THE RESULTS OF PHOTOMETRIC RECORDING OF THE OCCULTATION OF THE STAR HIP 97157 BY ASTEROID (41) DAPHNE WITH THE TELESCOPE OF THE GLOBAL MASTER ROBOTIC NET

The occultation of the star HIP 97157 = HD 186461 = SAO 162945 (V = 6.7, R = 5.8) in Aquila by asteroid (41) Daphne has been predicted to be observed from the Tunka Valley near Irkutsk, Russia on 2013 October 25 approximately at 12:32:25 UT by Steve Preston (President of the IOTA, USA; Preston 2013a).

We had the following information about asteroid (41) Daphne. Its approximate linear diameter was estimated as 210 km, and at the moment of the mentioned stellar occultation the distance of Daphne from the Earth should be equal to 2.50200 AU, so the angular size of this minor planet should be 0116. The magnitude of the asteroid is m_v = 12.3. Therefore in case of the occultation of HIP 97157, a magnitude drop should be about 56. Since the approximate speed of the asteroid's shadow at the moment of the occultation should be 23.6108 km s⁻¹, the maximum duration of the occultation process (taking into account the slope between the direction of the beam of light from the star and the earth's surface) should be about 90.

The most important information about the star HIP 97157 (as applied to the situation of asteroidal occultation) that we had is the value of its angular diameter d: it was estimated as 2 milliarcseconds (mas; Preston 2013b).

The photometric observations of the occultation of the star HIP 97157 by asteroid (41) Daphne were carried out automatically, in robotic mode on 2013 October 25 with the 40 cm telescope of the Global MASTER Robotic Net located at the Tunka Astrophysical Center of the Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, and of Irkutsk State University, which is situated in the Tunka Valley near Irkutsk at the coordinates: longitude = 103°04'026 E, latitude = +51°48'36'', Altitude = 680 m (Figure 1).

Zoom In Zoom Out Reset image size

The first Russian robotic telescope MASTER (http://observ.pereplet.ru.) came into operation in 2002 near Moscow (Lipunov et al. (2004)), with the help of private funding from the Moscow association on Optics (http://www.ochkarik.ru/master/). Construction of the all-Russia network MASTER began in 2008 (Lipunov et al. 2010). At present, the telescopes of the MASTER-Net are located at the observatories of Lomonosov Moscow State University (near Kislovodsk), Ural State University (in Kourovka near Ekaterinburg), Irkutsk State University (in the Tunka Valley near Baikal Lake), and Blagoveshchensk Pedagogical University (in Amursk region). These observatories span six time zones. A description of the MASTER II telescopes can be found in Kornilov et al. (2012) and Gorbovskoy et al. (2013).

The 40 cm telescope in Tunka is equipped with the CCD camera Alta U16M. In addition, we have used two wide field cameras, each with a field of view of 384 sq. degrees and an aperture of 72 mm, to monitor the occultation process in real time.

When preparing the observation of this occultation we had to solve the problem of choosing the most appropriate method of photometric recording of the phenomenon. In principle, it was possible to obtain a set of the images of the occulted star using sidereal tracking of the telescope, and then carry out the photometry of the obtained images of the star. However, in this case the CCD camera could not make more than two images per second. In addition, such quick operation could damage the shutter of the camera. So we could not get time resolutions better than 0.5 s when sidereal tracking was in operation.

But for further analysis of the data obtained it was desirable to have as high of a time resolution as possible. Therefore, an alternative option was the recording of the images in the telescope's field of view when sidereal tracking was off. In that case the track of each star would have the form of a long strip. This mode provides a higher precision of timing in determining the times for the start and the end of the occultation, as well as allows us to study in more detail a brightness variability during the eclipse of the star by the asteroid. The charge transfer by CCD during the exposure was not made, and only after finishing the exposure was the data obtained read into computer memory. Observers of occultation phenomena around the world have long called this method of photometric recording of the light curve of occultation "drift-scan." Naturally, of the two mentioned methods, we have chosen the latter method.

With the main 40 cm telescope we have obtained the photometric drift-scan of the occultation in broadband visible light (without any color filters). The duration of this drift-scan is equal to 600, and its recording was started at 12^h31^m59225 UT.

