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Acta Astronautica

Acta Astronautica 67 (2010) 279–283

0094-57

doi:10.1

$ A s

E-m1 IA2 Re

S.p.A. (R

journal homepage: www.elsevier.com/locate/actaastro

ACADEMY TRANSACTIONS NOTE

Effect of the total solar irradiance variations on solar-saillow-eccentricity orbits$

Giovanni Vulpetti 1,2

Via Casal De’ Pazzi 20, 00156 Rome, Italy

a r t i c l e i n f o

Article history:

Received 26 September 2009

Received in revised form

17 January 2010

Accepted 1 February 2010Available online 18 February 2010

Keywords:

Total solar irradiance

Solar sailing

Sailcraft

Lightness vector

65/$ - see front matter & 2010 Elsevier Ltd. A

016/j.actaastro.2010.02.004

elf-funded work.

ail address: giovanni.vulpetti@fastwebnet.it

A Full Member—Section 2.

tired, formerly at GalileianPlus s.r.l. (Rome, It

ome, Italy).

a b s t r a c t

Solar photon sailing is based on the radiation pressure of the total solar irradiance (TSI).

This note analyzes the effects of the actual irradiance on low-eccentricity orbits as an

introductory work to much more complex cases such as sailcraft-Mars rendezvous. Two

special orbits are considered: (1) a circular warning orbit and (2) a transfer orbit

between Earth and Mars. It turns out that TSI fluctuations can cause large perturbations

to these types of solar-sail trajectory.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In the last four or five decades of solar photon sailing(SPS) studies, the solar radiation pressure on the sail hasbeen assumed always constant, and virtually assigned bythe analyst each time.

Before the astronautical era, there was no universally-accepted evidence that the so-named solar constant mightdeviate from a strict constant value. Ground measure-ments sufficiently precise/accurate to reveal any realdeviation from the ‘‘solar constant’’ were not possible, inpractice, before space satellites (e.g. [1,2]). Many historicalconcepts, ideas and facts (from China, Europe, and UnitedStates) about the Sun and the investigation of the solarobjects affecting the Sun’s overall electro-magnetic emis-sion can be found in [3]. In the industrial era, mainly afterS.P. Langley (USA) and A. Secchi (Italy) in the 19thcentury, C.G. Abbot (USA) was pioneer in measuring fordecades the solar constant and its potential variability [4].

ll rights reserved.

aly), and Telespazio

However, his measurements (though performed fromobservation stations on mountains of USA, Chile, andArabia, and by means of new instruments) indicated a toomuch high (some percent) solar change. A remarkablesynopsis of his and his team’s prolonged measurementcampaign can be found in [5].

On November 16, 1978, the first cavity radiometer,named HF after Hickey–Frieden, was switched on aboardthe US satellite NIMBUS-7; this was followed by othersatellites endowed with modern instruments for observingthe Sun. One of the most striking results of thespace-era benefits has been the discovery that the totalsolar irradiance (TSI, standardized at 1 AU) is variable indeed.Let us be more precise: from the astrophysical viewpoint, theSun is a main-sequence star, and its exitance can beconsidered constant. From the viewpoint of the heliosphereevolution, including the heliospheric electric/magnetic fields,interplanetary plasma and its interaction with celestialbodies, planetary climates, etc., the Sun is notably variable.Furthermore, the spectral solar irradiance (SSI) is variable inits various bands, especially at low wavelengths.

As solar radiation pressure is the primary source ofmomentum change of a photon sailcraft, we shall analyze theeffect of its variations on trajectory. Because this paper ispreliminary to another paper [6] of this research line, let us

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G. Vulpetti / Acta Astronautica 67 (2010) 279–283280

focus on a couple of cases that should have a high relevanceto future in-space propulsion: a storm-warning orbit and aguessed sailcraft-Mars rendezvous (heliocentric arc only).Essentially, we compare trajectories for each case: one drivenby constant TSI and the other one under variable TSI. To thisaim, we use the Heliocentric Inertial Frame of reference (HIF),obtained by counterclockwise rotation of the InternationalCelestial Reference System (ICRS) about its X-axis by theecliptic’s obliquity at J2000. The independent parameter inthe motion equations is the Barycentric Dynamical Time(TDB). Compliantly, planetary perturbations are computed viathe DE405 ephemeris data distributed by Jet PropulsionLaboratory.

Among the TSI features, there is a stochastic behavior;hence, we adopt a particular method for comparing solar-sail trajectory arcs. This approach is useful in preliminaryinvestigations for pointing out a non-negligible problemin sailcraft design.

