Improved models for near-Earth asteroids (2100) Ra-Shalom, (3103) Eger, (12711) Tukmit & (161989) Cacus

We present 24 new dense lightcurves of the near-Earth asteroids (3103) Eger, (161989) Cacus, (2100) Ra-Shalom and (12711) Tukmit, obtained with the Instituto Astrofísico Canarias 80 and Telescopio Abierto Remoto 2 telescopes at the Teide Observatory (Tenerife, Spain) during 2021 and 2022, in the framework of projects visible NEAs observations survey and NEO Rapid Observation, Characterization and Key Simulations. The shape models and rotation state parameters ( 𝑃 , 𝜆 , 𝛽 ) were computed by applying the lightcurve inversion method to the new data altogether with the archival data. For (3013) Eger and (161989) Cacus, our shape models and rotation state parameters agree with previous works, though they have smaller uncertainties. For (2100) Ra-Shalom, our results also agree with previous studies. Still, we find that a Yarkovsky — O’Keefe — Radzievskii — Paddack acceleration of 𝜐 = ( 0 . 223 ± 0 . 237 ) × 10 − 8 rad d − 2 slightly improves the fit of the lightcurves, suggesting that (2100) Ra-Shalom could be affected by this acceleration. We also present for the first time a shape model for (12711) Tukmit, along with its rotation state parameters ( 𝑃 = 3 . 484900 ± 0 . 000031 hr, 𝜆 = 27 ◦ ± 8 ◦ , 𝛽 = 9 ◦ ± 15 ◦ ).