The image obtained (see Figure 2) was identified with the star chart, and the coordinates of the stars from the TYCHO catalog were used to calculate the plate scale, which was found to be 1878 pixel⁻¹.

Zoom In Zoom Out Reset image size

The direction of the star motion on the image was also determined (from east to west). The velocity of this motion was calculated as $(1296000^{\prime\prime} /86164\;{\rm{s}})\cdot \mathrm{cos}\delta$, where 1296000'' is the number of arcseconds in a circle, 86164 s is the length of the sidereal day, and δ is the declination of the star.

The resulting velocity is 14785 s⁻¹, or 7.873 pix s⁻¹. The star passes one pixel in 0.127 s. The recording time was 60 s, so the length of the star track is 472 pixels.

The image of the track was cropped from the full frame. The cropped image has dimensions of 500 × 200 pixels and covers the track and adjacent background. The obtained image of the track is shown in Figure 3.

Zoom In Zoom Out Reset image size

We shall call the lines of pixels parallel to the center line of the track "rows," and call the columns of pixels perpendicular to the direction of the mentioned center line "strips." The IRAF task "Apall" was applied for processing the image of the track. The intensity distributions along the strips, or one-dimensional cuts, were plotted, and the track centers were determined for every strip. These one-dimensional cuts across the star track were averaged over 200 rows, and the result is presented in Figure 4.

Zoom In Zoom Out Reset image size

The fitting of the intensity profiles was not performed; simply the sum of counts was computed for each row. These distributions allowed us to determine which zones should be used for calculating the signal from the star versus which zones should be used for the background measurement. If we assume that N is the number of pixels in a strip corresponding to the track center, then the sum of the counts from the star can be calculated in the interval {$N-7,N+7$}, and the background was computed as the median of the counts detected in the intervals {$N-50,N-20$}; {N + 20, N + 50}. The normalized background counts were then subtracted from the stellar track counts.

We estimated the signal-to-noise ratio as ∼10, using the data on mean flux from the star and from the background.

To determine the strip corresponding to the start of the recording, the approximate symmetry of the beginning and the end of the track was assumed. The error of the first strip determination can be estimated as 2–3 pixels. The time of the exposure start was recorded in the file header with a precision of 0.001 s, but the uncertainty in the determination of the time corresponding to the first pixel results in an error of the absolute time determination ≃0.3 s. The precision of relative time measurements is much higher, about 0.01 s. The relative time of passing each strip was calculated by multiplying the number of the strip by the time interval of passing one pixel (0.127 s), and the plot of the counts from the star versus time has been constructed.

As a result we have obtained the photometric curve of the occultation of the star by the asteroid. Then we applied a simple smoothing of the original curve in order to get rid of the excess influence of statistical noise; after that we have fitted each of five different portions of the resulting curve with the segment of an optimal straight line (see an explanation below). The original photometric occultation curve, as well as these segments of straight lines, are presented in Figure 5.

Zoom In Zoom Out Reset image size

We would like to emphasize that the photometric observations presented here, as far as we know, are only the second efficient photometric registrations of asteroidal occultation that have been conducted in the former Soviet Union. The first observation of this kind was carried out on 1983 May 4 at the Engel'gardt Astronomical Observatory of Kazan State University with the 48 cm reflector AZT-14 by the astronomer–observer V.B. Kapkov. He recorded the photoelectric curve of the occultation of the relatively faint star by the famous asteroid (2) Pallas with a time resolution of 01, using a photoelectric photometer operating in the photon-counting mode (Kapkov 1984).

We can see on the photometric occultation curve presented in Figure 5 some small drop of the light flux at the relative time ≃47–48 s. In order to verify whether this slight decline of the signal is not related to a secondary (faint) component of the occulted star, or (improbably) with a secondary (small) fragment of the asteroid, we also constructed similar light curves for two other relatively bright stars observed within the field of view of the camera. A similar drop of the signal, virtually at the same time, is also seen on the tracks for these stars, so it is not connected with any eclipse.

As it is known, when the occultation of a star by an asteroid occurs, diffraction effects arise. However, in the given case the effects of diffraction can be ignored for the following reasons.

