Automatic Sun Tracking with Solar Radiation Pressure

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A comparison for this sun following property with the available flight data from. Hayabusa is done. Nomenclature. CoM = Center of mass. SRP =Solar Radiation ...
Automatic Sun tracking with Solar Radiation Pressure in Interplanetary Missions F.Janssens1, Noordwijk, 2201KA, The Netherlands

Abstract This paper is about the use of the solar radiation pressure torque as a sun following device. The starting point is the averaged torque over a spin period. The attitude dynamics is investigated in terms of the angular momentum defined by the spin rate times the inertia about the spin axis. We assume that the nutation is negligible and show that the angular momentum precesses about an equilibrium direction that makes a constant angle with the sun when the spin rate is constant. In an elliptic orbit, the equilibrium angle remains constant because the variable distance to the sun is compensated by the rate change of the true anomaly. The precession period is a constant when expressed in true anomaly and independent of the radius of the precession circle. The equilibrium angle increases with the spin rate and has a maximum value of 45 ∞ from the Sun. For higher spin rates, the attitude does no longer follow the sun and is an inertially fixed direction (gyroscopic stabilisation). A comparison for this sun following property with the available flight data from Hayabusa is done.

Nomenclature CoM = Center of mass SRP =Solar Radiation Pressure Fe = Solar irradiance 1358 Watt/m2 at 1 AU = velocity of light 299792.5 km/s c fSRP, = Instantaneous SRP force on a flat panel fa , f r =force due to absorption, force dur to reflection = absorption coefficient. (dominant effect for solar panels) Ca = reflectivity coefficient. Cr =1 - Ca (dominant for solar sails) Cr σ = fraction of Cr that undergoes specular refection. 1-σ , fraction that undergoes diffuse reflection Crs, Crd = Cr σ , β Cr (1-σ) C = Maximal Principal moment of inertia p = [px ± py pz ]T , coordinates of the midpoint of a solar panel (centre of pressure) in a body fixed , principal axis system. The configurations considered have 2 identical panels , symmetrically mounted (± py ). Ω = spin rate h = Angular momentum of the satellite h =|h| = C Ω P = Solar radiation pressure at 1 AU , 4.5266 µN /m2 ( Fe/c), A = Surface of a solar panel. Many satellites (Roseta, Hayabusa) have two two identical solar panels. ( Ahayabusa = 11.5 m2 , ARosetta = 31.4 m2 ) t = Solar radiation pressure torque averaged over a spin period t = ts (1h x 1x) ts = torque coefficient = 2 PA cosθ l (θ) / r2 l = lever arm in the torque coefficient 1

Dynamicist, retired from ESA. AAS member. [email protected] 1 American Institute of Aeronautics and Astronautics

θ

= Angle between the angular momentum and the sun. { cos θ

= 1h • 1 x }

ν = true anomaly =rate of change of the true anomaly dν/dt ωs r = modulus of the radius vector r Sun-Satellite (in AU) 1h = h / (CΩ) unit vector of the angular momentum 1r = unit vector of the radius vector r 1s = unit vector from the satellite to the sun. x-axis of the orbital frame (-1r) = normal to a solar panel 1n 1z : z-axis of the reference frame , opposite to the orbit normal.

I. Introduction

T

he power of the first interplanetary missions was always delivered by radio isotope generators as the duration of those missions combined with the distance to the sun made batteries unsuitable (Ulysses) . The development in the solar panel technology enabled interplanetary missions with solar panels as power source (Rosetta, Hayabusa). The size of the panels brought the role of solar radiation pressure (SRP) to the foreground when it was discovered that this force could provide automatic sun tracking by putting the satellite in a slow spin (barbecue mode). In fact, this effect saved the Hayabusa mission after the loss of the ion propellant. In this paper we use SRP models for solar panels as given in [Wertz, Chobotov ]1,2 .These models take into account the specular and diffuse reflection of the photons as well as the absorption. From this solar radiation force, the corresponding torque for the attitude dynamics is obtained. More elaborate models, including the transmittivity and non-Lambertian diffusion are presented in [Green , McInnes ]3,4.These models are used for solar sails in the context of solar power and solar sailing missions. This paper is about the use of the SRP torque as a sun following device. The starting point is to calculate the averaged torque over a spin period. The length of these calculations depends on the geometry of the Solar Array panels [Sonnabend]6. The attitude dynamics is investigated in terms of the angular momentum defined by the slow spin rate and the inertia about the spin axis under the assumption that nutation can be neglected. We show that the angular momentum may rotate about an equilibrium direction that makes a constant angle with the sun direction. The equilibrium angle is constant when the spin rate is constant which is the case for appropriate orientations of the solar panels. The value of the equilibrium angle increases with the spin rate and has a maximum value of 45 ∞ from the Sun. For higher spin rates, the attitude is an inertially fixed direction (gyroscopic stabilisation). The result for the equilibrium angle shows also that this angle remains constant when the satellite is in an elliptic orbit were the distance to the Sun is variable. A comparison for this sun following property with the flight data from Hayabusa available in the public domain is done.

