The Sun Diver: Combining solar sails with the Oberth effect
TThe Sun Diver: Combining solar sails with the Oberth effect
Coryn A.L. Bailer-Jones
Max Planck Institute for Astronomy,K¨onigstuhl 17, 69117 Heidelberg, Germany
Accepted to the
American Journal of Physics
26 September 2020A highly reflective sail provides a way to propel a spacecraft out of the solar system using solarradiation pressure. The closer the spacecraft is to the Sun when it starts its outward journey, thelarger the radiation pressure and so the larger the final velocity. For a spacecraft starting on theEarth’s orbit, closer proximity can be achieved via a retrograde impulse from a rocket engine. Thesail is then deployed at the closest approach to the Sun. Employing the so-called Oberth effect, asecond, prograde, impulse at closest approach will raise the final velocity further. Here I investigatehow a fixed total impulse (∆ v ) can best be distributed in this procedure to maximize the sail’svelocity at infinity. Once ∆ v exceeds a threshold that depends on the lightness number of the sail (ameasure of its sun-induced acceleration), the best strategy is to use all of the ∆ v in the retrogradeimpulse to dive as close as possible to the Sun. Below the threshold the best strategy is to useall of the ∆ v in the prograde impulse and thus not to dive at all. Although larger velocities canbe achieved with multi-stage impulsive transfers, this study shows some interesting and perhapscounter-intuitive consequences of combining impulses with solar sails. I. INTRODUCTION
Sailing on the Earth uses the pressure of the wind topropel a vehicle such as a ship. The pressure come frommaterial particles, namely air molecules, imparting mo-mentum on the sail. Photons also possess momentumand therefore exert pressure, and these too can be usedto propel a vehicle. The pressure is much smaller, how-ever, and it can only be used as an effective means ofpropulsion in the near-vacuum of space.A photon of momentum p has energy E = pc , where c is the speed of light. As force equals rate of changeof momentum, the force exerted by a photon hitting asurface is ˙ E/c . If a beam of photons of intensity I isincident on a surface of area A , then ˙ E = IA so thepressure on that surface is I/c . This assumes the photonsare absorbed. If they are instead perfectly reflected thepressure is 2
I/c . This is a small number: The intensityof the Sun incident on the top of the Earth’s atmosphereis about 1370 W m − , so photons reflected from a surfacenormal to the Sun’s direction generate a pressure of just9.1 µ Pa. Yet if we build a low mass spacecraft with alarge reflecting surface, the resulting acceleration is non-negligible, and as it is continuous, large velocities canbe achieved. For example, if this pressure acted on a100 m sail of 1 kg mass, the sail’s velocity would changeby 80 m / s after one day (neglecting gravity). This is theprinciple of a solar sail.Solar sails are attractive because they free the space-craft from having to carry propellant. It is an un-avoidable consequence of the rocket equation that theamount of propellant a rocket must carry to change itsvelocity by ∆ v increases exponentially with ∆ v . The rea-son is that most of the propellant is used to acceleratethe unused propellant. Solar sails are being considered as a way to explorethe solar system. A few prototypes have in fact beenbuilt and launched, after having been brought above theEarth’s atmosphere by a conventional rocket. Of particu-lar interest is the Japanese mission IKAROS which useda 200 m sail of 16 kg mass. It was launched in 2010 andflew past Venus, and is so far the only solar sail to haveleft Earth’s orbit. Solar sails are particularly interestingfor missions that require a large ∆ v over a long duration,or that involve many maneuvers. As sails provide con-tinuous thrust, and can be tilted so that the net forceon them is no longer directed along the line to the Sun,they can produce non-Keplerian orbits, enabling trajec-tories that would be much more expensive to attain withimpulsive rocket maneuvers. Solar sails have also beeninvestigated as a source of thrust for deep space and inter-stellar missions. Although the radiation pressure fromthe Sun drops with the inverse square of the distancefrom the Sun, so does its gravitational pull, so that if asolar sail is light enough it can escape the Sun’s potentialwithout any additional assistance.It is this final application of solar sails – attaining thelargest possible velocity at infinity – that we will inves-tigate here. A large asymptotic velocity is paramount ifwe want to travel to the outer solar system or interstellarspace in the shortest time possible. We start from therealization that if a sail began its outward journey closerto the Sun than the Earth, it would gain extra acceler-ation due to the higher solar intensity in the first partof its journey, and so would achieve a larger velocity atinfinity.