Since in this case the angular size of the asteroid is more than 50 times greater than the estimated angular diameter of the star, when considering the occultation process in the first approximation one can apply a model of diffraction by a straight edge of the remote screen. The angular dimension of the first Fresnel zone for the light source having a very small angular diameter can be estimated as

$$\begin{eqnarray}&&{\alpha }_{1}^{\prime\prime }\simeq 206265\cdot {(\lambda /L)}^{1/2},\end{eqnarray} \tag{ 1 }$$

where λ is a characteristic wavelength of observation and L is a distance to the edge of the screen. Taking $\lambda \simeq 0.55\;\mu {\rm{m}}$ and $L=3.7429387074\cdot {10}^{11}\;{\rm{m}}$ we obtain ${\alpha }_{1}\simeq 0\buildrel{\prime\prime}\over{.} 00025$. Since the linear speed of the asteroid's shadow should be equal to 23.6108 km s⁻¹, its angular velocity will be 00130 s⁻¹, so the duration of passing of the first Fresnel zone over the line of sight of the observer should be about 0.019 s, and the duration of passage of the subsequent zones should be less. As stated above, the accumulation time for our measurements of the light flux is about 0.127 s, so all diffraction details of occultation process should be smoothed over the time of the count accumulation. In addition, the estimated angular diameter of the star d ≃ 0002 is almost an order of magnitude greater than the angular dimension of the first Fresnel zone, therefore the smoothing of the diffraction pattern must be so strong that in practice no signs of diffraction should be observed. Under such circumstances, we have reason to analyze the obtained occultation curve within the framework of the laws of geometric optics.

Considering the occultation curve under such a simple model, one can easily distinguish five different portions of it, and each of them can be approximated by a segment of a corresponding optimal straight line; we have performed the fitting of these portions with the appropriate segments of different optimal straight lines (Figure 5).

It is fairly clear in Figure 5 that the portions of the occultation curve corresponding to the fall of the signal at the beginning of the eclipse of the star's light by the asteroid's body, and its rise to about the previous level after the eclipse (these portions are numbered 2 and 4), are not sheer vertical as might be expected in case of the eclipse of a point source, but have a certain inclination to the vertical direction. For it to be seen more clearly, we have shown the relevant part of the occultation curve on a larger scale in Figure 6.

Zoom In Zoom Out Reset image size

Obviously, the presence of the inclinations to vertical to portions 2 and 4 of the occultation curve means that we are dealing with the eclipse of the remote light source having a finite angular size: at the beginning of the occultation this source is gradually eclipsed by the (straight) front edge of the asteroid, and at the end of the occultation the source is gradually emerging from the other (straight) edge of the asteroid. Note that Figure 6 clearly shows that the slopes of portions 2 and 4 are somewhat different from each other, which corresponds to different durations of the processes of the star's eclipse for the front edge of the asteroid and the star's reappearance from its other edge.

We have determined the following time intervals: (1) the duration of disappearance of the stellar disk behind the front edge of the asteroid ${\rm{\Delta }}{t}_{1}\simeq 1.40\;{\rm{s}};$ (2) the duration of the state of minimum brightness corresponding to complete absence of the light flux from the star ${\rm{\Delta }}{t}_{2}\simeq 6.22\;{\rm{s}};$ (3) the duration of the reappearance of the stellar disk from the other edge of the asteroid ${\rm{\Delta }}{t}_{3}\simeq 0.635\;{\rm{s}}$.

If we assume that the velocity vector of the asteroid is perpendicular to both of its aforementioned straight edges, where the disappearance of the star and its reappearance occurred, we can calculate the angular size of the occulted stellar source by multiplying the asteroid's angular velocity by the time intervals ${\rm{\Delta }}{t}_{1}$ or ${\rm{\Delta }}{t}_{3}$. In case of the front edge we obtain the value of the angular dimension of the source of ≃00182, and in case of the opposite edge we obtain ≃00083. These values are too large compared with the estimated angular diameter of the star d ≃ 0002, so one would assume that we are dealing with a very extended source, such as, for example, a close double star; however, we do not see any signs of the source duplicity on the occultation curve. In addition, the above estimates of the angular dimensions of the light source are very different from each other. In this situation, for a correct interpretation of the results obtained it is necessary to assume that the direction of the asteroid's velocity is far away from the perpendiculars to both mentioned edges, but, on the contrary, forms small angles with them.