INTRODUCTION
An asteroid is classified as a near-Earth asteroid (NEA) if it reaches its perihelion at a distance of less than 1.3 Astronomical Units (AU) from the Sun as stated in Center for Near Earth Object Studies (CNEOS) 1 .Therefore, NEAs are the subgroup of minor bodies that come closest to the Earth.According to CNEOS 2 , as of 04/24/2023 there are 31,756 confirmed NEAs, of which 10,398 have a typical size greater than 140 m and 851 are larger than 1 km (the largest confirmed to date is (1036) Ganymed, with a diameter of ∼ 41 km, while the smaller known NEAs, as 2015 TC25, have radii of ∼ 1 m).
Among all the objects in this group, there is a subgroup known as Potentially Hazardous Asteroids (PHAs), which according to CNEOS 13 are those that represent a potential risk of collision with ★ E-mail: rodriguezrjavier@uniovi.es † E-mail: jlicandr@iac.es 1 https://cneos.jpl.nasa.gov/about/neo_groups.html 2 https://cneos.jpl.nasa.gov/stats/totals.html 3 https://cneos.jpl.nasa.gov/glossary/PHA.html the Earth.More specifically, an asteroid is classified as PHA if its orbit has a Minimum Orbit Intersection Distance (MOID) with the Earth of 0.05 AU or less and its absolute magnitude is H < 22, which implies that the object is larger than ∼ 140 m.These objects are fundamental due to their proximity to Earth and the possibility of a collision.By monitoring and studying these asteroids, we can accurately characterize and make them a potential resource source if their composition is rich in any interesting element.From the asteroids presented in this work (161989) Cacus, belongs to this group since its MOID is 0.014085 AU and its H is 17.2 from data of European Space Agency (ESA) Near Earth Objects Coordination Centre (NEOCC) 4 .
To obtain the models, it's widely applied the Convex Inversion Method detailed in Kaasalainen & Torppa (2001); Kaasalainen et al. (2001), which generates a convex model and its corresponding spin state from a suitable set of lightcurves.In the process, both the spin state and the shape are fitted at the same time, searching for the set of parameters (complete spin state and the corresponding shape) that best reproduce the observed lightcurves of the asteroid.The lightcurves can be dense (that is, observations made at high cadence, of the order of minutes, and typically spanning a few hours) or sparse (a few observations per night but typically extending for years).Dense lightcurves are usually the result of specific follow-up programs, such as the Visible NEAs Observations Survey (ViNOS; Licandro et al. (2023)), while sparse lightcurves are usually obtained from surveys that periodically patrol the sky such as the Asteroid Terrestrial-impact Last Alert System (ATLAS; Heinze et al. (2018); Tonry et al. (2018)), the All-Sky Automated Survey for Supernovae (ASAS-SN; Kochanek et al. (2017)) or the Wide Angle Search for Planets (SuperWASP; Parley et al. (2005)) among many others.In the lightcurve inversion process, it's possible to work only with dense data (Torppa et al. 2003;Ďurech et al. 2007), only sparse data (Ďurech et al. 2016, 2019) or a well-balanced combination of both (Ďurech et al. 2009b).However, to obtain reliable results, the lightcurves must be acquired by covering the widest possible range of phase angles, which results in observations corresponding to different geometries that encode information related to the main features of the asteroids.A large number of asteroid models, along with their parameters, lightcurves and many other products, is available at the Database of Asteroid Models from Inversion Techniques (DAMIT 5 ; Ďurech et al. (2010)), operated by The Astronomical Institute of the Charles University (Prague, Czech Republic).
Small asteroids make up the vast majority of the NEA population (97.3% is estimated to have a diameter smaller than 1 km, according to CNEOS 2 ).Two critical mechanisms acting on these small bodies are the Yarkovsky (Yarkovsky 1901;Bottke et al. 2006;Vokrouhlický et al. 2015) and the Yarkovsky-O'Keefe- Radzievskii-Paddack (YORP;Yarkovsky (1901); Radzievskii (1952); Paddack (1969); O' Keefe (1976); Bottke et al. (2006); Vokrouhlický et al. (2015)) effects.The first consists of orbital changes due to thermal reemision of the absorbed solar radiation, increasing the orbit's semi-major axis if the asteroid is a prograde rotator and decreasing it otherwise.It also plays a crucial role in injecting new NEAs from the Main Asteroid Belt (Chesley et al. 2003;Morbidelli & Vokrouhlický 2003).The YORP effect is a constant change in the spin state caused by anisotropic thermal re-emission and the resulting torque.
There are several observations attributed to the YORP effect that are considered as indirect detections.One is the clustering in the directions of the rotation axes among members of the same asteroid family; for example, this clustering has been observed among the Koronis members (Slivan 2002).It is also thought to be responsible of the bimodalities observed in the rotation rates (Pravec et al. 2008) and obliquities (Hanuš et al. 2013b) for small asteroids.Furthermore, it is believed to be a prominent mechanism in the formation of small binaries (Walsh et al. 2008).
In Section 2 of this work, we present new dense lighcurves of the NEAs (2100) Ra-Shalom, (3103) Eger, (12711) Tukmit and (161989) Cacus, acquired at Teide Observatory.In Section 3 we ex-5 https://astro.troja.mff.cuni.cz/projects/damit/plain how these observations have been processed along with archival lightcurves to compute the shape models and rotational state applying the lightcurve inversion method.Results are presented and compared to previous published models in Section 4. Finally, our conclusions are presented in Section 5.
The IAC80 is a 82 cm telescope with  / = 11.3 in the Cassegrain focus.It is equipped with the CAMELOT-2 camera, a back-illuminated e2v 4K x 4K pixels CCD of 15 µm 2 pixels, a plate scale of 0.32 arcsec/pixel, and a field of view of 21.98 x 22.06 arcmin 2 .We used a Sloan  filter.Observations were done using sidereal tracking, so the asteroid's proper motion limited the images' individual exposure time.We selected exposure times such that the asteroid trail was smaller than the typical FWHM of the IAC80 images (∼ 1.0 ′′ ).The images were bias and flat-field corrected in the standard way; there was not needed to correct the dark current since it is almost 0 for these CCD, so correcting the bias is enough.
TAR2 is a 46 cm  / = 2.8 robotic telescope.Until July 2022 TAR2 was equipped with a FLI-Kepler KL400 camera, since then is equipped with a QHY600PRO camera.The FLI-Kepler KL400 camera has a back illuminated 2K x 2K pixels GPixel GSense400 CMOS with a pixel size of 11 µm 2 that in the prime focus of TAR2 has a plate scale of 1.77 arcsec/pixel and a field of view of ∼ 1 deg 2 .The QHY600PRO camera detector is a Sony back illuminated 9K x 6K pixels IMX455 CMOS of 3.76 µm 2 pixels, that in the prime focus of TAR2 has a plate scale of 0.65 arcsec/pixel and a field of view of ∼ 1.6 x 1.1 deg 2 .Both CMOS use a rolling shutter and have the advantage of zero dead-time between images.For a complete description of the QHY600PRO capabilities see Alarcon et al. (2023).The images were bias, dark and flat-field corrected in the standard way.With both cameras we obtained a continuous series of 10 seconds images without filter (Clear) or using a Johnson  filter with the FLI camera and a UV/IR cut L-filter with the QHY with the telescope moving in sidereal tracking.To increase the SNR, consecutive images were aligned and combined to produce a final series of images of larger exposure time.In general, the number of images used to obtain the final combined one is determined by the proper motion of the NEA.This is computed such that the total exposure time is shorter than the Table 1.Observational circumstances of new lightcurves acquired by ViNOS.The table includes the object, telescope and filters used (r-sloan, V, Clear and Luminance), the date and the starting and end time (UT) of the observations, the phase angle (), the heliocentric () and geocentric (Δ) distances and phase angle bisector longitude (PABLon) and latitude (PABLat) of the asteroid at the time of observation.time it takes for the asteroid trail to be equal to the typical FWHM of this telescope (∼ 3.6 ′′ ).
To obtain the lightcurves, we did aperture photometry of the final images using the Photometry Pipeline 8 (PP) software (Mommert 2017), as we did in (Licandro et al. 2023).The images obtained with the L-filter were calibrated to the  SLOAN band using the Pan-STARRS catalogue while the other images were calibrated to the corresponding bands for the filters used.
The new lightcurves are presented in Appendix C along with the synthetic models computed following the method explained in Section 3 (see Figures C1 for (2100) Ra-Shalom, C2 for (3103) Eger, C3 for (161989) Cacus, and Figure 10 in Section 4.3 for (12711) Tukmit).