But a spacecraft starting from a circular orbitof 1 au radius (1 astronomical unit, the mean Earth–Sundistance) would require an impulse in order to approachthe Sun. One way to achieve this is to use a rocket toapply a retrograde boost to decelerate the spacecraft by a r X i v : . [ phy s i c s . pop - ph ] S e p ∆ v . This will put the spacecraft on an elliptical orbitthat “dives” closer to the Sun (the larger ∆ v , the closerthe approach). Once the spacecraft reaches perihelionit opens its sail and uses radiation pressure to sail awayfrom the Sun. But assuming that we have a fixed budgetof ∆ v available to change the velocity of the spacecraft,is the best strategy to use all of this to dive as close tothe Sun as possible? The so-called Oberth effect, ex-plained in section II B, shows that applying the ∆ v whenthe spacecraft is moving faster transfers more kinetic en-ergy to the spacecraft than when it is moving slower. This suggests that it might be better to save some of theavailable ∆ v for a prograde boost at perihelion, which iswhen the spacecraft is moving fastest.We explore this scenario as a means of providing in-sight into the mechanics of solar sails and the use of im-pulsive boosts. The goal is not to identify the optimalset of orbital transfers that achieve the largest asymptoticvelocity for a spacecraft. Such problems have been ad-dressed in other articles and books. Indeed, becausesails can provide a continuous, variable, and non-centralthrust, their orbits can be very complex, so we generallyneed sophisticated procedures and numerical methods tofind the optimal trajectory for a given purpose. Thisarticle provides instead an introduction to the topic byfocusing on single-transfer orbits and analytic solutions,a topic that has not been covered comprehensively byother works. Some of the results are counter-intuitive,thereby providing insight into both Keplerian orbits andsolar sails. Broad introductions to solar sailing are pro-vided by Vulpetti et al. and, at a deeper level, McInnes. We will make some simplifying assumptions. We as-sume the sail is a perfect reflector, and that the Sun is apoint source that radiates isotropically. We will also ne-glect the gravity of any body other than the Sun. In prac-tice a spacecraft launched from the Earth would need toescape the Earth’s gravity, yet there are an infinite num-ber of solar orbits it could be placed on in that process,each of which would require additional impulses. Con-sidering these would be important in practice, but herewould only obfuscate the main issues.It is well known from orbital mechanics that booststangential to the orbit are more efficient at changingthe energy of the orbit than are boosts with a radialcomponent.
For this reason we will only consider pro-grade boosts – ones that increase the orbital velocity –and retrograde boosts – ones that decrease the orbitalvelocity. These boosts are assumed to be instantaneous.All distances and velocities are relative to the Sun, ex-cept for ∆ v , which is the change in velocity relative tothe spacecraft’s instantaneous reference frame. Velocitiesare non-relativistic, so a classical treatment suffices.We first go over some background physics in section II,before exploring the nominal sun diver scenario in sec-tion III. Some variations on this are considered in sec-tion IV. II. BACKGROUND PHYSICSA. Solar sails
We consider a sail with its normal kept pointed at theSun. As noted in the previous section, the pressure ofthe solar photons is 2
I/c , so the acceleration of a flatsolar sail of mass per unit area σ is a = 2 I/cσ . The solarintensity I drops off with the inverse square of the dis-tance r from the Sun, so may be written I = L s / πr ,where L s is the luminosity of the Sun (3 . × W). Wecan then write a = L s / πcσr . We will see momentar-ily that it is convenient to express this acceleration as afraction of the local acceleration due to the Sun’s grav-ity, g = µ/r , where µ = GM s , M s is the mass of theSun, and G is the gravitational constant. For orienta-tion, g = 5 . × − m s − when r = 1 au. The ratio a/g is called the lightness number of the sail, λ , and it followsfrom the above that λ = L s / πcµσ . It is a property ofthe spacecraft and the Sun only, and in particular is in-dependent of r . The acceleration of the sail may now bewritten a = λµ/r .It is interesting to note in passing that we obtain a = g ,i.e. λ = 1, when the mass per unit area of the sail is σ = L s / πcµ . Numerically this is 1 . × − kg m − ,which is about ten times less than plastic food wrap. Thisindicates how small the solar radiation pressure is, andhow light a solar sail needs to be to achieve an appreciableacceleration.The sail experiences two forces: (i) the radiation pres-sure pushing it away from the Sun; (ii) gravity pulling ittowards the