2.2.1 ADS 15184

The trapezium ADS 15184 (also known as WDS 21390+5729) is located at RA=21h 38m, Dec=+57° 29. It has four bright components denoted as A, B, C, and D. Their distribution is presented on the left of Figure 4. The right panel of this figure shows our image for ADS 15184 in the V filter. On the figure we have identified the components A, B, C and D. The A and B components are very bright and even on the short time exposure appear melded as shown in Figure 5. Due to this fact, the photometry of both these objects is carried out as if they represented one star only, which we denote as the AB component.

Fig. 4 The left image represents the components of the trapezium ADS 15184. The axes are given in arcsecs. On the right, we show our observed image in the V filter, with N-E orientation and scale in arsecs. Exposure time is 0.2 seconds 

Fig. 5 At the centre of the left image we show the blended AB components of ADS 15184. On the right panel, vertical and horizontal cuts are performed on the central star component AB. These cuts show that components A and B are not well resolved in our images. The colour figure can be viewed online. 

We obtained 70 images of this trapezium in the filters U - I. Two sets of images were taken with integration times of 0.1 and 8 seconds respectively, so that we could get images where the bright stars were not saturated and also images where the fainter stars could have a substantial number of counts.

To perform photometry on each one of the stars of a trapezium, it is necessary to vary the aperture radius to ensure that the same percentage of the light is included for each stellar image. Table 7 shows the radii for each component, where r ₁ is the radius of the central aperture, r ₂ is the inner radius of the sky annulus, and r ₃ is the outer radius of the sky annulus.

The photometric measurements we obtained are given in number of counts, which is directly proportional to the exposure time. We normalise all our measurements to a standard time of 10 seconds. To achieve this, we use the following equation:

Where I ₁₀ represents the intensity of the star at the normalised time (10 seconds), t is the observation time and I _obs is the observed intensity. The normalised intensity is transformed to magnitude in a standard manner.

Using equations 1-5 for each filter and taking the coefficients A, K and C for the night 10 - 11 December, shown in Tables 1-5, we calculate the values for the magnitudes and colours of the stars in the trapezium. We estimate the magnitude errors to be of the order of ±0.02 and ±0.03 for the colours.

Using the colour values, we may determine the Q index (see ^{Johnson & Morgan (1953)}), which we know is reddening independent: Q=(U-B)-0.72(B-V).

We can associate a different spectral type to different values of Q depending on whether we consider stars of Luminosity Class I (Supergiants) or V (Dwarfs) (see Table 9). We could also use for this classification the empirical calibration published by [^{Lyubimkov et-al.(2002)} in which they associate the effective temperature of stars of luminosity classes II-III and IV-V to the value of the Q-parameter (see their equations 6 and 7 and their Figure 11). We shall try this approach elsewhere.

Table 10 gives the spectral types associated to each star of this trapezium based on the values of Q. In the case of AB, the spectral type given is that of a star that results from the union of A and B.

From the U - B and B - V colours shown in Table 9 we make a two-colour diagram, where the red dots represent the components AB, C and D. The blue and green lines represent the intrinsic main sequence and the intrinsic supergiant sequence respectively. We also see the reddening line (red colour line, see Figure 6). This line has a slope equal to ≈0.72 (see Figure 6). This line allows us to deredden the observed colours of a star shifting the points in a direction parallel to this line until we intersect one of the intrinsic sequences (green or blue lines).

Figure 6: Two colour diagram (U - B vs B - V) for ADS 15184. The green line is the supergiant intrinsic sequence (SGIS), in blue the main sequence (MSIS) intrinsic sequence and the red dots represent the observed colours of the components of this trapezium. The red line and the red arrow represent the reddening line and direction. The colour figure can be viewed online. 

Finding the intersection of a line parallel to the reddening line that also passes on the observed points with the intrinsic sequences allows us to find the intrinsic colours of each star, which permits us to calculate the colour excess E (B - V), the absorption A _V and the distance in parsecs to the star in question (d(pc)=10(mv-MV-Av+5)/5). We have assumed the ratio of total to selective absorption AVE(B-V) to be equal to 3.1.

Table 11gives the values for the intrinsic colours, the excesses, the absorption and distances assuming the stars are supergiants (top panel), while in the bottom panel it gives the same information assuming the stars belong to the main sequence.

A comparison of our results with the SIMBAD Astronomical Database [^{Wenger et-al.(2000)} is presented in Table 13. The magnitudes of the stars listed in SIMBAD come from the following references: [^{Fabricius et-al.(2002)}, [^Reed(2003)], [^{Mercer et-al.(2009)}, and [^{Zacharias et-al.(2012)}. Table 12 shows that our magnitudes differ in the worst case by one magnitude.

Using Curve 15 of van de Hulst’s [^{Johnson(1968)}] we can obtain the dereddened values of the magnitudes for each component. The results of this exercise is shown in Table 15, these values coincide with those reported in Table 11.