METHODS
When discussing asteroid characterization, some basic parameters are needed to create the asteroid's model that we further describe next.First of all, the sidereal rotation period (), is the time the asteroid takes to complete a single revolution over its rotation axis and adopt the background stars as the reference frame.It is derived from the asteroid lightcurves applying periodogram-type tools.Lambda () and Beta () are the ecliptic coordinates towards which the spin axis of the asteroid points, being  the ecliptic longitude (0 • <  ≤ 360 • ), and  the ecliptic latitude (−90 • ≤  ≤ 90 • ).With the pole solution (, ) and the asteroid's inclination (i), longitude of ascending node (Ω) and the argument of pericenter (), the obliquity () is then obtained.In the case of 0 • ≤  ≤ 90 • , the asteroid will have a prograde rotation and retrograde otherwise (90 • <  ≤ 180 • ).It is possible to obtain a pole ambiguity for , that is, we could obtain two solutions with almost the same value for , and a pair of values for  that differ ∼ 180 • between each other.
In this work, we used our new lightcurves presented in Section 2, along with available sets of archival lightcurves.All the archival lightcurves were obtained from the DAMIT and Asteroid 8 https://photometrypipeline.readthedocs.io/en/latest/Lightcurve Data Exchange Format (ALCDEF; Stephens & Warner (2018)) databases.In Tables A1, A3, A5 and A6 we summarize the archival lightcurves used for each asteroid.
We applied the lightcurve inversion method to the set of lightcurves for each asteroid with two codes.The first one (No YORP code) was utilized.e.g., in Ďurech et al. (2010) or Hanuš et al. (2011).It generates models with constant  and is publicly available at the DAMIT website.The second code used (YORP code) is a modification of the former, which allows for linear evolution in  over time, thus allowing to detect if the asteroid exhibits the YORP effect.It was gently provided by Josef Ďurech in personal communication; since it is not publicly available, the code was used in previous studies as Ďurech et al. (2012).
For each asteroid we applied the following procedure independently with the No YORP and the YORP codes; Firstly, we obtained a medium resolution solution searching for  and  values in all the sphere (0 • <  ≤ 360 • , −90 • ≤  ≤ 90 • ) with 5 • steps and adopting as initial value for  the previously accepted value (except for (12711) Tukmit, for which we used the  found with the period search tool implemented in the DAMIT code).Secondly, we performed a fine pole search with 2 • steps in a 30 • x 30 • square centered on the previous solution and starting with the  obtained in the previous search.The initial parameters for modelling were set to their default (and recommended) values; in the case of the YORP code, the YORP value was set to  = 1 × 10 −8 .Only the convexity regularization weight was modified in order to maintain the dark facet area below 1% when needed.After running both codes, we reduce the solution's  2 given by the code, to the number of measurements for each asteroid, obtaining a  2 red value, selecting as a final solution the one with the lowest  2 red value.To obtain the uncertainties of the solution we opted for creating 100 subsets from the main set of measurements that was used to obtain the best-fitting solution in terms of  2 red .To create this subsets, we removed randomly 10% or 25% of the measurements from the initial set depending on its measurement number.We then recalculated the best-fitting solution for each of this new subsets, repeating the fine pole search, thus obtaining 100 solutions.With this 100 solutions, we then calculated the mean (which is almost identical to the best-fitting Furthermore we applied the method proposed in Vokrouhlický et al. (2017) to alternatively obtain the uncertainty in the YORP effect at the 3 level.For that we iterated the YORP code with all parameters, besides the YORP effect, fixed at the initial best-fitting solution values, modifying only the  parameter and finally adopting as the final solution the one corresponding to the lowest  2 red value (see Figure 7 as an example).