Sun. Using r to denote the position vectorof the sail relative to the Sun, and ˆ r to denote its unitvector, then from Newton’s second law the dynamicalequation of the sail is¨ r = − µr ˆ r + λµr ˆ r = − µ (1 − λ ) r ˆ r . (1)This is the equation for Keplerian orbits in which thestandard gravitational parameter is µ (1 − λ ) as opposedto µ . Thus for 0 < λ < λ = 1 there is no net force, so the sailmoves in a straight line or remains at rest. For λ > < λ < / λ = 1 / / < λ < λ > λ ≥ / r i fromthe Sun with a velocity v i , what velocity does it achieve atinfinity ( v ∞ )? As the force is conservative the directionof v i is irrelevant, provided it is not directly towards theSun. Using conservation of energy we find v ∞ = v i + 2 µ ( λ − r i . (2)Clearly, the closer the sail is to the Sun initially, thelarger v ∞ will be, provided v i and/or λ are large enoughto permit escape at all. This suggests that to maximize v ∞ we should maneuver our spacecraft close to the Sunbefore opening its sail.The lightness number λ is key to determining the ve-locity that the sail can achieve. Sails launched to datehad small lightness numbers. The IKAROS solar sailthat went to Venus was primarily a technology demon-strator with a lightness number below 0.01. LightSail 2,deployed into Earth’s orbit in 2019 by a private organi-zation (the Planetary Society), had a sail area of 32 m and mass of 5 kg, giving it a theoretical lightness num-ber of 0.01. One of the interstellar exploration concepts proposes a 400 m diameter (125 000 m area) sail weigh-ing just 100 kg. Together with a payload mass of 150 kg,this implies a lightness number of 0.78. B. Oberth effect
Consider a spacecraft travelling with a velocity v i insome inertial reference frame S . The spacecraft’s (spe-cific) kinetic energy is (1 / | v i | . If it uses its rockets toincrease its velocity by ∆ v , its kinetic energy becomes(1 / | v i + ∆ v | . The increase in the spacecraft’s kineticenergy is therefore (1 / | ∆ v | + v i · ∆ v . This is max-imized when v i and ∆ v are parallel and then increasesmonotonically with increasing | v i | . That is, the fasterthe spacecraft is moving initially, the larger the increasein its kinetic energy for a given ∆ v . Even though thespacecraft expends the same amount of energy in its restframe to produce a given ∆ v , independent of v i , moreof this energy goes into the kinetic energy of the space-craft in S – and thus less into the kinetic energy of thepropellant – the faster the spacecraft is moving in S .This observation can be exploited by a spacecraft tooptimize the use of rocket propellant to escape from agravitational field. If a spacecraft is on an elliptical or-bit, then while its energy (kinetic plus potential) is con-served along the orbit, its velocity will vary, being largestat periapsis. Thus we will maximize the increase in theenergy of the spacecraft if we apply the impulse ∆ v atperiapsis. Specifically, we fire the rockets tangentially tothe orbit to increase its velocity – a prograde boost. Thisprinciple of maximizing kinetic energy increase is some-times known as the Oberth effect, after the pioneeringrocket scientist Hermann Oberth who first described itin the 1920s. If a spacecraft has a velocity v i at a pointon an elliptical orbit that is a distance r i from the central body, then if a prograde boost of ∆ v is applied at periap-sis where the velocity is v p , it is straightforward to showthat the velocity v f of the spacecraft when it returns to r i (but now on a different orbit) is given by v f = v i + (∆ v ) (cid:18) v p ∆ v + 1 (cid:19) . (3)Clearly, the larger v p for given v i and ∆ v , the larger v f . III. SUN DIVER
Equations (2) and (3) suggest that we can maximizethe velocity of our spacecraft at infinity if we maneu-ver our spacecraft as close to the Sun as possible beforeopening its sail and/or applying a prograde boost. If ourspacecraft starts on a circular orbit of radius r i and veloc-ity v i , we can use a retrograde boost to lower the orbitalvelocity and thus drop into an elliptical orbit with peri-helion less than r i . This is the classic Hohmann transferorbit. At perihelion we open the sail and apply an in-stantaneous prograde boost as motivated by the Obertheffect.In practice our spacecraft will carry a fixed amountof propellant which, according to the rocket equation,corresponds to fixed total ∆ v budget. This presents uswith a dilemma. Do we(a) Use the full ∆ v in the retrograde boost to drop asclose as possible to the Sun, but leave no propellantfor a prograde boost at perihelion, i.e. just rely onthe sail from there?(b) Forego the dive entirely and apply a full progradeboost on the initial circular orbit as we open thesail?Or do we perform a combination of the two? This isthe general case, shown in Fig. 1. The spacecraft startson the circular orbit (cid:172) . With 0 ≤ f ≤
