Elsevier

Icarus

Volume 213, Issue 1, May 2011, Pages 345-368
Icarus

Stardust-NExT, Deep Impact, and the accelerating spin of 9P/Tempel 1

https://doi.org/10.1016/j.icarus.2011.01.006Get rights and content

Abstract

The evolution of the spin rate of Comet 9P/Tempel 1 through two perihelion passages (in 2000 and 2005) is determined from 1922 Earth-based observations taken over a period of 13 year as part of a World-Wide observing campaign and from 2888 observations taken over a period of 50 days from the Deep Impact spacecraft. We determine the following sidereal spin rates (periods): 209.023 ± 0.025°/dy (41.335 ± 0.005 h) prior to the 2000 perihelion passage, 210.448 ± 0.016°/dy (41.055 ± 0.003 h) for the interval between the 2000 and 2005 perihelion passages, 211.856 ± 0.030°/dy (40.783 ± 0.006 h) from Deep Impact photometry just prior to the 2005 perihelion passage, and 211.625 ± 0.012°/dy (40.827 ± 0.002 h) in the interval 2006–2010 following the 2005 perihelion passage. The period decreased by 16.8 ± 0.3 min during the 2000 passage and by 13.7 ± 0.2 min during the 2005 passage suggesting a secular decrease in the net torque. The change in spin rate is asymmetric with respect to perihelion with the maximum net torque being applied on approach to perihelion. The Deep Impact data alone show that the spin rate was increasing at a rate of 0.024 ± 0.003°/dy/dy at JD2453530.60510 (i.e., 25.134 dy before impact), which provides independent confirmation of the change seen in the Earth-based observations.

The rotational phase of the nucleus at times before and after each perihelion and at the Deep Impact encounter is estimated based on the Thomas et al. (Thomas et al. [2007]. Icarus 187, 4–15) pole and longitude system. The possibility of a 180° error in the rotational phase is assessed and found to be significant. Analytical and physical modeling of the behavior of the spin rate through of each perihelion is presented and used as a basis to predict the rotational state of the nucleus at the time of the nominal (i.e., prior to February 2010) Stardust-NExT encounter on 2011 February 14 at 20:42.

We find that a net torque in the range of 0.3–2.5 × 107 kg m2 s−2 acts on the nucleus during perihelion passage. The spin rate initially slows down on approach to perihelion and then passes through a minimum. It then accelerates rapidly as it passes through perihelion eventually reaching a maximum post-perihelion. It then decreases to a stable value as the nucleus moves away from the Sun. We find that the pole direction is unlikely to precess by more than ∼1° per perihelion passage. The trend of the period with time and the fact that the modeled peak torque occurs before perihelion are in agreement with published accounts of trends in water production rate and suggests that widespread H2O out-gassing from the surface is largely responsible for the observed spin-up.

Research highlights

► Changes in a comets spin rate are followed through two perihelion passages. ► Chages in spin rate and rotational phase are used to estimate the torques that operate during perihelion passage. ► Analytic models of the observed spin state are used to predict the spin state at a future spacecraft encounter. ► The observed spin changes and modelled torque profile is consistent with trends seen in water production rates.

Introduction

In their assessment of the spin state of Comet 9P/Tempel 1 following the Deep Impact encounter A’Hearn et al. (2005) found a sidereal period of 1.701 ± 0.014 dy (40.82 h; 211.6 ± 1.7°/dy). Thomas et al. (2007) determined the sense of spin to be direct and the preliminary pole position was RA = 5°, Dec = +78° (±10° on the sky) roughly 11° from one of the two, photometrically degenerate, i.e. photometrically, but not geometrically, equivalent, directions (46°, +73°) found earlier by Belton et al. (2005a) using ground-based photometry. A’Hearn et al. (2005) also noted that the pre-impact rotation period of 1.744 ± 0.006 dy (41.86 h; 206.42°/dy) determined by Belton et al. (2005a) differed significantly from the Deep Impact value and suggested that the difference might be explained by an inadvertent shift in the analysis of Earth-based data by a half or whole cycle between observing runs. Later, Belton et al. (2006) improved the Deep Impact sidereal period estimate to 1.6976 ± 0.0096 dy (40.74 h; 212.06 ± 1.2°/dy) and Thomas et al. (2007) revised the pole position (J2000) to RA = 294°, Dec = 73° (±5° on the sky). Thomas et al. also set the prime meridian as W(t) = 252.63° + 212.064°d, where d is the number of days since the standard epoch (JD 2451545.0). The chosen prime meridian passed through a 350 m, unnamed, crater just west of the impact site (Thomas et al., 2007). W(t) is the angle between the chosen prime meridian and the intersection of the body equator and the standard Earth equator and defines the rotational phase of the nucleus at time t. This latter formula assumes a constant rotation period of 1.6976 dy between the time of impact (JD 2453555.73928) and the standard epoch.