RESULTS AND DISCUSSION
We proceed now to show the results obtained following the methods proposed in Section 3 with a discussion for each asteroid (see Table 2 for a summary of the values obtained).
We applied the inversion algorithm to 93 archival lightcurves and our 7 new lightcurves acquired during 2022 (see Tables 1 and A1).First of all we ran the No YORP code since no linear evolution of  was previously reported.Figure 1 shows the shape model obtained with this code, corresponding to a pole solution  = 278 • ,  = −60 • ,  ≃ 164 • and a rotation period of  = 19.820056hr.The fit between the model and the data results in  2 red = 1.66 normalized to the 4987 data points (See Figure B1).
Next we performed the inversion with the YORP code, obtaining the shape model presented in Figure 2  and the data was slightly better, resulting in  2 red = 1.64 normalized to the 4987 data points (See Figure B2).In Figure 3 we show the fits between the constant period (No YORP) and linearly increasing period (YORP) models for Ra-Shalom and the data corresponding to several seasons of observations.
The photometric data set is large (∼ 5000 measurements), so as explained before, we estimated the mean final values of the rotation state parameters (, , ) with their uncertainties repeating the modelling around the best solution with 100 subsets, removing 25% of the points in each subset. .We also estimated the uncertainty of the YORP effect in the event it is present at the 3 level iterating the YORP code with all parameters, besides the YORP effect, fixed in the previous best solution.In this particular case, we decided to run it from 0 to 0.5 × 10 −8 in 0.02 × 10 −8 steps, in accordance with the low  value derived from the computed model.With this method, we obtain  = (0.29 ± 0.05) × 10 −8 rad d −2 (see Figure 4). .
Following Rozitis & Green (2013), it is possible to estimate the expected YORP acceleration acting on a NEA from a statistical approach knowing its diameter (in km), semi-major axis (in AU) and eccentricity computing Adopting for Ra-Shalom a mean diameter of D=1.76 km from NEO-WISE data (Masiero et al. 2021), a semi-major axis  = 0.8321 and eccentricity  = 0.4365, we obtain an estimated value for the YORP acceleration of  = 4.7 +6.5 −3.3 × 10 −8 rad d −2 , one order of magnitude greater that the estimated value from the linearly increasing period code.If we use the diameter estimated from radar physical models (Shepard et al. 2008a) of D=2.9 km, we obtain an estimate of  = 1.7 +2.4  −1.2 × 10 −8 rad d −2 , which is again one order of magnitude greater than our obtained value.Obviously, more observations are necessary to confirm or discard our preliminary result.Anyway, for our estimated value of  it is worth computing the characteristic timescale  yorp = /, which is the time needed to change the rotation rate of the asteroid significantly.We find that Ra-Shalom may decrease its rotation period to one-half (∼ 10 hr) in about 400  Myr.As this rotation rate is well above the breakup limit, (2100) Ra-Shalom should not experience structural changes in the next 500 Myr due to this effect.
Both linear increasing period and constant period models are a good fit with the data, being slightly better considering an acceleration of the period.It is believed that the YORP effect is responsible of the bimodality in the rotation periods observed in small asteroids, showing greater populations of fast and slow rotators (Pravec & Harris 2000).Interestingly, all asteroids with reported YORP effect to date show acceleration, which could be a bias since they all have fast rotation periods and are therefore easier to study.However, Ra-Shalom is a case of interest because it has a considerably slower rotation period (∼19 hours).Yet, the data suggests an acceleration instead of deceleration, being deceleration a result that would not be unusual given its slow rotation rate.This could also suggest that the YORP effect is more efficient at accelerating than decelerating (Statler et al. 2013).Another hint of the presence of this effect on Ra-Shalom is the value of the ecliptic latitude for its spin pole; we know that another consequence of this effect is to bring the rotation axis to extreme obliquity values (Hanuš et al. 2013a), so a value of  = 165 • suggests that this effect could be taking place.
We computed a model with the YORP code since the effect was already reported.For that, we used 90 lightcurves with a temporal span of 36 years (1986 -2022), finding as best solution:  = 214 • ,  = −71 • ,  ≃ 177 • , rotation period corresponding to July 6 1986 (date of the very first observation in the data set)  = 5.710148 hr, and a YORP acceleration  = 0.847 × 10 −8 rad d −2 .The fit between model and data corresponds to a value of  2 red = 1.74 normalized to the 6034 data points (see Figures 6 and B3).In Figure 5 we show the shape model of (3103) Eger.
We recomputed the model around the best solution with 100 subsets, each removing 25% of the points (∼ 6000 measurements).We obtained the following final values:  = 5.710148 ± 0.000006 hr, We employed also the 3 method to obtain a second estimation of the uncertainty of , iterating the  value from 0 to 3 × 10 −8 in 0.05 × 10 −8 steps, and maintaining the rest of the values fixed at the best solution values (see Figure 7).In this way we obtained  = (0.85 ± 0.08) × 10 −8 rad d −2 , which is in agreement with the previous computed value.
We also computed a shape model with constant period obtaining the following values:  = 218 • ,  = −71 • ,  ≃ 178 • , rotation period  = 5.710136 hr with  2 red = 2.95 (Figure 8 shows the fit of both models to some example lightcurves).The  2 red value is higher than the linearly increasing period shape model solution ( 2 red = 1.74) previously obtained, thus we conclude that our linearly increasing period model for (3103) Eger confirms and refines the previous values for its spin parameters and their uncertainties.
For (3103) Eger we estimated a value  yorp = / of ∼ 8 Myr, time it would take the asteroid to decrease its rotation period to ∼ 2.8 hr, close to the critical rotation period of ∼ 2 hr, meaning that significant structural changes could take place within this typical time scale.