1, a retrogradeboost of f ∆ v is applied at A to put the spacecraft on theelliptical transfer orbit (cid:173) . At this orbit’s perihelion P,a prograde boost of (1 − f )∆ v is applied and the solarsail is simultaneously deployed to put the spacecraft onorbit (cid:174) . It is not obvious what value of f will producethe largest asymptotic velocity. The retrograde boost atA will reduce the energy of the spacecraft, yet we hopeto more than recover this from the higher intensity solarradiation closer to the Sun.To resolve this we will now compute the velocity of thesail for the general case. The initial circular orbit (cid:172) ofradius r i has an orbital velocity v i given by v i = µr i . (4)At point A on this orbit the retrograde boost is applied,leaving the spacecraft with a velocity u i = v i − f ∆ v (5) −1.0 −0.5 0.0 0.5 1.0 − . − . . . . x [au] y [ a u ] l l A › f D v l P › (1−f) D v FIG. 1. The orbits in the nominal sun diver scenario: (cid:172) an initial circular orbit; (cid:173) an elliptical transfer orbit afterthe retrograde boost of f ∆ v of A; (cid:174) the final orbit after theprograde boost of (1 − f )∆ v and sail deployment at P. Theseorbits are shown to scale with r i = 1 au, ∆ v = 10 km / s, f = 0 .
5, and λ = 0 .
3, which makes orbit (cid:174) a hyperbola. Thedot in the centre is the Sun (not to scale). which puts it on elliptical orbit (cid:173) . Immediately after theboost the spacecraft has not yet moved, so its (specific)energy is E A2 = − µr i + 12 u i (6)where the first subscript (A) refers to the position and thesecond subscript (2) to the orbit. The spacecraft cruisesfrom its aphelion at A to its perihelion at P, where theradius is r p , the velocity is v p , and the energy is E P2 = − µr p + 12 v p . (7)By conservation of energy E A2 = E P2 . The (specific)angular momentum, r × v , is also conserved. Evaluatingthis at A and P is simple because the velocity is perpen-dicular to the radial vector. Thus r i u i = r p v p . (8)Equating Eqs. (6) and (7) and substituting for r p fromEq. (8) gives a quadratic equation in v p , v p − µr i u i v p + 2 µr i − u i = 0 . (9)This has two solutions. One is v p = u i , which only makessense when f = 0, i.e. the spacecraft makes no retrograde boost and so no dive. This can be treated as special caseof the other solution, which is v p = 2 µr i u i − u i . (10)When the spacecraft arrives at P on orbit (cid:173) , it si-multaneously opens its sail and applies a prograde boostof (1 − f )∆ v , putting it onto orbit (cid:174) . As the sail isnow open and its normal directed towards the Sun, thegravitational parameter is reduced by the factor (1 − λ )(section II A), so the spacecraft’s energy is E P3 = − µ (1 − λ ) r p + 12 [ v p + (1 − f )∆ v ] . (11)The spacecraft will cruise away from the Sun and will getat least as far as r i . Its energy at some distance r whereits velocity is v s is E r3 = − µ (1 − λ ) r + 12 v s . (12)From conservation of energy we can equate E P3 and E r3 and then substitute for v p from Eq. (10). After a few linesof algebra we get the following expression for v s in termsof the initial orbit, λ , ∆ v , and f (via u i from Eq. 5), v s = 4 λ v i u i +2 µ (1 − λ ) (cid:18) r + 1 r i (cid:19) +( v i − ∆ v ) − v i u i ( v i − ∆ v ) . (13)Note that r i and v i are not independent quantities: Theyare related via Eq. (4) because we assumed a circularinitial orbit when using angular momentum conservation.Equation (13) is not very intuitive, but we can checkthat it gives the right results in certain limiting cases.For example, f = 0 corresponds to not doing any diveand applying the full ∆ v in the initial orbit at the sametime as the sail is deployed. Equation (13) then gives v s = ( v i + ∆ v ) at r = r i for all λ , as we would expect.What value of f gives the largest value of v s at somedistance r ? This is potentially a function of all theother parameters in Eq. (13). We consider the space-craft starting at the Earth’s orbit of r i = 1 au, in whichcase v i = 29 . / s. While the numerical results will ofcourse change when selecting a different initial orbit, thestrategy that we should adopt to achieve the maximumvelocity at infinity is independent of this. Furthermore,it is sufficient to consider the velocity of the sail uponits return to r = r i , as we will see that the strategywhich maximizes the velocity here will also maximize itas r → ∞ , if the spacecraft can reach infinity at all.Figure 2 shows how v s ( r = r i ) varies with f for a sailwith λ = 0 . of ∆ v . We see that forall ∆ v > v s increases monotonically with f . In otherwords, the largest velocity is achieved by diving as closeto the Sun as possible. When ∆ v = 0 then of course v s = v i .Figure 3 shows the same situation but for a smallerlightness number of λ = 0 .