The duration of the disappearance and reappearance of the star is determined by the angular velocity component v_x perpendicular to the corresponding edge of the asteroid. If v is the value of the asteroid's angular velocity, and φ is the angle between the velocity vector and the corresponding edge, then we have

$$\begin{eqnarray}&&{v}_{x}=v\cdot \mathrm{cos}(90^\circ -\varphi )=v\cdot \mathrm{sin}\varphi .\end{eqnarray} \tag{ 2 }$$

The actual angular size of the star should be equal to $d={v}_{x}\cdot {\rm{\Delta }}t=v\cdot \mathrm{sin}\varphi \cdot {\rm{\Delta }}t$, where ${\rm{\Delta }}t$ is the duration of the phenomenon of disappearance or reappearance of the star. Therefore, if we know the values of v, ${\rm{\Delta }}t$, and d, we can calculate the angle between the velocity vector of the asteroid and its corresponding straight edge:

$$\begin{eqnarray}&&\mathrm{sin}\varphi =d/(v\cdot {\rm{\Delta }}t)\end{eqnarray} \tag{ 3 }$$

or

$$\begin{eqnarray}&&\varphi =\mathrm{arcsin}(d/(v\cdot {\rm{\Delta }}t)).\end{eqnarray} \tag{ 4 }$$

Unfortunately we do not know the exact value of d: after studying several available catalogs of direct measurements of stellar angular diameters (from the analysis of photoelectric observations of occultations of stars by the Moon and from the modern interferometric observations), we have found that the data on the star HIP 97157 are not in any of these catalogs. Therefore, in order to obtain rough estimates of angles φ, we take the expected value of the angular diameter of the star d ≃ 0002. Then for the front edge of the asteroid, we find ${\varphi }_{1}\simeq 6^\circ \pm 3^\circ$, and for the opposite edge we find ${\varphi }_{2}\simeq 14^\circ \pm 3^\circ$, so these edges of the asteroid are inclined toward each other at an angle of about 8°. Thus, the occultation of the star occurred at very small angles between the edges of the asteroid and the direction of its velocity in space.

Finally, we can calculate the length of the "chord" of the asteroid in the direction of its motion, corresponding to the time interval during which the asteroid completely "screened" the star's light. For this we should use the simple formula:

$$\begin{eqnarray}&&a(\mathrm{km})=V\cdot {\rm{\Delta }}{t}_{2},\end{eqnarray} \tag{ 5 }$$

where V = 23.6108 km s⁻¹ is the linear velocity of the asteroid, and ${\rm{\Delta }}{t}_{2}\simeq 6.22\;{\rm{s}}$. Considering that the error in determining the V value can be estimated as 0.0001 km s⁻¹, and the error in determining the ${\rm{\Delta }}{t}_{2}$ value should be about 0.01 s (see above), the error in determining the a value will be equal to 0.24 km. As a result, we obtain a ≃ 146.86 ± 0.24 km.

This result does not contradict the available data on the dimensions of the asteroid Daphne. For example, in a recent paper (Conrad et al. 2008) the dimensions of the asteroid fitted in a model of a triaxial ellipsoid amounted to 239 × 183 × 153 km. The authors of that paper pay attention to the fact that the object is highly irregular in shape.

When processing observational data we have obtained useful estimates of the geometric parameters of the process of occultation of the star HIP 97157 by asteroid Daphne and some information on its shape. Also, we have found a useful estimate of the linear dimensions of the asteroid.

The project MASTER is partly supported by the Program of Development of Lomonosov, Moscow State University.

This work was partly supported by the grants of the Russian Foundation for Basic Research, 15-02-07875 and 14-02-31546.

This work was also partly supported by the State Order of the Ministry of Education and Science of the Russian Federation, No. 3.615.2014/K.

The work was supported by the Russian Science Foundation Agreement Number 16-12-00085, http://www.rscf.ru/sites/default/files/docfiles/KD_002.pdf.

We are grateful to the anonymous referee for valuable discussions that gave us the opportunity to make useful changes to our article.