ICARUS 131, 167–170 (1998)ARTICLE NO. IS975867

Kuiper Belt Constraint from Pioneer 10

John D. Anderson and Eunice L. Lau

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

Klaus Scherer

Max Planck Institut fur Aeronomie, P.O. Box 20, D-37189, Katlenburg-Lindau, Germany

and

Doris C. Rosenbaum and Vigdor L. Teplitz

Physics Department, Southern Methodist University, Dallas, Texas 75275E-mail: teplitz@phyvms.physics.smu.edu

Received June 5, 1997; revised November 6, 1997

on the order of the age of the Solar System or longer;hence KB bodies may well be unprocessed planetesimalsWe extract information from the failure of small objects toor fragments thereof.cause detectable damage to Pioneer 10, which has been inside

the Kuiper Belt for a decade. Belt objects too small for telescope The Pioneer 10 spacecraft is special among human-madedetection and too large for IR emission visibility are addressed. objects in that it has since 1986 been in the likely regionThis is a size range with few other potential techniques, short of the belt past 35 AU and at 28 from the plane of theof a new space mission, for direct detection. Results, based on ecliptic. The object here is to consider what can be learnedan 8-inch radius propellant tank, are bounds of about 1/10th from the absence of failures of selected systems in thisof an Earth mass on low-mass, low-density objects. Implications period about the abundance of small centimeter-sized ob-of Poynting–Robertson drag and ISM erosion, and potential

jects in the KB. The largest spacecraft component is theimprovements to the bounds, are discussed. 1998 Academic Pressmain antenna; however, it was wisely designed to continueKey Words: comets; interplanetary dust; planetesimals;to function even if an object passes through it. A morespacecraft.vulnerable component is the spherical propellant tank,about 21 cm in radius, which contains the hydrazine usedfor both course corrections and maintaining the period of1. INTRODUCTIONthe spin-stabilized spacecraft. Because we have not had

Recently in a beautiful series of observations, objects access to its design specifications, we assume conserva-have been detected in the region postulated (Kuiper 1951, tively, in line with recent experimental data from Yanagi-Fernandez 1980, and Duncan et al. 1988) to be the source of siwa et al. (1996), that an object with a kilojoule of energylow-inclination short-period comets. Objects of diameter incident within the 21-cm radius will cause the tank toover 100 km have clearly been detected (Jewitt and Luu be penetrated. We neglect the extent to which secondary1995); objects of 10 km diameter may have been detected objects from outside the 21-cm radius might penetrate the(Cochran et al. 1995). Investigation of the Kuiper Belt tank. Other components of Pioneer 10 might be included(KB) (see Weissman (1995) and Weissman and Levison in more detailed calculations.(1997) for reviews and extensive references) is of particular Why should Pioneer 10 be affected by passage throughinterest since it gives potential insight into the current the KB when it was designed to pass through the asteroidmodel of planet formation through, roughly, a two-step belt? The total densities in the two belts are roughly similarprocess in which interstellar grains adhere to form kilome- but: (i) First, Pioneer 10 has spent 10 times as long in theter-size ‘‘planetesimals’’ and planetesimals then agglomer- KB as it did in the asteroid belt; (ii) there is observationalate gravitationally to form planets (see, for example, contri- evidence (e.g., Soberman et al. 1974) for the asteroid beltbutions to Levy and Lunine (1993)). In the region that the distribution in radius of particle size leans toward

larger particles; however the distribution in radius in thesufficiently past Neptune, orbits become stable for times

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Copyright 1998 by Academic PressAll rights of reproduction in any form reserved.

168 ANDERSON ET AL.

KB, for the size range of interest of around a fraction of is an incident kilojoule. This is in line with recent experi-mental results of Yanagisawa et al. (1996) assuming thata centimeter, is not known from direct observation (see

Mann et al. (1996) and references therein for recent work the titanium tank is about 10 mil (0.25 mm) thick. Weassume that the results of Backman et al. (1995) and Sternon dust in the asteroid belt); (iii) there is also a factor of

about 2.5 in relative velocity. Note that Tremaine (1990) (1996) from IR limits on Kuiper Belt emissions precludesignificant mass in objects less than 0.5 cm. Densities downhas reviewed the general question of dark (not easily de-

tected) matter in the Solar System. In Section 2 we calcu- to 0.02 g cm23 for this size are capable of tank destruction.Equation (4) gives the upper limit to the mass of KB objectslate an upper limit to the amount of matter in the form of

particles of size greater than r 5 0.5 cm, with density r, such that: (i) radii greater than 0.5 cm; (ii) uniform densityr; (iii) distributed in radius with index c . 4. This equationand with a power law size distribution function, r2c, c .

4. In Section 3, we discuss our results. follows from the knowledge that no objects hit Pioneer’shydrazine tank under the assumption that there is sufficientmass in the objects described that NC such hits were to2. CALCULATIONSbe expected.