2. Now we see that for smaller initial burn fraction, f s a il v e l o c it y a t r = r i [ k m / s ] D v = 5 D v = 10 D v = 15 D v = 20 l = 0.5 FIG. 2. The solid lines show the sail velocity v s from Eq. (13)for r = r i (i.e. upon return to its initial altitude) and λ = 0 . f for different values of ∆ v (in km / s). Thedashed lines show the corresponding sail velocity for f = 0,i.e. when no dive is performed; these lines are horizontal. initial burn fraction, f s a il v e l o c it y a t r = r i [ k m / s ] D v = 5 D v = 10 D v = 15 D v = 20 l = 0.2 FIG. 3. As Fig. 2 but for λ = 0 . values of ∆ v , v s ( r = r i ) decreases monotonically with in-creasing f . In these cases, therefore, the largest velocityupon return to r = r i is achieved by not diving at alland simply doing a prograde boost of ∆ v from the initialorbit. This boost is formally applied at point P in Fig. 1,although of course it could be applied at any point onthe initial orbit. It appears that for a given λ , there is avalue of ∆ v below which the best strategy is to apply thefull boost prograde ( f = 0, no dive) and above which thebest strategy is to apply the full boost retrograde ( f = 1,full dive).This conclusion can also be obtained from Fig. 4, whichplots v s ( r = r i ) for ∆ v = 10 km / s and lightness numbersfrom 0.0 to 0.5, and moreover in Fig. 5 which zooms inon the transition region for lightness numbers between0.24 and 0.28. The curves in the latter figure all show aminimum. It can be shown by differentiation that if thefunction v s ( f ) has a turning point at all within the range initial burn fraction, f s a il v e l o c it y a t r = r i [ k m / s ] l = 0 l = 0.1 l = 0.2 l = 0.3 l = 0.4 l = 0.5 D v = km s10 FIG. 4. As Fig. 2 but now for ∆ v = 10 km / s for lightnessnumbers ranging from 0.1 to 0.5. The dashed line is for f = 0,i.e. no dive performed, which is the same for all lightnessnumbers. . . . . . initial burn fraction, f s a il v e l o c it y a t r = r i [ k m / s ] l = 0.24 l = 0.25 l = 0.26 l = 0.27 l = 0.28 D v = km s10 FIG. 5. As Fig. 4 but for a narrower range of lightness num-bers. ≤ f ≤ r , andindeed any values of the other parameters. Hence thereis never an intermediate value of f which will maximize v s : The optimum strategy is either f = 0 or f = 1. Inother words, to achieve the maximum velocity at infinitywe must use all the propellant in one go, either all at P( f = 0) or all at A ( f = 1) in Fig. 1. Partitioning thepropellant use between these points (or indeed any otherspoints) will yield a lower velocity at infinity. Which ofthe two strategies we should adopt depends on the valuesof the parameters. For example, in Fig. 5 we see that forthe three smallest values of λ shown (0.24, 0.25, 0.26), f = 0 maximizes v s , whereas for the two largest valuesof λ shown (0.27 and 0.28), f = 1 maximizes v s .We can use an inequality to determine the relationshipbetween the parameters that governs the transition from f = 0 to f = 1 being the best strategy. We ask for what . . . . . . D v / v i li gh t n e ss nu m b e r l f = 1 (full dive) is optimalf = 0 (no dive) is optimal3.0 8.9 14.9 20.8 26.8 D v [km/s] for v i = 29.8 km/s FIG. 6. The full retrograde boost ( f = 1, full dive) achievesa larger sail velocity v s than the full prograde boost ( f = 0,no dive) for values of λ and ∆ v/v i above the line (inequal-ity (15)). The opposite is true for values below the line. values of the parameters is v s ( f = 1) > v s ( f = 0) . (14)Using Eq. (13) and a little manipulation this inequalitycan be written λ > (1 − ∆ v/v i ) − ∆ v/v i . (15)This holds for all r because the term involving r inEq. (13) does not have a factor of f in it, so cancels out.This inequality is plotted in Fig. 6, and is one of the mainconclusions of this study. It shows that for a given ∆ v the optimal strategy is not to dive if λ is too small. Thereason is that with a smaller lightness number, the extrakinetic energy provided by the solar radiation from mov-ing close to the Sun does not compensate for the energylost by performing the dive.Using Eq. (13) we can compute the velocity the sailwill attain as r → ∞ . This is shown for case λ = 0 . v for no dive and for a fulldive. The transition between the strategies yielding thelarger velocity occurs at ∆ v = 13 . / s in accordancewith inequality (15). Note that if ∆ v is too small thespacecraft cannot reach infinity at all.Figure 8 shows the velocity at infinity for several differ-ent