A. Evolution Angular Momentum A spinning spacecraft equipped with large solar panels is in a direct trajectory about the Sun. In the reference frame depicted in Fig.1, the change of its angular momentum under influence of the SRP torque is given by :

h + ω s 1 z × h = t

(1)

The origin is at the CoM of the satellite, the x-axis 1x remains sun-pointing. As the satellite orbits the sun, the x-axis remains in the orbit plane. The z-axis is parallel but opposite to the orbit normal such that the rate of change of the true anomaly ν as seen from the satellite is positive: ωs = dν/dt. The y axis completes a right handed system ( 1y = 1z x 1x ) and is also in the orbital plane. This reference frame can be compared to the to the standard orbital frame : the origin is not an inertial point but the CoM of the satellite. The x-axis is opposite to the standard radius vector from the Sun and the z-axis is opposite to the orbit normal of a direct trajectory about the sun. The velocity has a positive projection on the y-axis as both the x- and z-axis are reversals of the orbital frame. The rotation

Fig.1 - Reference frame matrix from an inertial frame to this frame is easily expressed in the orbital elements.

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The components of the satellite angular momentum h in the reference frame are h =[ hx hy hz ]T . When the nutation of the satellite is negligible, the magnitude of h is : |h| = Ω C .This assumption was used for calculating the right hand side t . h Also : ωs1z = [ 0 0 ωs ] T and the angle θ between h and the sun is cos θ = 1 h • 1 x = x ΩC The averaged torque in Eq.(1) follows from the solar radiation pressure force averaged over a spin period T:

t=

1 dt T ∫T

∫ dA A

ρ × f SRP

(2)

where ρ is a generic point from the CoM to a point of the solar panel. The required calculations for t can be quite tedious and depende mainly on the directions of the normals to the solar panels. Details of most of the practical cases are given in [Sonnabend]6. In this paper we consider only flat panels with a normal perpendicular to the maximal axis of inertia in which case the resulting averaged torque remains perpendicular to the angular momentum h and the direction to the sun (no propeller torque that changes the spin rate). The derivation is given in Appendix A. The torque t is time varying via the angle between the angular momentum and the sun. The total SRP force, fSRP is partly due to absorption (fa) and partly to reflection ( fr ). The force due to the absorption on a flat solar panel with surface A is opposite to the direction to the Sun [Wertz, Chobotov, McInnes, Green, Terauchi ]1-4,14 : fa = -PA Ca cos θ 1s. As solar panels are dark blue, the major part of the torque is due to the absorption of photons. The remaining photons are reflected and contribute as a specular and diffuse (Lambertian) reflection. For a solar panel, a further split-up of the diffuse reflection in Lambertian and non-Lambertian is a secondary effect and is not included in the model of SRP force used in this paper. The force due to the reflected photons is : fr = -PA cos θ (Crd 2/3 1n +Crd 1s+2 Crscosθ 1n) (3) The absorption was 70% for ROSETTA , 80% or 76% for SELENE [Kubu-oka, Ping]8 and 90% for HAYABUSA [Mimasu and&]11 . For other surfaces as solar sails and gold foil, the reflectivity is the dominant effect. The influence of the SRP force on an orbit about the Sun, is often investigated with models that take only the force in the direction of the sun into account [Scheeres]12 For the torque t we introduce the following notations , t = ts 1h x 1x with ts(r,θ) = ( 2 P A / r2) l(θ) cosθ

(4)

Notice that the magnitude |t| is ts sinθ . The lever arm l depends on the configuration of the solar panels and the model of the solar radiation pressure. Introducing these notations in Eqn.(1) :

h x − ω s h y = 0 h h y + ω s h x = t s z ΩC hy h z = −t s ΩC

and in the form

h = A( h x , t ) h

:

⎡ hx ⎤ ⎡ 0 ⎢ ⎥ ⎢ ⎢h y ⎥ = ⎢−ωs ( t ) ⎢ hz ⎥ ⎢ 0 ⎣ ⎦ ⎣

(5a-c)

ωs ( t ) 0 −

ts

ΩC

⎤ t s ( r ,θ ) ⎥ ΩC ⎥ 0 ⎥ ⎦ 0

⎡ hx ⎤ ⎢h ⎥ ⎢ y⎥ ⎢⎣ hz ⎥⎦

(6)

Eq. 6 is not a linear system of differential equations with constant coefficients as ts contains hx and ωs is time varying for an elliptic orbit . However the anti-symmetric structure confirms that |h| is constant and justifies the substitution of h by ΩC under the assumption of negligible nutation (See Appendix) .