We parametrize the density of KB particles with the Equation (4) is of interest in two special cases. For largeradius distribution c, effectively all objects are of the minimum radius (0.5

cm). If we take NC 5 4 (2s effect), Eq. (4) then gives abound of M%/3 or lower for r , 0.4 or M%/10 for r ,n(r) 5 n0 r2c. (1)0.133. If c 5 5, Eq. (4) gives (with NC 5 4) a bound ofM%/3 or lower for r , 0.1 g cm23 and a bound of M%/10Note that n0 in Eq. (1) has dimensions of rc23, but that n0for r , 0.03.does not appear in our result and also that both n0 and c

If the observations of Cochran et al. (1995) are con-may well be different for centimeter-sized objects than forfirmed, a distribution in radius of the form of Eq. (1) withkilometer-sized ones. With VK taken as the volume of thec sufficiently greater than four must change form beforeKB, we relate n0 to the total mass, M12 , in the KB byr approaches the kilometer range. If it did not, there wouldbe too little mass in the form of objects with sizes greater

M12 5 (4/3)fVKn0 r Er2

r1

dr r2c r3. (2)than 10 km to satisfy their preliminary result that the totalmass in such objects exceeds 0.04 M% . It can also be shownthat (c 2 4) must be less than 0.22 to satisfy these twoWe investigate the region c . 4 for which more mass isconstraints simultaneously without any change of form.in the form of small objects than large; this is the regionEven for c so close to four (4.22), Eq.(4) constrains M toin which Pioneer can provide the most information. It isbe less than an Earth mass if r , 0.08. A calculation byconservative from the point of view of estimating total KBStern and Colwell (1997) shows that collisional equilibriummass. The condition relating the three parameters M12 , r,which, as shown by Dohnanyi (1969), implies c 5 7/2,and c to the absence of NC collisions capable of destroyingshould apply in the Kuiper Belt. Our ‘‘observational’’ re-Pioneer issult is consistent with that prediction.

NC . neff vs pT, (3)3. DISCUSSION

where T is time spent in the KB, s is the cross section forWe turn now to the implications of (a) Poynting–destruction, v is Pioneer’s velocity, and neff is the density

Robertson drag (absorption and re-radiation of sunlightof particles energetic enough to destroy Pioneer if incidentwith net loss of angular momentum resulting in the bodywithin cross-sectional area s.spiraling inward toward the Sun) and (b) erosion of KBEquations (1)–(3) give our principal result:objects due to interaction with the interstellar medium(ISM). Micrometer-sized bodies in the Kuiper Belt areblown outward by radiation pressure while bodies in theM12 , 0.2 Sc 2 1

c 2 4D rNC M% . (4)size and density ranges for which Pioneer limits are inter-esting are subject to the drag and spiral inward if they arenot first destroyed by mutual collisions and/or collisionsEquation (4) is based on a constant (or appropriately aver-

aged) density for the KB out to heliocentric distance 65 with interstellar grains (see Flynn 1994, Backman et al.1995, Liou et al. 1996, Stern 1996, Weissman and LevisonAU (the approximate distance to which Pioneer has actu-

ally gone) with total wedge angle 2/5 rad. We take the 1997). Using the form in Lang (1980) for Poynting–Robertson drag, we have for a body of density r and radiusvelocity of Pioneer to be approximately 2.5 AU year21 or

12 km s21. T is 10 years. Our criterion for tank destruction r, after time t,

KUIPER BELT CONSTRAINT FROM PIONEER 10 169

c @ 4 and M%/3 for density r , 0.1 g cm23 when c 5 5R2

0 2 R2 53L((1 2 a)tPR

4f c2 rr, (5) (see Eq. (4)). If the result of Cochran et al. are taken at

face value and if Eq. (1) were valid for the full range1 cm , r , 10 km, then c , 4.22. Finally, centimeter-sizedwhere R0 is the initial heliocentric distance and R the final,

while L( is the solar luminosity and we assume that the particles are removed by both the Poynting–Robertsoneffect and erosion by the intersteller medium so that ouralbedo a is not too close to one. For r p 0.1 g cm23, the

radius r p 0.5 cm, and distance R p 50 AU, Eq. (5) gives bounds apply to particles produced (from collisions of largeKB objects) over the past billion years or since we lasttPR p 109 years.

Objects that are centimeter-sized will also be lost from passed through a giant molecular cloud, whichever ismore recent.the KB from erosion by the ISM, including bombardment

by dust grains when the sun passes through a molecularcloud. ISM induced erosion has been computed by Stern ACKNOWLEDGMENTS(1990), who finds that Oort cloud bodies suffer erosion at

Two of us (D.C.R. and V.L.T.) appreciate extremely helpful conversa-a rate such that, in the age of the Solar System, total masstions with the members of the Pioneer Program Office, M. Wirth and D.loss from an average surface amounts to 360 g cm22. It isLozier, and others with W. J. Dixon, who worked on Pioneer design, and

therefore likely that passage through a molecular cloud D. Kessler and E. Christiansen of the Johnson Space Center Space Debriscleans the Kuiper Belt of dust particles up to sizes greater Office. We have also benefited from conversations with R. Braibanti, G.

Cleghorn, G. Pitman, M. Smith, and A. Stern (who also made a numberthan 1 cm. Thus, our limit in Eq. (4) may be a limit on lowof helpful suggestions on an early draft) as well as helpful suggestionsdensity, centimeter-sized objects produced by collisions infrom an anonymous referee. We thank three potential Solar Systemthe KB in a time as short as the past 108 years. Finally,astronomers, Kendall Bryant, Jennifer Schake, and Melissa Urban of

note that most of the KB objects discovered so far are Loretto High School, Loretto, Tennessee, both for focusing our attentionbelieved to be in mean motion resonances with Neptune on this problem and for assisting with the numerical calculations.and that this resonance structure could affect the R depen-dence of the density distribution; our results can bear only REFERENCESon the space average of the density distribution.

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