Astron. Astrophys. 362, 746-755 (2000)

3. Examples

We illustrate the theory of the orbit perturbations due to the radiation effects, as formulated in the previous sections, with two examples, namely two near-Earth asteroids: 6489 Golevka and 1566 Icarus. These objects have been found potentially interesting targets for detection of the Yarkovsky effect (Vokrouhlický et al. 2000). The orbital eccentricity is large in both cases, 0.599 and 0.827, which suggests the possibility of large long-term orbital changes according to the formulas given in this paper, especially (24).

The technique used in this text is fairly similar to that in Vokrouhlický et al. (2000), but we add a preliminary step in the analysis to obtain a first information about the orbit displacement produced by presence of the radiation forces in the dynamical model. At this stage we disregard planetary and more subtle (e.g., relativistic) perturbations of the asteroid orbit and consider the solar gravitational influence and the radiation forces only ("perturbed two-body problem").

Given the initial orbital elements at an epoch corresponding to the weighted mean of the available observations, we numerically integrate the asteroid orbit with the radiation forces (7), (19) and/or (23) included in the model. We readjust the initial orbital elements to match the integrated orbit by a Keplerian ellipse in the sense of the least squares technique. Residuals after this fitting procedure give an information about the "true" order of magnitude of the radiation-effect perturbations (keeping the initial elements unchanged the residuals would be polluted by unobservable free-Keplerian terms).

Since the radar ranging yields the most precise observations, we project the orbit perturbation onto the geocentric line-of-sight direction of the asteroid. We thus obtain the perturbation of the geocentric distance and of the corresponding rate . The first is related to the radar delay measurement and the second to the Doppler shift between the transmitted and received signal. At this stage of our procedure we also check validity of the formulae (24) - (26) and (27) - (29) for the long-term effects in the Keplerian elements.

At the second, and a more precise, stage we use the OrbFit program that allows the orbit determination from the optical and radar astrometry data. An information about this software, and a free download, may be obtained from http://newton.dm.unipi.it/~asteroid/orbfit/ . We proceed in exactly the same way as in Vokrouhlický et al. (2000); a complete dynamical model to the post-Newtonian level is used for the orbit determination from the available data. For the given asteroid, the initial state vector and the covariance matrix constructed at the weighted mean of the observations is propagated to the next close approach to the Earth (when the radar observation might be taken). The confidence region, as determined by the fit to the current data, is projected onto the space of the radar observables, notably distance from the Earth and rate of change of this quantity (with aberration effects and other small corrections included). This projection is performed with the algorithms discussed in Milani (1999) and implemented in OrbFit both for optical astrometry and radar astrometry. Such projection is constructed for two models: (i) a "nominal" model, not including the solar radiation perturbation, and (ii) an "extended" model, including the solar radiation perturbation. Position of the confidence regions predicted by the two models is compared. When no overlapping at the level is observed, we can conclude that the radiation effects might be detected during the next close approach of the asteroid. If the two confidence regions partially overlap, we can evaluate the probability (less then unity) of this detection.

The results obtained with the less accurate perturbed two body method are consistent, as far as the size of the perturbations is concerned, with the results obtained with the more accurate procedure. This implies that nonlinear coupling of the radiation forces with gravitational perturbations is not important, at least over time spans of the order of tens of years.

3.1. Golevka

Golevka has been observed by radar in June 1991, 1995 and 1999. Unfortunately, the 1999 data cannot be used as astrometric data; thus the 91-95 baseline is rather short to detect subtle non-gravitational phenomena in Golevka's orbit. Nevertheless, the next close approach to the Earth occurs on May 20, 2003. Vokrouhlický et al. (2000) considered the possibility to detect the Yarkovsky perturbation using the radar data which we presume will be taken at this approach. Here we are going to investigate whether these data could reveal existence of the direct solar radiation pressure perturbations on the orbit of Golevka.

We consider the physical parameters of Golevka as derived by Hudson et al. (2000): notably surface albedo () of 0.15, mean radius of 265 meters and spin axis orientation with ecliptic longitude and latitude and . These values superseed the previous model of Golevka by Mottola et al. (1997) and is consistent with indications Zaitsev et al. (1997). The shape model of Golevka, as derived from the radar observations, is very complex and impossible to fit with a ellipsoidal model (to which our theory is limited). We can only obtain an order of magnitude of the non-sphericity effect by adopting , a rather conservative value since the estimate of the longest to shortest geometric axes of Golevka is about 1.4 (Hudson et al. 2000).