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B . Equilibrium Solution

 = 0 . Eqn. 5a and 5c give: hyE = 0 which means that this Eq.6 has an equilibrium solution hE that follows from h solution is in the xz-plane. Eqn. 5b gives following relation between hxE = h cosθ and hzE = h sinθ : ω s h xE =

ts ω h h hzE ⇒ sin θ E ≡ zE = s xE ΩC ΩC ts

(7)

This expression for θE is, a priori, not a constant as ωs (t), ts [r(t),θ]. However, inserting the result for θE in (6) , we can rewrite this system as :

⎡ ⎡ hx ⎤ ⎢0 ⎢ ⎥ h t ω ( ) = ⎢ −1 s ⎢ y⎥ ⎢0 ⎢ hz ⎥ ⎣ ⎦ ⎣⎢ and changing to the the true anomaly

⎡h ⎢ ⎢h ⎢h ⎣

' x ' y ' z

⎤ ⎥ 1 tanθ E ⎥ ⎥ 0 ⎦⎥

⎡ hx ⎤ ⎢h ⎥ ⎢ y⎥ ⎢⎣ hz ⎥⎦

0

1

0 1 − tanθ E

(8)

as independent variable : dν = ωs dt

⎤ ⎡ ⎥ ⎢0 ⎥ = ⎢−1 ⎥ ⎢0 ⎦ ⎣⎢

1

0 1 − tan θ E

⎤ ⎥ 1 ⎥ tan θ E ⎥ 0 ⎦⎥ 0

⎡ hx ⎤ ⎢h ⎥ ⎢ y⎥ ⎢⎣ hz ⎥⎦

(9)

where (.)' = d(.)/dν . Then the system matrix is constant, provided θE is constant. Substituting the expression for ts as given by Eqn.(4) in Eqn.(7) :

sin θ E ≡ ( ω s r 2 )

ΩC 2 PAl

(10)

The term ωsr2 is the orbital angular momentum per unit mass which is constant for a Keplerian orbit. Its dimensions are T-1 as PA is the SRP force at 1 AU and r is the ratio of the distance to 1 AU. The unit for ωsr2 is rad/year = 3.026 10-7 RPM or 1.8156 10-6 deg/s. For flat solar panels without a dihedral angle ψ 2, the lever arm is :

l(θ ) = px ( Ca + Crd ) cosθ + Crd

py

(11)

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Combining Eqns. 10 and 11 gives an equation for θE that contains only constant parameters : the orbital angular momentum, the spin rate and spin inertia, the SRP force at 1 AU, the solar panel coefficients and the coordinates of the midpoint of a panel from the CoM. All these constant quantities combine into an equilibrium angle which is the only parameter needed for the motion of the angular momentum w.r.t. the true anomaly. The variation of the distance to the sun combined with the variable rate of the true anomaly results in a constant equilibrium angle. The value of py relative to py depends on the configuration of the solar panels. Fig. 2 and 3 show that py is smaller for Hayabusa than for Rosetta. For a small equilibrium angle we have :

θE =

C ωsr 2 Ω

(12)

py ⎞ ⎛ ⎟ 2 PA ⎜⎜ p x ( C a + C rd ) + C rd 3 ⎟⎠ ⎝

Solving Eq.10,11 for Ω (θE) :

Ω= 2

py ⎞ ⎛ 2 PA ⎟ sin θ E ⎜⎜ p x ( C a + C rd ) cos θ E + C rd 2 C (ω s r ) 3 ⎟⎠ ⎝

(13)

called the residual torque by [Sonnabend]. The existence of this torque made the return trajectory of Hayabusa possible. 4 American Institute of Aeronautics and Astronautics