4 Briefly, the lightness vector L� ðlr lt lnÞT is defined as the

sunlight-pressure acceleration measured in the sailcraft orbital frame,

and normalized to the local solar gravitational acceleration. It controls

2. A short reminder of TSI

The thrust acceleration due to the solar radiationpressure can be defined via either classical concepts in SPS[7] or recent ones such as [8] that we use here for SPStrajectory computation. Regardless which formalism onelikes to use, the key point to grasp is that the sailcraftthrust acceleration is proportional to TSI. Even if the sailattitude were constant (with respect to the local sunlight)and the sailcraft moved at the same distance from theSun, thrust would not be constant because TSI varies withtime. TSI changes may be thought as a mixing of (i)secular variations, (ii) cyclic variations, and (iii) stochasticfluctuations (e.g. [9]).

Reconstruction of TSI to before the Maunder Minimum(occurred approximately from 1645 to 1715 AD) are nowsufficiently reliable. Since such historical minimum, thecycle-averaged TSI has shown an overall increase of1:3þ0:2�0:4 W=m2 [10] or, equivalently, 0.029 percent and per

century, with a max error range of 0.013 percent and percentury. In contrast, the typical variations of TSI over one

solar cycle (around 11 years) amount to about 0.1–0.2percent. In this paper, item (i) is considered to be not relevantto solar sailing for this reason, and the fact that the envisagedsolar-sail missions conceived hitherto exhibit flight and/oroperational times much less than a century (of course).

A big amount of data from satellite instruments are beingprocessed since 1978 (e.g. [11]). The construction ofcomposites of TSI time series is an active research area.One of the three TSI composites, the PMOD3 composite, isshown in Fig. 1, where one can see the daily changes since1975 because the TSI reconstruction has been extended tothe minimum between cycles 20 and 21. The black curverepresents the smoothing process with a bandwidth of 81days (i.e. three solar Carrington rotations). Of greaterimportance is the analysis of TSI in the frequency domain,some examples of which are reported in Fig. 2, where the

3 Physikalisch-Meteorologisches Observatorium Davos (PMOD),

Switzerland, http://www.pmodwrc.ch/pmod.php?topic=tsi/composite/

SolarConstant.

relative power spectrum is plotted vs. TSI-variabilityfrequency [12]. Such type of analysis contributed to manysignificant areas for which a deeper understanding of thesolar variability is a key for the physics of the heliosphere,and the Earth’s global atmospheric phenomena.

Three important general items can be gathered bysimple inspection of Fig. 2: (a) there are characteristicperiods as short as a few minutes to as long as a couple ofsolar cycle durations, (b) variability is in terms of differentfunctions of the frequency, (c) high frequencies exhibitthe lowest spectral variation but a peak at 5 min. Changesof the averaged slopes are ascribed to various features ofthe solar-surface’s magnetic field, e.g. the granulation,mesogranulation, and supergranulation.

3. Sailcraft trajectory with variable TSI

Using the time series of TSI composite over the lastthree complete solar cycles (i.e. 21, 22, and 23), it is easyto compute the first statistical moments (includingskewness and kurtosis), which are reported in Fig. 3showing the histogram of the daily means in thementioned cycles. Considering the ratio of four standarddeviations to the mean, one obtains 0.0016. One maywonder whether such a small change might cause someperturbation to sailcraft trajectories with respect to usingthe mean value. We are going to verify just whether suchperturbation is or is not small.

The radiation-pressure thrust acceleration is inverselyproportional to the sailcraft sail loading, say, s, i.e. theratio of the sailcraft mass to the effective sail area. Oneexpects that a low-s sailcraft may be sensitive also to thehigh frequencies in Fig. 2; this sensitivity should increaseas technology allows lower and lower s�values. Oneexpects also that the overall influence of TSI on sailcrafttrajectories depends on the mission class.

As TSI exhibits stochastic features as well, it is notpredictable in the details, especially considering thatmany years normally elapse from the phases A–B, wherea space mission is designed, to the time-frame where thespacecraft actually flies. Consequently, in order to ascer-tain whether the actual TSI may or may not perturbsailcraft orbits appreciably, we did not use any of the(still-unreliable) TSI prediction models, but rather we haveconsidered sailcraft trajectories as occurring in past years,specifically during cycle-23; thus, we have employed theactual TSI measurements.

The mission analysis code was modified to computethe lightness vector4 [8] as dependent on TSI(JDTDB),where the argument is the Julian Date scale associated toTDB. Between the TSI daily means, a linear interpolationwas employed. In so doing, we have assumed (for this

the evolution of any sailcraft trajectory. Its components were named the

radial, transversal, and normal (lightness) numbers, respectively. ln is

measured along the sailcraft’s orbital angular momentum. Sailcraft’s

orbital energy and its time rate, and the orbital angular momentum rate

depend explicitly on such components.