lightness numbers. In each case the optimal strategy( f = 0 or 1) at each ∆ v has been adopted according toinequality (15). Only if λ > . v = 0. As we saw before, for λ < . v is not todive at all. Recall that λ = 0 corresponds to no sail.How fast and close to the Sun does the spacecraft getin a full dive? This is shown in Fig. 9 as a function of ∆ v .For ∆ v larger than about 20 km / s the spacecraft will ap-proach within 10 solar radii (0.047 au), which is approx-imately the closest a spacecraft has ever approached the D v[km/s] s a il v e l o c it y a t i n f i n it y [ k m / s ] f = 0f = 1 FIG. 7. The velocity of the spacecraft at infinity as a functionof ∆ v with λ = 0 . f = 0, thick line) and a fulldive ( f = 1, thin line). D v[km/s] s a il v e l o c it y a t i n f i n it y [ k m / s ] FIG. 8. The velocity of the spacecraft at infinity as a functionof ∆ v for various lightness numbers λ (indicated on the right).For each combination of ∆ v and λ the optimal boost strategyis selected according to Fig. 6. The line is thin where theoptimal strategy is f = 1 (full dive) and thick where it is f = 0 (no dive). Sun (the Parker Solar Probe, although it achieved thisthrough a series of gravity assists).
Thermal consid-erations, i.e. not melting the spacecraft, would probablyset the limit on the closest approach.We have only considered ∆ v < v i in order to avoidthe singularity caused by the spacecraft dropping intothe centre of the Sun. If we had enough propellant fora larger boost, then the optimal strategy would be todive as close to the Sun as possible and to use all of theremaining propellant in the prograde boost at perihelion.We will nonetheless look at the idea of applying progradeboosts higher in the potential in the next section. . . . . . . D v [km/s] p e r i h e li on d i s t a n ce [ a u ] D v[km/s] p e r i h e li on v e l o c it y [ k m / s ] FIG. 9. Variation of the perihelion distance (top) and peri-helion velocity (bottom) as a function of the size of the ret-rograde boost for a full dive ( f = 1). The horizontal line inthe upper panel indicates 10 solar radii. IV. VARIATIONS ON THE SUN DIVERSCENARIOA. Boost at infinity
In the nominal scenario in the previous section we con-sidered applying rocket boosts in only two places, namelyon the initial circular orbit and/or at perihelion. Wechose perihelion because the Oberth effect tells us to ap-ply the boost when the spacecraft is moving fastest. Yetwhen λ > v in this case? We saw that in order to achieve thelargest final velocity when λ > / v retrograde to do a full dive (samescenario (a) as in section III).(c) Open the sails on the initial circular orbit to sail to D v[km/s] s a il v e l o c it y a t i n f i n it y [ k m / s ] FIG. 10. The solid lines show the velocity of the spacecraftat infinity after performing a full dive (Eq. (13), f = 1) as afunction of ∆ v for various lightness numbers λ (indicated onthe right). The dashed lines show the corresponding velocityachieved if no dive is performed, and the ∆ v is applied atinfinity instead (Eq. (16)). infinity, then apply the full ∆ v .The velocity at infinity for scenario (a) we obtain fromEq. (13) with r → ∞ , f = 1. The velocity at infinity forscenario (c) before we apply the final boost is obtainedfrom Eq. (13) with r → ∞ , ∆ v = 0. We then add ∆ v toget v c = v i √ λ − v Scenario (c) . (16)A comparison of the expressions for the two velocities isnot very informative, but a plot makes it clear which sce-nario is superior. The two velocities are shown in Fig. 10as a function of ∆ v for various lightness numbers. Weonly show λ > / v = 0 where they are of course equiva-lent. The reason is that for such large lightness numbers,more energy is gained from the Sun by diving close tothe Sun, than is lost by performing this dive.In practice a significant part of the mass of a rocketis its propellant. Thus the mass of the spacecraft,and therefore its lightness number, depends strongly onwhether the propellant has been expended. In our nom-inal scenario, which includes scenario (a), the sail is onlydeployed after all the propellant has been depleted – andthe propellant tanks and engines would be jettisoned too– so the spacecraft would have a small mass and thuslarge lightness number. Scenario (c), in contrast, requiresthe solar sail to accelerate all of the propellant to infin-ity, and so the spacecraft with the same sail would havea smaller lightness number than in (a). So in practicescenario (c) would be even worse. B. Sail towards the Sun