Fig.2 - Hayabusa

Fig.3 - Rosetta

The relation {θE, Ω} is qualitatively shown in Fig.4. There is a maximal value of Ω above which no equilibrium exists : Ωmax = PA [px (Ca+Crd) + Crd py ÷2/3] / ( C ωsr2). (14) The corresponding maximal equilibrium angle follows from dΩ/dθE = 0 and equals 45∞ when Crd is zero or neglegible. When Ω is larger as Ωmax, the angular momentum is gyroscopic stabilised and does no longer follow the sun. When Ω is smaller as Ωmax, there are 2 solutions for the equilibrium angle. As the solar panels must be about perpendicular to the sun to avoid loss of power, θE must be small and the practical usable spin rate is much smaller as Ωmax. Fig.4 - Equilibrium angle and spin rate We have assumed ωs > 0, so for Ω > 0, the right-hand side of Eqn.7 is positive and the equilibrium angle is below the orbital plane as the projection on the z-axis is positive. A reversal of Ω changes the sign of θE. C. Equations about the equilibrium A rotation - θE about the y-axis takes the x-axis to the equilibrium direction. In this equilibrium frame, h = [ hx1 hx2 hx3]T with the x-axis on the equilibrium direction. After using :

⎡ hx1 ⎤ ⎡cos θ E ⎢h ⎥ = ⎢ 0 ⎢ y1 ⎥ ⎢ ⎢⎣ hz 1 ⎥⎦ ⎢⎣ − θ E The system (5a-c) becomes :

sin θ E ⎤ ⎡ hx ⎤ 0 ⎥⎥ ⎢⎢h y ⎥⎥ 0 cos θ E ⎥⎦ ⎢⎣ hz ⎥⎦

0 1

(15)

hx' 1 = 0 hz 1 sin θ E h y1 hz' 1 = − sin θ E h'y 1 =

(16)

In this frame, the component on the equilibrium direction is constant . The angular momentum describes circles about this direction with a period Pprec:

Pprec = 2π sin θ E ( true anomaly )

(17)

For an equilibrium angle of 2∞, the precession period is 12.56∞ in true anomaly. For a circular orbit, this also sin 2∞ = .0349 of the orbital period. For an elliptic orbit, the precession period varies around the orbit. The minimal value 5 American Institute of Aeronautics and Astronautics

occurs for a symmetric arc Pprec/2 around perihelion and the maximum value for a symmetric arc around the aphelion. For example, for an eccentricity e = .272727 and a period of 605 days (a = 1.4 AU), the precession period varies between 11 and 35 days. This period is independent of the amplitude or radius of the circle.

D. Application to Hayabusa Although this model neglects the SRP from the HGA and side and top panels of Hayabusa, a comparison with flight data is of interest. The following data of Haybusa's return trajectory to Earth were found in the open literature: [Kawaguchi]7 : Equilibrium angle : ~2 deg, Period : ~ 28 days, spin rate ~ .6 deg/sec [Kitajima]15 : C inertia : 428.3 kgm2 11 [Mimasu&al] : only px can be reconstructed In the many publications and progress reports on Hayabusa, the osculating elements of this scientific satellite in the period 20 Aug-13 Oct were not made available. To complete the numerical application, the orbital period (from the semi-major axis a) , angular momentum (from the parameter or semi-latus rectum p) and the true anomaly (conversion to time) must be known as well as py to complete the calculation of the lever arm.

Fig.5 Sun tracking Hayabusa , [Kawaguchi]7 The only relevant data available for this application are shown in Fig.5. The equilibrium angle is about 2º, so the precession period of about 28 days must be compared to 12.5º in true anomaly. For the conversion to time, the osculating elements in Sep 2006 must be used. Only the fact that the precession period is independent of the amplitude is confirmed.

II.

Conclusion

It is well known that a slowly rotating spacecraft equipped with solar panels on an heliocentric trajectory can track the Sun by exploiting the torque caused by the solar radiation pressure. Investigating the dynamics of this problem in an orbital frame, it is shown that the angular momentum describes a circle with a center that is offset from the Sun by a constant angle (equilibrium angle) when the solar radiation torque, averaged over a spin period, is perpendicular to the Sun and the angular momentum (no spin effect). We show that this angle is also constant in an elliptic orbit. The maximum equilibrium angle is 45º and puts a limit on the allowable spin rate. When the angle reaches this value, the transition to inertial gyroscopic stabilisation takes place. The precession period about the center is constant in true anomaly and independent of the radius of the circle.

