Interplanetary Travel

Transcription

1 Interplanetary Travel Interplanetary Travel Concept Patched Conic Hypothesis Departure & Arrival Manoeuvres

2 Interplanetary Travel Concept Interplanetary travel is concerned with motion of manmade objects when these travel through outer space, passing many planets in the process. Such motions need clear understanding of the changing nature of forces as well as strength of the gravitational field of the planets involved. In reality, we need to model such motions using the general n-body equations, but as we have solutions only for the -body and restricted 3-body problems, such an approach is not feasible.

3 Interplanetary Travel Concept In practice, it is found that good solutions to interplanetary trajectories are possible by considering the complete trajectory as a sequence of multiple -body segments, joined together at a suitable common point. Thus, the motion of a spacecraft starting from earth and going to moon, mars, jupiter etc. can be captured by solving successively, a number of -body problems. Concept of patched conics, for a smooth transition between segments, is used to synthesize total trajectory. Sphere of activity (SOI) represents the common point.

4 Patched Conic Hypothesis Patched conic hypothesis can be investigated as follows. A spacecraft, when escaping from a planet on a hyperbolic path, reaches the edge of SOI of that planet. At this point, the spacecraft is assumed to become heliocentric, which represents a point of patching between the two trajectories (Departure). Next, when the spacecraft reaches the SOI of the target planet, it again becomes planeto-centric and this point now represents the second patch (Arrival). At each patch point, velocity is the patch parameter.

5 Departure Concept Departure from a planet needs a change of reference frame at the boundary of the SOI. Consider the schematic of departure as given below.

6 Patched Conic Relations In escaping from earth, V is the scalar velocity at infinity, (or the edge of SOI). Assuming circular parking orbit & outgoing asymptote aligned with earth s orbital vector, we get the patch relations as follows. R R = R + r V = V + v = V + v a = ; ; ; 0 t 1 ε 0 0 µ µ 1 µ 1 VHelio = ; = v = v ; R at r µ µ µ v = v + ; v V V ; V = v + ; 0 Helio 0 r0 r0 r0 µ r a = = θ = ψ = ε a 0 1 ; e 1 ; cos 1 e + R

7 Departure Solution Features A 3-d view of the possible departure trajectories is given below, with locus of all possible points being a cone.

8 Patch Condition Example A spacecraft is required to escape from surface of nonrotating earth so that its V = 700 m/s. Determine the nominal V 0. Also, what would be the new V if the V 0 is higher by 10%. (µ Earth = 3.986x10 14, R E = 6378 km). V esc parabola = = 11,180 m / s V = V + V = 11,570 m / s 0 nomin al esc parabola dv µ dr V dv dv = = = V rv r V V V E 0 nomin al nomin al 0

9 Arrival at a Planet Once a spacecraft is put on a heliocentric Hohmann transfer ellipse through departure manoeuvre, it will arrive at the destination planet on this ellipse. The first point of contact with the planet is the SOI of that planet at which point, the spacecraft comes under the influence of the gravity of the target planet. From this point onwards, planeto-centric analysis will be required to determine whether the spacecraft will have a flyby, will form an orbit or will impact its surface. Usually, our interest is for a flyby or for an orbit (called capture), so we first determine the impact conditions.

10 Conditions for Impact on Planet To derive conditions for impact, consider the case when the spacecraft just grazes planet s surface, as below. b Impact Parameter or Stand-off Distance

11 Impact Parameter Solution Following relations provide the solution for the impact parameter b. b cos φ = ; h = V r cosφ = V b = Vprp r Vp µ V µ = V = + V r p p rp V V = V = V + V b = r + V µ ; esc p esc ; p 1 rp esc

12 Conditions for Impact on Planet Impact parameter b is the minimum distance that is permitted for no impact. We can now arrive at conditions with respect to an approach distance d.

13 Impact Condition Formulation Following relations provide conditions for impact. d = rsoi cos φ ; d > b there will be a flyby d = b there will be surface graze; d < b there will be an impact lim b = rp ; V lim b = 0; V 0 The collision (or capture) cross-section is shown below.

14 Impact Condition Example An approaching spacecraft reaches the SOI of venus with V = 700 m/s and φ = -85 o. Determine whether or not the spacecraft will impact. (µ Venus = 3.40x10 14, r SOI- Venus= AU, 1AU = 1.497x10 11 m, r Venus = 605 km). V esc = = 10,360 m / s b = = ( o ) 7 d cos AU m d = = = > b It will be a flyby. m

15 Impact Condition Example (b) In case it does not impact, calculate the minimum distance from the planet surface. 11 h = V d = rsoiv cosφ = m / s V 6 ε h ε = = m / s ; e = 1+ = µ µ venus 7 7 a = = m; rp = a( e 1) = m ε = = 6 hplanet m 19, 070km venus

16 Summary Interplanetary travel model is essentially an extension of the -body conic solution, by patching different segments of the trajectory. Departure hyperbola demonstrates the constraints that are implicitly applied on interplanetary mission starting point. Arrival manoeuvre is the counterpart of departure, with patch condition applied at the planet SOI.