In the first step, we use the perturbed two-body formulation discussed above. Fig. 4 shows the orbit perturbation for Golevka projected onto the geocentric line-of-sight for the effect of variable albedo (acceleration from (7); note that the absorbed radiation pressure effect - the first term in (7) - is also included). We have assumed and , which means a 2% difference of the albedo values between the southern and the northern hemisphere. Such a small albedo variation cannot be measured from the Golevka lightcurve data. In fact, Magnusson (1991) indicates that smaller asteroids show in average larger variation of the surface albedo. Thus the value adopted in this text seems to be a conservative estimate. Nevertheless, the dynamical effect is rather large - up to 150 meters during the 1995 close approach. Moreover, the effect accumulates with time so that the perturbation with still increase in the future. Notice the rapid change in the sign of the range perturbation during the close approaches (especially in 1995). A typical time scale of this change is  days. During the 1995 closest approach time the perturbation is close to zero. Since the observations cover only about 12 days around the close approach (and only 10 days in 1991), the maxima of the perturbation in range might not be covered by the observations. In 2003, the close approach perturbation is somewhat smaller, about 75 meters. This value is larger than the uncertainty of the observations (of the order of 40 meters), but smaller than the orbit determination uncertainty from the current data (about 2.9 km).

Fig. 4. Simulated orbit displacement along the line-of-sight from the Earth for the asteroid 6489 Golevka vs time between 1988 and 2006. An odd-symmetry zonal term of the Golevka's albedo assumed (i.e. , in the text). Radius ( meters) and the spin axis orientation from Hudson et al. (2000). The four close approaches to the Earth denoted by shaded strips.

The results of the simpler model are confirmed by the analysis of the complete solution shown in Fig. 5. The projection of the uncertainty ellipsoid onto the plane of the radar observables, range R and range-rate , is shown for two models: (i) the nominal model that do not include the radiation effects (dashed lines), and (ii) the extended model that includes the acceleration (solid lines). The axes origin is always referred to the nominal-model solution. The same parameters of the albedo anisotropy as above. The thicker lines indexed 0 correspond to the time of the closest approach of the nominal orbit (20 May, 2003). Similar confidence boundaries for five and ten days before and after this data are shown by thinner lines, labeled and . The confidence levels are computed from the least squares fit to the currently available astrometric data (both optical and radar). Note that the mean displacement of about hundred meters is in a good agreement with the previous simpler analysis. However, the rather large orbit uncertainty prevents detection of the radiation effect: the uncertainty regions overlap to a large extend. We have checked that the results are not much sensitive on the degree k of the albedo distribution, provided k is not too large.

Fig. 5. Projection of the confidence ellipses of the Golevka orbit uncertainty onto the space of radar observables: the geocentric distance R (in km) and the rate-of-change of the geocentric distance (in km/day). Results of the nominal model (without the radiation effects) given by the dashed lines, while results of the extended model (including the radiation effects) given by the solid lines. Origin of axes referred to the corresponding values of the nominal model. Data at the nearest future close approach of Golevka (20 May, 2003) are given by thick lines. Similar results 5 and 10 days before and after the close approach of the nominal orbit are shown by the lighter curves with labels and . The extended model is obtained by adding an odd-symmetry zonal term in the albedo of Golevka (i.e. , in the text). Radius ( meters) and spin axis orientation from Hudson et al. (2000).

Fig. 6 shows the result of the perturbed two-body approach for the radiation pressure on a flattened Golevka ( acceleration from (19)). A constant albedo of 0.15 is assumed, and the flattening parameter as discussed above. The effect is smaller, but comparable, to the perturbation due to the radiation pressure on a spherical body with variable albedo (above). In Fig. 6 we have assumed that the Golevka spin axis is fixed in space. However, we have verified that free precession with a cone aperture up to 15 degrees does not change our conclusions. We do not report the PR effect perturbation of the Golevka orbit, since it is quite small (smaller than 5 meters in range variation).

Fig. 6. Simulated orbit displacement along the line-of-sight from the Earth for the asteroid 6489 Golevka vs time between 1988 and 2006. Golevka is approximated with a spheroid with oblateness parameter and polar radius  meters. A constant surface albedo is assumed, and the spin axis orientation from Hudson et al. (2000). The four close approaches to the Earth are denoted by shaded strips.

In general, we can conclude that though larger than the observation uncertainty, the radiation effects could hardly be detectable from the radar data taken during the next close approach. The main reason is a too short observed time span (the 1988-2003 interval covers little more than 4 revolutions of the asteroid around the Sun). Thus the effect of the long periodic perturbations in a and e thus cannot accumulate to large orbit displacements. In both cases reported above, the short periodic effect due to the elementary radiation pressure (purely radial force) contributes largely to the perturbation.

3.2. Icarus

Icarus is the first asteroid observed by the radar technique (June 1968). It has been also observed at the next close approach to the Earth in June 1996 and returns back in June 2015. These dates define a suitably long time span over which we have a good quality orbital data (despite the fact that all radar data available so far are Doppler measurements only). Moreover, Icarus' high eccentricity (0.827) results in high rates of long term drifts in the element, especially a, as it is clear from (24). For that reason Vokrouhlický et al. (2000) considered a possibility to detect the Yarkovsky effect in the motion of Icarus with the 2015 data. Here, we complement their analysis by the investigation of other radiation effects acting on the same orbit.