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Appendix - Average torque over a spin period In this section we use two axis systems B and I. B is a body fixed principal axes system with origin at the CoM. The x-axis is the nominal spin axis and carries the angular momentum. At t = 0, the Sun is in the xy-plane and makes an angle θ with the x-axis. Frame I coiincides with the body system at t = 0 but the xy-plane is frozen. Frame I does not follow the spin motion. The body fixed system rotates about x with the angular velocity Ω(spin). The rotation angle is α(t) = Ω t . Fig.A1 shows the normals n± to the solar panels when they are in the xy-plane of the B system. When the angle ψ , called dihedral angle, is zero (as for HAYABUSA) the panels are in the yz plane and the normal is parallel to the x-axis. We have (Fig.A1) : ( n± )B = [ cosψ±sinψ 0]T ( p± )B = [ px ±py pz]T ( s )B = [ cosθ sinθ cos α -sinθ sinα ]T

Fig. A1 - Geometry solar panels The instantaneous SRP force , up to a factor -PA/r2 is given by :

f± = n n± + s s

(A1)

1 ⎧ ⎫ n = 2C r ⎨σ cosθ + ( 1 − σ )⎬ cosθ 3 ⎩ ⎭ s = (C a + C r ( 1 − σ ) )cosθ

The torque generated by this force is : t = p + × f + + p − × f − . With the notations above, the torque in the B frame is given in the I frame by applying the rotation matrix :

⎡1 0 R = ⎢⎢0 cα ⎢⎣0 sα

0 ⎤ − sα ⎥⎥ cα ⎥⎦

(A2)

In the result for t in the I frame, all terms in cosα, sin α, (cosα sin α) average to zero while terms in cos2α, sin2 α average to 1/2. (A3) The result is : tav = [ 0 0 tzi]T where tzi= -PA/r2 sinθ cosθ l(θ) and where

l(θ ) = [ l E C rs sin 2ψ + p x ( C a + C rd ) cosψ ] cos θ + l E C rd / 3 l E = − p x sinψ + p y cosψ

(A4)

For a zero dihedral angle ψ = 0 (Hayabusa , Rosetta as flown ):

l(θ ) = p x ( C a + C rd ) cos θ + p y C rd / 3

(A5)

So, the torque is perpendicular to the xy-plane of the I frame which is defined by the direction of the angular momentum and the direction to the sun. When the z-direction of the I frame is written as nâ s the sin θ factor in Eq. A3 is taken into account. As the torque is perpendicular fo h, its modulus is conserved. This is a consequence of the fact that the normals n± are in the xy-plane in the B frame. When the 2 panels rotate opposite about the y-axis, the normals are : ( n± )B = [ cosη 0 ±sinη ]T The average torque is then no longer perpendicular to the angular momentum and has a spin effect (propeller torque) . In general, 4 independent angles for the 2 normals are needed. 7 American Institute of Aeronautics and Astronautics

The torque is averaged over a spin period. In the presence of a small nutation, the motion of the normal vectors is not a pure spin. In the case of a symmetric satellite and zero dihedral angle, they still rotate at the spin rate about h , and do simultaneously a small transverse circular motion with frequency { spin rate x inertia ratioλ, λ= C/A } which is clearly a second order effect.

References 1

J.Wertz , Spacecraft Attitude Determination and Control, Kluwer, 1978, pp129-132, 570-573 2 V. Chobotov, Spacecraft Attitude Dynamics and Control Kroeger Publishing Company, 1991, pp.77-80 3 C. McInnes, Solar Sailing Technology Dynamics and Mission Applications Springer, 1999,pp. 46-54 4 A. Green, Optimal Escape trajectory from a high Erath Orbit bu use of Solar Radiation Pressure thesis, 1977, pp.7-12. 5 J. Van der Ha,L. Liebig, Ulysses Solar Radiation Force Model, OAD Working paper 207,1984 6 D.Sonnabend, Deep Space Passive Sun Tracking, AAS 98-342 7 J.Kawaguchi,K,Shirakawa, A Fuel-free Sun-tracking Control Strategy and the Flight Results in Hayabusa (Muses-C), AAS 07-176 8 Gyroscopic Hybernation Mode Rosetta, Estec Memo, 1997 9 T.Kubo-oka, Solar Radiation Pressure Model for the relay satellite on Selene, Earth Planets Space,51 pp979-986. ,1999 10 .Ping & al.,How Solar radiation acts on RSAT and VSAT with a small evolving tipp-off in Selene, Earth Planets Space,53 pp919-925. ,2001. 11 Kubo-oka "SRP model for the relay satellite SELENE", Earth Planets Space, 51 (1999),pp.979-986 12 Dan Scheeres, Satellite dynamics of small bodies averaged SRP effects, JAS 1999 13

Y.Mimasu &al., Solar Radiation Pressure Model for Attitude Motion of Hayabusa in Return Cruising ,

IAC - 09-C.19.9 M.Terauchi and &, Effect of Thermal Radiation Force for Trajectory during Swing-by,2008-d-60, 26th ISTS, Hamamatsu 2008 15 A.Kitajima and & "Instability of Hayabusa attitude control by using one momentum wheel", 17th JAXA workshop, 2007 14

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