(12711) Tukmit
Previous studies of this NEA only measured its rotation period, obtaining  = 3.4848 ± 0.0001 hr in Warner & Stephens (2022) and Pravec (2000web) 9 .With our three new dense lightcurves (see Table 1), and two archival lighcurves from ALCDEF (see Table A5), we derived the first spin and shape model for (12711) Tukmit.Due to the short temporal window of the observations (less than one year), we computed a constant period model, obtaining a period of  = 3.484895 hr with a pole orientation  = 27 • ,  = 11 • and  ≃ 118 • .In Figure 9 we show the shape model for this solution.The fit between model and data has in this case  2 red = 1.06 (see Figures 10 and B4).
To estimate the mean values and their uncertainties, since the main data set for Tukmit is smaller compared to the others (∼ 150 measurements), we decided to remove 10% of the main data to obtain each subset instead of 25%.We obtained  = 3.484900 ± 0.000031 hr  = 27 Since the time span of the observations is so small (∼ 1 year) it is extremely unlikely that we would detect the YORP effect, if it were present, unless being extremely strong.Anyway, we computed a linear increasing period model, but as expected, the obtained bestfitting model was unsuccessful to improve the constant  model.We note that the aforementioned obliquity expected in a YORP affected asteroid is not present in the best-fitting model obtained ( ≃ 118 • )).Anyway, according to Rozitis & Green (2013) we could expect a YORP acceleration of  = 1.8 +2.5 −1.3 × 10 −8 rad d −2 , assuming D=1.94 km (Trilling et al. 2010), a=1.1863AU and e= 0.2721.If so, the value  yorp = / would be ∼ 8 Myr, time at which the asteroid would reach a rotation period of ∼ 1.7 hr, well beyond the critical rotation limit.More observations are needed to confirm and refine our results for (12711) Tukmit.
We added to those previous observations our three new lightcurves acquired during 2022 (see Table 1), increasing to 44 years the temporal window of the observations.We computed a linearly increasing period model since the YORP effect has been previously reported for (161989) Cacus.The best-fitting solution corresponds to a pole orientation of  = 251 • ,  = −61 • ,  ≃ 178 • ,  = 3.755067 hr (corresponding to February 28 1978) and a YORP acceleration  = 1.91 × 10 −8 rad d −2 .The fit between the model and data corresponds to a value of  2 red = 1.31 normalized to the 1534 data points.In Figure 11 we show the associated shape model (see Figure 12 for a graphical representation of the fit).
To obtain the final mean values and their uncertainties for each parameter of the model, we recomputed the model for 100 subsets obtained removing randomly 25% of the data from the main set (in this case the number of measurements is large enough ∼ 1500 measurements).We obtained  = 3.755067 ± 0.000001 hr,  = 251 We also used the 3 method to estimate the uncertainty of the YORP effect, iterating in this case the  value between 0 and 3×10 −8 with 0.01 × 10 −8 steps.In this way we find  = (1.92 ± 0.08) × 10 −8 rad d −2 (see Figures 13 and B5), in good agreement with the bestfitting model.
As for (3103) Eger, we also computed a shape model with constant period, obtaining the following values:  = 245 • ,  = −61 • ,  ≃ 176 • rotation period  = 3.755052 hr and  2 red = 13.65 (Figure 14 shows the fit of both models to some example lightcurves).The  2 red value is much higher than the linearly increasing period shape model ( 2 red = 1.31) previously obtained, thus, we conclude that our results for (161989) Cacus confirm previous works and significantly decrease the uncertainty of the  value.
We also estimate  yorp = / ∼ 8.2 Myr, time scale at which the asteroid would reach a rotation period of ∼ 1.9 hr that is beyond the critical rotation period.