The small difference between the pre-impact rotation rate and the spacecraft value would have been of little concern had it not been for the recognition by J. Veverka and his colleagues that the Stardust spacecraft, which had recently encountered Comet 81P/Wild 2 (Brownlee et al., 2004), and was hibernating in deep space, could be revived, and had enough propulsion capability to reach 9P/Tempel 1 for an encounter on 2011 February 14. This mission, now called Stardust-NExT, was selected as a Discovery mission of opportunity by NASA (www.astro.cornell.edu/next/Science.htm). A Level 1 science requirement of this mission is to “Image 25% of the surface previously observed in the Deep Impact mission at better than 80 m/pixel” in order to look for changes in the condition of the surface that might have occurred during the perihelion passage (2011 January 12.2) previous to the encounter. A secondary science goal is to image the, as yet unseen, artificial feature formed during the Deep Impact mission that is located at 350°.4W, −29°.1. This was designated as a secondary goal because, even if it were not attained, the part of the surface imaged could, especially if it were previously unseen by Deep Impact, lead to new insights for cometary science. To ensure that these objectives can be met, a high-precision rotational ephemeris and an assessment of its stability is required and it is for this reason that the present study was initiated.

In subsequent preparations for the Stardust-NExT mission it was noted that the Deep Impact rotation rate calculated by Belton et al. (2006) did not correctly phase the light curves obtained some 14 months earlier from the Hubble and Spitzer Space telescopes (Lamy et al., 2007, Lisse et al., 2005). This was the first quantitative indication that Comet 9P’s rotation might be changing as it approached perihelion.

The theoretical basis for short timescale changes in cometary spin has been emphasized by Jewitt, 1997, Jewitt, 2004, Samarasinha et al., 2004 and exploratory calculations of excitation timescales have been carried out by Gutiérrez et al., 2002, Jorda and Gutiérrez, 2002, Gutiérrez and Davidsson, 2007. For a small (effective radius = 3.0 ± 0.1 km; Thomas et al., 2007), underdense (bulk density ∼400 kg m−3; Richardson et al., 2007) nucleus with a water production rate of 6 × 1027 molecules/s at the 2005 epoch (Schleicher et al., 2006) the timescale for substantial changes in the spin state is ∼90 year based on Jewitt’s formulation of spin-up time and his conjecture that the typical dimensionless moment arm for torques is ∼0.05. Thus, from a theoretical point of view it should not be surprising if 9P/Tempel 1 was changing its current period by ∼1% (0.4 h) in a single perihelion pass or if the direction of the rotation pole drifted by a degree or two.

There is also a growing observational base to support the measurable presence of this effect in comets. Drahus and Waniak (2006) have shown through the introduction of a novel photometric time-series analysis technique that the rotation rate of the distant Comet C/2001 K5 (LINEAR) was perceptibly spinning-down as it receded from perihelion passage. In addition, earlier studies have found evidence of possible changes in spin rate in Comets 10P/Tempel 2 and 6P/d’Arrest (Mueller and Ferrin, 1996, Gutiérrez et al., 2003). Other evidence of the action of rotational torques includes the cases of Comets 1P/Halley, 2P/Encke and 29P/Schwassmann-Wachmann 1 each of which has been found to be in rotationally excited states (Belton et al., 1991 and references therein; Samarasinha and A’Hearn, 1991, Meech et al., 1993, Belton et al., 2005b).