2.2.2 ADS 4728

Trapezium ADS 4728 (also known as WDS 06085+13358) located at RA=6h 8m, Dec=+13°. 58 It has eleven components: A - K, which are distributed as shown on the left panel of Figure 7.

Fig. 7 The left image represents the components of the trapezium ADS 4728. The axes units are seconds of arc. On the right, we show our observed image in the V filter, with N-E orientation and scale in arsecs. Exposure time is 1 second. 

We observed 110 images for this trapezium in filters U - I.

The images for this trapezium were obtained with three different integration times 0.5, 1 and 8 seconds.

Figure 7 also shows an image of this trapezium in the V filter (right panel). The stellar components as well as the orientation and plate scale are indicated on this figure.

Table 14 shows the values of the radii we used for performing the photometry for each component. As done previously, all the measurements are normalised to 10 seconds with Equation 6.

Using the values of A, K and C we obtain the magnitudes and colours for the stars of trapezium ADS 15184 (see Table 15).

Table 16 shows the spectral type associated to each star from the Q parameter calibration

Using the U - B and B - V colours from Table 15 we plot the points on the two-colour diagram (see Figure 8) where the same conventions as those followed for the previous trapezium are followed.

Fig. 8 Two-colour diagram (U - B vs B - V) for ADS 4728. The colour figure can be viewed online. 

The first part of Table 17 presents the values for intrinsic colours, colour excesses, absorption and distances assuming the stars are supergiants, while the second part shows the same results assuming the stars to be in the main sequence.

The magnitudes of the stars listed in SIMBAD come from the following references: [^{Høg et-al.(2000)}, [^{Zacharias et-al.(2003)}, [^Reed(2003)], [^{Zacharias et-al.(2009)}, [^{Krone-Martins et-al.(2010)}, and [^{Gaia Collaboration (2020)}]. Table 18 shows our results compared with those listed in SIMBAD (shown with *).

Table 19 shows the dereddened values for magnitudes and colours for the stars in ADS 4728.

2.2.3 ADS 2843

Trapezium ADS 2843 (also known as WDS 03541+3153) is at RA=3h 54m, Dec=+31° 53 and has five components: A, B, C, D, and E, distributed as seen on the left of Figure 9.

Fig. 9 The left image represents the components of the trapezium ADS 2843. The axes units are arcseconds. On the right, we show our observed image in the V filter, with N-E orientation and scale in arsecs. 3 seconds exposure time. 

We obtained 148 images for this trapezium in filters U - I, with exposure times of 0.2 and 0.5 seconds.

The right panel of Figure 9 shows the image of this trapezium in the V filter. Component A is clearly saturated, and its brightness affects component B making it impossible to obtain good photometric measurements.

In Table 21 we show the photometric radius for each component.

In a similar manner as before, we use the transformation coefficients in Tables 1-5 and equations 1-5 to transform the number of counts for each star to intrinsic magnitudes and colours (see Table 22).

Table 23 shows the spectral type associated to each star based on the value of its Q parameter.

Figure 10 shows the two-colour diagram for ADS 2843, where the red dots represent component C, D and E.

Figure 10: Two-colour diagram (U - B vs B - V) for ADS 2843. The colour figure can be viewed online. 

From the two-colour diagram we obtain the values for the intrinsic colours, excesses, absorption and distance. However, in this case, the reddening lines for the components cross the intrinsic lines more than once. We have taken the excesses that appear to be more similar. Table 24 show the results for each case.

The magnitudes of the stars listed in SIMBAD come from the following reference: ^{Høg et-al.(2000)}. Table 25 shows the comparison of our results with SIMBAD (*).

We show the magnitudes dereddened with van de Hulst curve 15 in Table 26. This table assumes respectively supergiant stars and main sequence stars.

2.2.4 ADS 16795

Trapezium ADS 16795 (also known as WDS 23300+5833) is located at RA=23h 30m, Dec=+58° 32 and has six components: AB, C, E, F, G and I, whose location is shown in Figure 11.

Fig. 11 The left image represents the components of the trapezium ADS 16795. The axes units are arcseconds. On the right, we show our observed image in the V filter, with N-E orientation and scale in arcsec. Exposure time is 5 seconds. 

For this trapezium we obtained 144 images in filters U - I. This trapezium was observed on the night 2019, 14-15 December. For this night the transformation coefficients A, K and C for the filter U have no physical meaning (see Table 1), so no calculation of the U magnitude was possible.

Figure 11 shows the image of this trapezium in the V filter.

It is clear that the component AB is saturated so that the photometric measurements will only be performed on the other components (C, E, F, G and I).

Table 27 shows the photometric aperture radii for each component.

Table 28 gives the observed magnitudes and colours for the components of this trapezium. We obtained only those for the filters B, V, R and I because the results for U have no physical meaning and have been omitted.

Since we cannot obtain a value of U, it is impossible to calculate the value of the Q parameter. The magnitudes of the stars listed in SIMBAD come from the following reference: ^{Zacharias et-al.(2012)}. In Table 29 we compare our results with those listed in SIMBAD.