CONCLUSIONS
In this work, we computed models, spin state and shape, including period changes due to YORP for asteroids (2100) Ra-Shalom, (3103) Eger, (12711) Tukmit and ( 161989   the Instituto de Astrofísica de Canarias in the Spanish Observatorio del Teide.
The work has been funded by HUNOSA through the collaboration agreement with reference SV-21-HUNOSA-2.
This work uses data obtained from the Asteroid Lightcurve Data Exchange Format (ALCDEF) database, which is supported by funding from NASA grant 80NSSC18K0851.Table A1.Archival observations for (2100) Ra-Shalom.The information includes the date, the starting and end time (UT) of the observations, the phase angle (), the heliocentric () and geocentric (Δ) distances, phase angle bisector longitude (PABLon) and latitude (PABLat) of the asteroid at the time of observation.References: HAR92: Harris et al. (1992); OST84: Ostro et al. (1984)  Table A3.Archival observations for (3103) Eger.The information includes the date, the starting and end time (UT) of the observations, the phase angle (), the heliocentric () and geocentric (Δ) distances, phase angle bisector longitude (PABLon) and latitude (PABLat) of the asteroid at the time of observation.References: WIS87: Wisniewski (1987); VEL92: Velichko et al. (1992)

Figure 3 .
Figure 3. Fits between sets of lightcurves of (2100) Ra-Shalom corresponding to the 1997, 2000, 2003 & 2016 seasons and the best-fitting models.Dashed blue: best constant period model (C Model).Solid black: best linearly increasing period model (L Model).Data for each observation represented by the colour and shapes shown in each legend.

Figure 6 .
Figure 6.Four examples of the fit between dense lightcurves of (3103) Eger and the best-fitting linearly increasing period model (L Model).The data is plotted as red dots for each observation, meanwhile the model is plotted as a solid black line.The geometry is described by its solar phase angle .