Deep Impact photometry and imaging data from the ongoing worldwide Earth-based campaign on 9P/Tempel 1 Meech et al., 2005, Meech et al., 2011) plus an early Hubble Space Telescope study by Lamy et al. (2001) provide an unprecedented set of data with which to investigate the stability of the spin state of 9P/Tempel 1. The data that we use from ground-based and HST sources are described in Section 2 where we separate them into three groups: Region A (1997–1999), Region B (2001–2004) and Region C (2006–2010). This grouping allows us to document the changes that occurred during the 2000 and 2005 perihelion passages. In Section 3 we present the Deep Impact approach photometry that we use to obtain direct evidence for an acceleration of the spin rate. In Section 4 we provide the theoretical basis and assumptions used in the analysis of the data. In Section 5 we outline the rotational analysis and present the basic results on the spin rate of the nucleus and its rotational phase. In Section 6 we discuss the dynamical evolution of the comet’s spin state and construct analytical models for its changes through perihelion passage. In Section 7, we provide a general discussion of the relationship of our results with previously published studies of the comet’s H2O production rate. We also use our results to predict the rotation state we expect will be experienced by the Stardust-NExT mission at its encounter with 9P/Tempel 1 on 14 February, 2011. Section 8 contains a summary of our primary conclusions.

Section snippets

Earth-based observations and two independent methods of analysis

In an accompanying paper, Meech et al. (2011, in press) provide a detailed description of the Deep Impact and Stardust-NExT international observing campaign, its goals, participant contributions, observations and results. In Fig. 1 we show R(1, 1, α) magnitudes for the entire data set after reduction to unit heliocentric and geocentric distance and where α is the solar phase angle. The rise and fall of the coma brightness around perihelion dominates the figure and the substantial effect of

Deep Impact approach photometry

As noted in the introduction a number of estimates have been made of the rotational period of the nucleus based on the Deep Impact approach photometry. However, all of these were based on an early form of the photometry that was subsequently found to have short-comings that may have affected the results in the earliest parts of the approach sequence. These problems, which could possibly affect the accuracy of the estimated period, include estimation of the bias correction to the nearest DN

Rotational equations of motion and assumptions

The action of forces generated by the momentum of gas leaving the surface of an active comet nucleus can at any instant of time be decomposed into two parts: those which act at the center of mass of the body, F, and those which apply torques, T. Both forces are functions of time, t, and vary rapidly on a rotational time scale but are thought to vary relatively smoothly on an orbital time scale.

Occasionally major cometary events such as a splitting, or a major outburst, or the appearance or

Rotational analysis

We determine the values of W0j and Sj (Eq. (9)) for each Region j using the two methods described earlier. They share identical data sets, and the combined ground-based and HST data for each region are shown in Fig. 11. The Deep Impact data are already displayed in the bottom panel of Fig. 9. Descriptive information on the data is collected in Table 1.

The dynamical evolution of the spin rate of 9P/Tempel 1

Table 2, Table 3 contain our best estimates of the overall spin state of 9P/Tempel 1 and the changes that occurred during the perihelion passages in 2000 and 2005. Table 3 focuses on spin rates and orientation of the polar axis while Table 2 gives information on rotational phase. The spin rates and acceleration in Table 3 are the average of the values found in the JPL and Tucson studies.

In Fig. 18 we plot the observed spin rates and acceleration as a function of time. The spin rates derived

Discussion and predictions for Stardust-NExT encounter

The observations collected during the Deep Impactand Stardust-NExT international observing campaign cover an interval of thirteen years and two perihelion passages and clearly imply (Fig. 18) a roughly “stepwise” increase of the spin rate of 9P/Tempel 1 as the comet passes through succeeding perihelia. In addition, most of the torque must have been applied well before perihelion. The spin rate and its acceleration measured from the Deep Impact approach photometry imply even greater complexity

Conclusions

In this paper we have provided a detailed analysis of light curve information from the Deep Impact and Stardust-NExT international observing campaign and data obtained from the Hubble and Spitzer Space Telescopes and the Deep Impact mission that were obtained between 1997 and 2009. This analysis shows:

  • 1.

    The spin rate (period) changed in an approximately stepwise manner through the 2000 and 2005 perihelion passages, from 209.023 ± 0.025°/dy (1.7223 ± 0.0002 dy; 41.335 h) prior to 2000, to 210.448

Acknowledgments

We thank J. Giorgini for assistance with the JPL Horizons system and Aron Wolff and the NExT navigation team for making available trajectory information of the Stardust-NExT spacecraft for the 2011 encounter with 9P/Tempel 1. We also thank D.K. Yeomans for insights into the non-gravitational force modeling. This research was performed with the University of Maryland under Contract NNM07AA99C, with Cornell University under agreement 51326-8361, and through a Grant, HST-GO-11998.03-A, awarded to

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