If the normal of the sail is not kept parallel to the radialvector pointing from the Sun to the spacecraft, solar pho-tons will exert a non-central force on the spacecraft. Thisleads to non-Keplerian orbits, the properties of which de-pend on how the pitch angle α between the radial andnormal vectors varies. One particularly interesting so-lution to the dynamical equations, and one of the fewanalytic ones, occurs when α is kept fixed. This givesrise to logarithmic spiral orbits which are described inpolar coordinates ( r, θ ) as r = r i exp( θ tan γ ) (17)where γ is a quantity (the spiral angle) that depends on α and λ only. The sail describes a spiral path aroundthe Sun. The velocity of the sail, v spiral , at a point ( r, θ )in its orbit is v = µr [1 − λ cos α (cos α − sin α tan γ )] (18)where the radial and tangential components are v spiral sin γ and v spiral cos γ respectively. This trajectoryis interesting for our application because if α is nega-tive, the solar photons act to decelerate the spacecraftcompared to the non-sail Keplerian orbit, and so thespacecraft will spiral inwards towards the Sun. This isachieved without expending any propellant, and so al-lows us to apply the entire ∆ v at perihelion. Can this beused to achieve a larger velocity at infinity than the fulldive scenario of section III? We compare the followingtwo scenarios, both of which start from a circular orbitwith the sail folded away.(a) Apply the full ∆ v retrograde on the initial orbit inorder to dive to a distance r = r p , at which pointwe open the sail and then keep it pointed at theSun (same scenario (a) as in section III);(d) Tilt the sail in order to spiral towards the Sun untildistance r = r p , at which point we simultaneouslyapply the full ∆ v prograde and turn the sail to keepit pointed at the Sun.This is not an ideal comparison because the velocity onthe spiral orbit in (d) immediately after the sail has beentilted is not equal in magnitude or direction to the ve-locity of the initial circular orbit. An additional impulseor maneuver would therefore be required to put the sailonto the spiral trajectory. We can ignore this, however,because we will see that it does not change the answer tothe above question.To see which of these scenarios give us the largest ve-locity at infinity (or indeed any distance r > r p ) we com-pare the velocities at r = r p , which are v p from Eq. (10)with u i = v i − ∆ v for scenario (a), and v spiral + ∆ v fromEq. (18) for scenario (d). Let us refer to these as the“closest approach velocities”. A comparison at r = r p issufficient because in both scenarios the force experienced −0.5 0.0 0.5 1.0 − . . . x [au] y [ a u ] l FIG. 11. A logarithmic spiral orbit as described by Eq. (17)(where x = r cos θ , y = r sin θ ) with r i = 1 au, γ = − . ◦ ,shown from θ = 0 rad to θ = 8 π rad. The dot in the center isthe Sun (not plotted to scale). by the spacecraft after closest approach is the same andis conservative (gravity plus photon pressure, both di-rected radially). The magnitude of the velocity at anylater point is therefore determined entirely by the energyat r = r p .By equating Eqs. (6) and (7) and eliminating u i usingEq. (8), we may express v p in scenario (a) in terms of r p and r i only v p = 2 µ r i r p r i + r p ) . (19)Although we are only interested in a full dive here, thisexpression actually holds for any value of f . The corre-sponding value of f ∆ v , and therefore the ∆ v we use inscenario (d), is computed using Eqs. (10) and (5).For the sake of illustration let us adopt the followingparameters for the spiral orbit: λ = 0 . α = − ◦ ,which correspond to γ = − . ◦ . The orbit for four revo-lutions around the Sun is shown in Fig. 11. The magni-tude of the velocity, v spiral , as a function of radial distanceis shown by the lower solid line in Fig. 12. For the spiralorbit of scenario (d), the closest approach velocity thatwe achieve is v spiral + ∆ v , shown by the dotted line. Forthe full dive of scenario (a), the closest approach velocityis shown by the upper solid line. We see that the clos-est approach velocity for scenario (a) is above that forscenario (d) for any given r p (which is equivalent to anygiven ∆ v ). Although this is shown here for scenario (d)with a certain λ and α , we find that it holds for any λ and α .If only a limited ∆ v were available, say 5 km / s, then inscenario (a) we achieve r p = 0 .