Figure 10 .
Figure10.Fits between all the lightcurves of (12711) Tukmit with the best-fitting constant period model (C Model).The data is plotted as red dots for each observation, meanwhile the model is plotted as a solid blue line.The geometry is described by its solar phase angle .

Figure 12 .
Figure12.Fits between five lightcurves of (161989) Cacus and the best-fitting linearly increasing period model (L Model).The data is plotted as red dots for each observation, meanwhile the model is plotted as a solid black line.The geometry is described by its solar phase angle .

Figure 14 .
Figure 14.Example of lightcurves showing the offset of the fit of constant period model (C Model) to both the linearly increasing period model (L model) and the data for (161989) Cacus.The data is plotted as red dots for each observation, meanwhile the C Model is plotted as a solid black line and the L Model as a solid blue line.The geometry is described by its solar phase angle

Figure B1 .
Figure B1.Statistical quality of (2100) Ra-Shalom pole solutions obtained with the constant period code.The solutions are shaded by its  2 red value.The best solution obtained is shown as a white square ( = 278 • ,  = −60 • ) with  2 red = 1.66 (normalized by the 4987 data points).The solutions within a margin of 5.7% (3) are highlighted with a red border.

Figure B2 .
Figure B2.Statistical quality of (2100) Ra-Shalom pole solutions obtained with the linear increasing period code.The solutions are shaded by its  2 red value.The best solution obtained is shown as a white square ( = 283 • ,  = −62 • ) with  2 red = 1.64 (normalized by the 4987 data points).The solutions within a margin of 5.7% (3) are highlighted with a red border.

Figure B3 .
Figure B3.Statistical quality of (3103) Eger pole solutions obtained with the linear increasing period code.The solutions are shaded by its  2 red value.The best solution obtained is shown as a white square ( = 214 • ,  = −71 • ) with a  2 red = 1.74 (normalized by the 6034 data points), the solutions within a margin of 5.2% (3) are highlighted with a red border.

Figure B4 .
Figure B4.Statistical quality of (12711) Tukmit pole solutions obtained with the constant period code.The solutions are shaded by its  2 red value.The best solution obtained is shown as a white square ( = 27 • ,  = 11 • ) with a  2 red = 1.06 (normalized by the 263 data points), the solutions within a margin of 25% (3) are highlighted with a red border.

Figure B5 .
Figure B5.Statistical quality of (161989) Cacus pole solutions obtained with the linear increasing period code.The solutions are shaded by its  2 red value.The best solution obtained is shown as a white square ( = 251 • ,  = −61 • ) with a  2 red = 1.31 (normalized by the 1534 data points), the solutions within a margin of 10% (3) are highlighted with a red border.

Figure C1 .
Figure C1.Fit between lightcurves from (2100) Ra-Shalom presented in this paper and the best-fitting linearly increasing period model (L Model).The data is plotted as dots for each observation, meanwhile the model is plotted as a solid black line.The geometry is described its solar phase angle .

Figure C2 .
Figure C2.Fit between lightcurves from (3103) Eger presented in this paper and the best-fitting linearly increasing period model (L Model).The data is plotted as red dots for each observation, meanwhile the model is plotted as a solid black line.The geometry is described its solar phase angle .

Figure C3 .
Figure C3.Fit between lightcurves from (161989) Cacus presented in this paper and the best-fitting linearly increasing period model (L Model).The data is plotted as red dots for each observation, meanwhile the model is plotted as a solid black line.The geometry is described its solar phase angle .

Table 2 .
Results obtained in this work for each asteroid, we show type of model (linearly increasing period (L) and constant period (C)), rotation period, geocentric ecliptic coordinates of the spin pole (, ), obliquity ( ) and YORP acceleration () if the model has linearly increasing period (L).

Table A4 .
Continuation of table A3

Table A5 .
Archival observations for (12711) Tukmit.The information includes the date, the starting and end time (UT) of the observations, the phase angle (), the heliocentric () and geocentric (Δ) distances, phase angle bisector longitude (PABLon) and latitude (PABLat) of the asteroid at the time of observation.

Table A6 .
Archival observations for (161989) Cacus.The information includes the date, the starting and end time (UT) of the observations, the phase angle (), the heliocentric () and geocentric (Δ) distances, phase angle bisector longitude (PABLon) and latitude (PABLat) of the asteroid at the time of observation.