As for the physical data about Icarus we refer to the work of Veeder et al. (1989), De Angelis (1995) and Mahapatra et al. (1999). Veeder et al. give a radius of about 450 meters with a surface albedo of 0.4 (these values are used in this paper). Harris (1998) estimates a little larger radius ( meters) corresponding to a somewhat lower albedo, but reanalysis of the 1996 radar data by Mahapatra et al. (1999) supports Icarus' small size. De Angelis (1995) reports a triaxial shape model with ratios of the semi-axes and . Since we cannot yet model the radiation effect on a triaxial ellipsoid we approximate Icarus' shape by a biaxial ellipsoid with a flattening parameter . The spin axis orientation parameters ( and ) were taken from De Angelis (1995).

We again start our analysis by considering the perturbed two-body problem with the perturbation given by the radiation acceleration from (7). The following parameters of the surface albedo anisotropy are assumed: and . The 2% amplitude of the north/south asymmetry is very conservative and may even underestimate the real albedo variation. Fig. 7 shows the perturbation of the geocentric distance to the asteroid. Contrary to the Golevka example, the perturbation is now much larger and is dominated by the secular effect in the semimajor axis due to the albedo asymmetry (the short-periodic effect of the absorbed radiation pressure is negligible). The 2015 range perturbation may be as large as 26 km, again with a rapid change during a time span of about one month around the closest approach. This perturbation is significantly larger than the expected observation uncertainty (Mahapatra et al. 1999), but little smaller than the current orbit uncertainty propagated to 2015. These facts indicate that the albedo variation effect might be important for precise analysis of the 2015 radar data. We also mention that the range-rate perturbation is smaller than the range perturbation. In both previous close approaches to the Earth (1968 and 1996) the maxima of the range-rate perturbation ( km/day) were either comparable or smaller than the formal uncertainty of the observations ( km/day for the 1968 observations and even 2 km/day for the 1996 observations).

Fig. 7. Simulated orbit displacement along the line-of-sight from the Earth for the asteroid 1566 Icarus vs time between 1966 and 2018. An odd-symmetry zonal term is assumed in the albedo of Icarus (i.e. , in the text). The radius ( meters) and the spin axis orientation are taken from De Angelis (1995). The three close approaches to the Earth are denoted by shaded strips.

The importance of the perturbation due to the Icarus non uniform albedo is confirmed by the detailed analysis using the OrbFit program. Fig. 8 shows the uncertainty ellipsoids projected onto the radar observables for the 2015 close approach of Icarus (dates before and after the close approach are also considered as before in the Golevka's case). Comparison of the nominal model (no radiation effects) and the extended model (including the perturbing acceleration with the albedo asymmetry parameters as before) shows partial separation of the uncertainty ellipses. Though even in this case the radiation effect will not be possibly "detected" during the next close approach in 2015, it may potentially produce important orbit perturbation on a long-term because new radar data will potentially shrink the orbit determination uncertainty. We also note that the observed mean separation of the confidence intervals of the two models also confirms results of the simplified approach from the Fig. 7.

Fig. 8. Projection of the confidence ellipses of the Icarus orbit uncertainty onto the space of radar observables: the geocentric distance R (in km) and the range-rate (in km/day). Results of the nominal model (without the radiation effects) given by dashed lines, while results of the extended model (including the radiation effects) are given by solid lines. The origin of the axes refers to the corresponding values of the nominal model. The results for the nearest future close approach of Icarus (16 June, 2015) are shown by thick lines. Similar results for 5 and 10 days before and after the close approach of the nominal orbit are shown by the lighter curves with labels and . An odd-symmetry zonal term is assumed for the Icarus albedo (i.e. , in the text). The radius ( meters) and the spin axis orientation are from De Angelis (1995).

Secondly, we consider the effect of Icarus' nonsphericity for the resulting radiation pressure - the acceleration (19). The flattening parameter are noted above. Fig. 9 shows the geocentric range perturbation as results from the perturbed two-body analysis. The effect is very small, if compared to the non uniform albedo case studied above.

Fig. 9. Simulated orbit displacement along the line-of-sight from the Earth for the asteroid 1566 Icarus vs time between 1966 and 2018. Icarus is approximated with a spheroid with the oblateness parameter and the polar radius  meters. A constant surface albedo taken into account and the spin axis orientation from De Angelis (1995). The three close approaches to the Earth denoted by shaded strips.

As a final example we consider the Icarus' orbit perturbation due to the Poynting-Robertson effect. Fig. 10 shows the geocentric range perturbation within the perturbed two-body problem. Though smaller than in the Fig. 7, the orbit displacement is still of an appreciable order of magnitude ( km during the 2015 close approach). Surprisingly thus, the PR effect must be taken into account for the orbit analysis of Icarus including and beyond the 2015 approach, at least for consistency.

Fig. 10. Simulated orbit displacement along the line-of-sight from the Earth for the asteroid 1566 Icarus vs time between 1966 and 2018 due to the Poynting-Robertson effect. A radius  meters considered. The three close approaches to the Earth are denoted by shaded strips.

© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
helpdesk.link@springer.de