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Sol

ar Ir

radi

ance

(Wm

−2)

75 77 79 81 83 85 87 89 91 93 95 97 99 01 03 05 07 09Year

1366

1368

1362

1364

0 2000 4000 6000 8000 10000Days (Epoch Jan 0, 1980)

Min20/21 Min21/22 Min22/23 Min23/24

Mod

el

HF

AC

RIM

I

HF

AC

RIM

I

HF

AC

RIM

II

VIR

GO

Average of minima: 1365.465± 0.016 Wm−2 Difference of minima to average:+0.047; +0.105; +0.048; −0.200 Wm−2 Cycle amplitudes: 0.953± 0.019; 0.915 ± 0.020; 1.031 ± 0.017 Wm−2

0.1%

Fig. 1. The extended PMOD composite of the TSI (October 2009 update). Gray lines represent daily means (from C. Frohlich, PMOD—World Radiation

Center, Switzerland).

0.001 0.01 0.10 1 10 100 1000.Frequency [μHz]

100

102

104

106

Pow

er [p

pm2 /μH

z]

10000 1000 100 10 1.00 0.10 0.01Period (days)

11−year Period

1−year Period 27−day Period

5−minute Oscillations

VIRGO TSI (6_000_0312) 1996.1−1997.5VIRGO TSI (6_000_0312) 2000.8−2002.2TSI Composite (26_00_0312)

Fig. 2. TSI in the frequency domain using the PMOD composite from

1978 to 2002. The 2000–2002 spectrum regards high solar activity in

cycle-23. The 1996–1997 spectrum pertains to the cycle-23 minimum.

(from [12], courtesy of C. Frohlich, PMOD—World Radiation Center).

−5 −4 −3 −2 −1 0 1 2 30

200

400

600

800

1000

1200

0.1−W/m2 bins

Freq

uenc

y

Total Solar Irradiance Distribution around the Mean

cycles 21−22−23mean=1365.94 W/m2mode=1365.63 W/m2stdv= 0.56 W/m2skew= 0.118kurt= 4.19

Fig. 3. Histogram of TSI daily means of cycles 21–23.

G. Vulpetti / Acta Astronautica 67 (2010) 279–283 281

introductory work) that, in practice, the sailcraft isinsensitive to TSI changes due to phenomena with(pseudo) periods shorter than about 0.25 days or, equiva-lently, frequencies higher than 50mHz (see Fig. 2). Thus,the sailcraft thrust acceleration is sufficiently smoothedthat high-precision adaptive-stepsize integrators for(initial-value problems for) non-stiff equations may beapplied. We employed the well-known variable-ordervariable-stepsize Adams–Bashforth–Moulton method (e.g[13] for a good introduction); a Fortran-90 optimized codehas been used [14]. A further assumption has been made

here: TSI measurements, which are adjusted at 1 AU, havebeen used here beyond the Earth–Moon orbits withouttime delay. Only the 1/R2-scaling has been considered.

Two trajectory examples are presented in the next twosections: (1) a warning mission via a circular orbit with1-year period and 0.66-AU radius, and (2) a simple Earth-to-Mars transfer.

4. Warning mission via SPS

Coronal mass ejections and large flares often inducegeomagnetic storms, some of which have caused seriousmalfunctions of terrestrial facilities (in addition to variousproblems to Earth satellites). They are also a hazard to

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G. Vulpetti / Acta Astronautica 67 (2010) 279–283282

astronaut life. On August 25, 1997, US NOAA and NASAlaunched the spacecraft ACE that orbits around theL1 point of the Sun–Earth gravitational system; L1 islocated 0.01 AU sunwards. Besides many scientific goals,ACE provides an advance warning for space weather in therange 0.5–1.5 h, typically (e.g. see http://www.swpc.noaa.gov/rpc/costello/). A way to improve future Sun-monitoring missions is to use sailcraft, thus spending nofuel for controlling its halo orbit. There is though anotherway, which encompasses much higher forewarning timesand propellantless maneuvers. It is known that if a sail isoriented orthogonally to sunlight, i.e. L� ðlr 0 0ÞT;0olro1, then there exist circular orbits with periodsP¼ a3=2=