53 au and v p = 46 . / s, perihelion distance [au] v e l o c it y [ k m / s ] spiral spiral (radial component)spiral+ D vfull dive17.1 9.5 5.5 2.8 0.8 D v [km/s] FIG. 12. Sail velocities as a function of perihelion distance.The vertical solid line indicates 10 solar radii. The lower solidline is the velocity v spiral from Eq. (18) with r = r p for thelogarithmic spiral orbit shown in Fig. 11. The dashed lineis its radial component (which is negative). The tangentialcomponent is not show as it is almost equal to v spiral . Thedotted line is v spiral + ∆ v , i.e. the velocity of the sail afterthe prograde boost (scenario (d)). The upper solid line is theperihelion velocity for the full dive (scenario (a), Eq. (19)),which establishes a one-to-one relationship between r p and∆ v (Fig. 9) and is shown along the top axis. We see that thevelocity achieved in scenario (a) is always larger at a given∆ v than using this same ∆ v in scenario (d). whereas in scenario (d) we could perhaps spiral in muchcloser to the Sun, e.g. to 10 solar radii where v spiral =117 . / s, and then apply the same ∆ v to get a closestapproach velocity of 122 . / s. But the main point ofthe comparison in this section is to show that if we haveenough ∆ v to dive as close to the Sun as is thermallypossible, then scenario (d) is always inferior to scenario(a): Spiraling towards the Sun to this distance and thenapplying that ∆ v always results in a smaller velocity thana full dive.It was mentioned above that an additional impulsewould in practice be needed to put the spacecraft ona logarithmic spiral trajectory in the first place. Thiswould take away some of the available ∆ v , making sce-nario (d) even less favorable. C. Multi-stage transfer orbits
In section III we only considered a single ellipticaltransfer orbit, from the initial circular orbit to the pointwhere the sail is deployed. Multi-stage transfers can alsobe considered, and these are sometimes used in prac- tice because they sometimes need a smaller total ∆ v tomove between the same two orbits. An example is the bi-elliptic Hohmann transfer orbit. Starting on a circularorbit of 1 au radius, a prograde boost is used to put thespacecraft on an elliptical orbit of higher energy and thuslarger semi-major axis than the initial orbit. When thespacecraft reaches aphelion, a relatively small retrogradeboost is sufficient to put the spaceraft on a less eccentric,lower energy orbit, meaning it will dive close to the Sun.A final retrograde boost could be applied at perihelionto put the spacecraft on a low circular orbit around theSun, but in our application we would now open the solarsail to move away from the Sun at high velocity. Sucha maneuver can be set up to require less ∆ v than oursimple one-transfer maneuver to reach a given perihelion(or to reach a smaller perihelion for a given ∆ v ). V. CONCLUSIONS
We have examined the consequences of combining im-pulsive boosts with solar sails in Keplerian orbits as a wayof maximizing the velocity of a spacecraft at infinity. Oneof the main conclusions of this study may appear counter-intuitive: decelerating a solar sail by some ∆ v can resultin a larger velocity at infinity than accelerating it by thesame ∆ v . This is always the case for sufficiently large∆ v or sail lightness number λ (Fig. 6), and is true forany ∆ v when λ > /
2. In these cases the largest velocityat infinity is achieved by using the entire ∆ v in a retro-grade burn to dive as close to the Sun as possible beforeopening the sail at perihelion. This is because the extraenergy acquired from the solar radiation by diving closeto the Sun more than compensates for the energy lost byperforming the dive. For smaller ∆ v or lightness number,a larger velocity at infinity is achieved by instead usingthe entire ∆ v in a prograde burn at the moment the sailis deployed to move away from the Sun without perform-ing any dive. A combination of retrograde and progradeburns is always suboptimal. Tilting the sail to spiral into the Sun before applying the ∆ v , or using a sail (withlightness number above 1 /
2) to travel directly to infinityand then applying ∆ v , are also inferior in terms of finalvelocity achieved. We have looked here only at the use ofa single transfer orbit. A natural extension of this workwould be to examine the optimal combination of threeimpulsive boosts for two transfer orbits that maximizethe spacecraft velocity at infinity. ACKNOWLEDGMENTS
I thank Thomas M¨uller and Markus P¨ossel for usefulcomments on the manuscript.0 U. Walter,
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