IMPRIMACION DE PARCHE DESCRIPCION

The heliocentric leg is always tangent to the departure planet velocity vector, in order to exploit as much as possible its momentum upon injection on the transfer orbit. On the converse, the transfer orbit usually intersects that of the target planet with an angle ϕ2, unless a Hohmann

transfer is selected.

The velocity v2 at intercept is given by

v2 = s 2 µ⊙ r2 +_Et

while the angle ϕ2 satisfies the equation

cosϕ2=ht/(r2v2)

The velocity relative to the target planet is ~

v₃ =~v₂₋~v_T

where~vT is the planet orbital speed.2 From the law of cosines it is

v2₃ =v₂2+v2_{T −}v2vTcosϕ2

The angleϑ between the planet’s velocity and~v3 is obtained from the law of sines

sinϑ= v2 v3

sinϕ2

If a direct hit to the target planet is sought, the phase angle at departure must be chose in such a way that γ2 = 0. In this case, the intercept point on the planet’s orbit will be achieved

when also the planet is there. This means that the spacecraft enters the planet’s sphere of influence with a velocity vector parallel to the local vertical to the planet centre, so that the approach trajectory to the planet is a straight line along which the probe flies at hyperbolic speed.

If a fly–by trajectory is sought, the phase angle at departure must be changed accordingly, in order to miss the planet by a certain amount distance x(measured along the planet’s orbit). In this case it is

γ1 = (ν2−ν1)−ω2(t2−t1)±x/r2

where the + sign means a passage behind the planet, while the - sign is for passaged ahead of it. It should be noted that the former is typical of missions that requires injection into a planetary orbit, while the latter allows the exploitation of the swin–by effect, where the spacecraft gains energy from the planet.

Letting the offset distance of~v3 be defined asy=xsinϑ=x(v2/v3) sinϕ2 and remembering

that the velocity ~v3 represents the hyperbolic excess speed upon entering the planet’s sphere

of influence, the (constant) magnitude of the angular momentum is h =yv3, while the energy

of the approach orbit is _Ee ≈ v23/2. As a consequence, the other orbital elements can easily

be found as p = h2_/µ

T (parameter) and e =

q

(1 + 2_Eeh2/µ2_T, where µT is the gravitational

parameter of the target planet.

The periapsis radius is thus given by rp = p/(1 +e) and from conservation of angular

momentum, the velocity at periapsis passage is vp=yv3/rp.

G. Colasurdo, G. Avanzini - Astrodynamics – 7. (Introduction to) Interplanetary trajectories 99

In general, the design variable is a given periapsis radius (e.g. for injection onto a planetary orbit), which allows for the evaluation of a certain offset distange, y = vprp/v3. But from

conservation of energy it is also

Ee= v₃2 2 = v2 p 2 − µT rp =_⇒ vp = s v2 3+ 2µT rp

so that the offset distance is given by

y = rp v3 s v2 3+ 2µT rp

From the practical point of view, it is important to determine the offset distance which results into a periapsis radius equal to the radius of the target panet. This value is called impact parameter, b, inasmuch as any offset minor than b will result with an impact on the planet surface. By letting rp =rT, it is b= rT v3 r v₃2+ 2µT rT

It should be noted that, being 2µT/rT >0, it is always b > rT.

The circle of radiusbplaced at the boundaries of the sphere of influence and perpendicular to the direction passing through the planet in the direction of the asymptote of the entry hyperbola is the effective collision cross section. If the probe must enter the planet atmosphere, which is always a thin layer of width ha several order of magnitude smaller than the planet radius,

ha≪rT, the entry corridoris an anulus of inner radiusband width db, wheredb= (db/drT)ha

Bibliography

1. Richard Battin, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education Series, New York, 1987.

2. Roger R. Bate, Donald D. Mueller, Jerry E. White,Fundamentals of astrodynamics, Dover, New York, 1971

3. Fred P.J. Rimrott, Introductory orbit dynamics, Vieweg, Braunschweig, 1989

4. J.W. Cornelisse, H.F.R. Schyer, K.F. Wakker,Rocket propulsion and spaceflight dynamics, Pitman, London, 1979

5. B. Wie, Space vehicle dynamics and control, AIAA Education Series, Reston, 1998. 6. M.H. Kaplan, Modern Spacecraft Dynamics and Control, J. Wiley & Sons, New York,

1976.

7. G. Mengali, Meccanica del Volo Spaziale, Edizioni Plus, Pisa, 2001