ffiffiffiffiffiffiffiffiffiffiffi1�lr

p, where the radius a is expressed in AU,

and P is in years. Thus, considering an ecliptic-coplanarcircular orbit with P=1, the warning time to Earth is givenby twC ð1�aÞ=u, where u is the bulk speed of the particlesunder monitoring. As such a mission is useful especiallywhen the Sun emits energetic/intense particle streams,one could think of a sailcraft for which tw may be 30 timesthose ones of ACE, and in such a way the operational orbitis not too perturbed gravitationally. The result is thefollowing nominal circular orbit: a=0.65948, lr ¼ 0:71318,and s¼ t2:15633 g=m2, where t is the thrust efficiency[8]. In principle, this may be realized in sailcraft with asignificantly thin bi-layer sail. This lightness number istied to the TSI reference value of 1367 W/m2, a value oftenused in literature. We computed two orbits of exactly oneyear each from the initial Keplerian state (a e 0 0 0 0) att0=2003.06.17 0-TDB, and with the perturbations (ordisturbances) from the inner planets and Jupiter: oneorbit begins with (a=0.65948, e=0) that corresponds tothe reference TSI; the second orbit starts with the sameposition and velocity that imply (a=0.65871,e=0.0011677) under TSI(t0)=1366.4 W/m2, namely, theobserved value at t0. The evolution of the osculating-orbitperiod variation is plotted in Fig. 4, where thecontributions (amplified by 100) of the five planets havebeen reported as well. It is apparent how and how muchthe sailcraft senses the changing TSI. Whereas the (much

0 0.2 0.4 0.6 0.8 1−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

time (year)

sailc

raft ΔP

(da

y)

Sailcraft’s Orbital Period Change vs. Time

change due to planets x 100

change due to variable TSI

g−field = Sun + inner planets + Jupiter reference orbit with TSI =1367 W/m2: R = 0.65948 AU, λr= 0.71318 P = 1 year warning time = 17−35 hours

Fig. 4. Warning-orbit sailcraft DP under variable TSI and constant TSI.

weaker) gravitational-perturbation period variations passthrough zero several times, there is no such similarbehavior for the TSI-change variations, which depend onthe particular time series taking place.

One may object that this sailcraft is significantlyperturbed because it is close to the Sun and rather light.What happens for heavier and more distant sailcraft? Arather different case is shown in the next section.

5. Earth-to-Mars transfer via SPS

Let us consider a simple starting trajectory (effi0:07) fora sailcraft’s heliocentric transfer from Earth to Mars in the2003–2004 time-frame. The nominal transfer, under TSI(1AU)=1367 W/m2 again, has constant L¼ ð0:05395375;0:03381874; 0:02324687ÞT for 460 days starting from2003.05.29 09:58:26 TDB. In other words, one thrustingarc with constant sail attitude is considered. In the Mars’orbital frame, the sailcraft arrives at rf=85.0 Martian radiiwith speed vf=553 m/s. The sailcraft initial state, with thesame control and flight time, has been then propagatedunder the variable TSI occurred in the same period from2003/May/29 to 2004/August/31. The changing TSI causessmall thrust-acceleration changes: Each of the threecomponents of L follows the fluctuations of the solarirradiance. Because the sail attitude angles are keptconstant, the combined effect of the orbital energy andangular momentum (unbalanced) variations accumulates astime goes by. This is apparent from Fig. 5, where theposition difference (in HIF) DRðtÞ �RðTSI�varÞ

ðtÞ�RðTSI�conÞðtÞ

is plotted vs. time. Two coordinate deviations ultimatelyalter the ensuing Mars-approaching low-thrust trajectoryenormously, thwarting the TSI-constant control, no matterhow well it may be designed and implemented onboard.Thus, some different control strategy appears to be required;this is detailed in another paper [6], where a multi-arccontrol is carried out to balance the TSI fluctuations.

0 100 200 300 400 500−200

−150

−100

−50

0

50

time (day)

Iner

tial C

arte

sian

Cha

nges

(mar

tian

radi

us)

ΔX

ΔY

ΔZ

Mars Orbital Frame: missing position = [−64.1 , 158.1 , 4.0 ] m.r.

Fig. 5. Sailcraft-Mars rendezvous case: variable TSI relative to constant

TSI position in HIF vs. time.

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G. Vulpetti / Acta Astronautica 67 (2010) 279–283 283

6. Conclusions

The key point is the following: by the most sophisticatedTSI models available nowadays (Spring 2009) one couldderive the solar radiance from the solar-surface magnetic-field (B) evolution. In principle, one may compute the TSIvariations if the change of B were known; however,predictions of DB are still very unreliable [15].

The above trajectory results suggest us preliminarily:(1) TSI fluctuations, though 0.1–0.2 percent of the meanvalues, produce large perturbations on sailcraft dynamics,(2) mission analyst should use proxy-prediction modelsfor computing nominal TSI-variable trajectories duringthe flight design (with a careful comparison to TSI-constant computations), and (3) perhaps, the sailcraftshould carry a TSI-measurement package as part of theattitude and orbit control system.

Acknowledgments

The author thanks the IAA full members who acceptedto endorse this Note; their suggestions have been greatlyappreciated. Thanks go also to Mr. L. Kos of NASA/MSFCfor his useful comments on the whole paper.

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