R + r The rocedure is similar when considering the aroach to lanet. The sacecraft will have then a heliocentric velocity equal to the aoaxis (aohelion) velocity r v a r r + r and a relative velocity with rese c t to the lanet r v rel, (! ) r r + r which is also the excess hyerbol ic veloc ity w ith resect to lanet. It is worth noting a this oint that the sacecraft heliocentric velocity is less than that of the lanet itself, so that, as seen from the lanet, the sacecraft will be aroaching from its advancing side. For cature into a circular orbit of radius R c, the geometry is shown below: To un ΔV r c, V rel, 3

4 Just before the insertion rocket firing, the energy er unit mass relative to lanet is equal to ½ (v rel, ), and it is also equal to the sum of the kinetic energy at that oint of closest aroach, lus the otential energy: (v closest a. )!. Thus we must have r c v r + (! ) closest a. r c r + r and the insertion velocity increment must be this, minus the orbital velocity around lanet : r!v ( ) + " " r c r + r r c Comarison to simle Escae+Transfer+Cature. A simle-minded aroach to the same mission would be to first aly an imulse at the surface of lanet to achieve escae (!Vesc, / R ), then, after slowing down to zero velocity with resect to lanet, aly a second imulse to enter the ellitic transfer orbit towards lanet (this would be our v, ), then, in the vicinity (but still outside the OI of ) lanet, aly a third imulse to m rel atch the heliocentric velocity of lanet (this would be our v rel, ), and finally, starting from zero relative velocity far from lanet, aly a fourth imulse to cature the craft into orbit about lanet (this is equal to the escae velocity from a distance R to the lanet,!v e, / R ). You can easily check that the two imulses we derived before are, res sc ectively,!v (!V esc, ) + (v rel,)!v (!V sc,) e + (v rel, ) and so our revious scheme is definitely more effective. These two strategies are called sometimes the Hohmann (simle, four imulses) and the Oberth (combined, two imulses) maneuvers. 4

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