3 The flux-size and surface brightness-size relationships

The early stars with H$\alpha$ emission we have isolated may have either intrinsic emission (the objects of our search), or be stars in HII regions. In the nebulae the central ionizing star visibility in V band depends on the contribution of the nebula lines to this band. After spectroscopy of the objects it will be possible to answer the question, however, examination of Fig. 2 can considerably clear up the situation. On the plot H$\alpha$flux -- size for all measured objects of the image No.6 (small circles) the objects are readily seen to be divided into two (or probably three) branches. It is natural to connect this division with different morphological types of the objects: the lower sequence represents "stars'', the upper (or two upper sequences) -- "nebulae'' or "stars plus nebulae''. On the image there appeared to be about 60% extended objects, 20% of stellar objects and also about 20% of the objects located in the lower-left corner of the diagram of Fig.2 at the point of a junction of all branches.

It is known that a compact HII region can imitate a hot star in the U-B, B-V colours and may also be a star-like source (in M33 $1{}^{\prime\prime}\ =\ 3.5$ pc). Nebulae (or nebulae with a visible star) may well be included in the catalogue of hot stars IFM. For star-like objects the flux must be ${F\propto 10^{\alpha d}}$, where d is the size of a star on a plate (Zickgraf & Humphreys 1991). In the case of a supernova remnant or a bubble-type nebula, ${ F\propto d^2}$, and in the case of a filled or a diffuse nebulae, ${F\propto d^3}$. An intermediate power sequence is also possible depending on the morphology of a nebulae and a star contribution to the H$\alpha$ band.

Figure 2: An object size FWHM as a function of ${\rm H}\alpha$ flux for the same stars as in Fig. 1 (small circles). There are also shown brighter stars (triangles), obvious nebulae -- bubbles (squares) and diffuses (large circles) from the same image. Bold lines show approximations of the sequences with bright stars and nebulae included, thin lines are those for the studied objects only

The results of testing this interpretation are contained in Fig. 2. We plot there from the same image a number of objects, whose morphology is obvious: apparent but not bright stars (triangles), large bubble-type nebulae (squares), "core plus halo''-type nebulae and diffuse nebulae (circles). The small circles indicate the objects from Fig.1. It is seen from Fig. 2 that the stars indeed fall in the lower stellar sequence, bubbles -- in the uppermost sequence (to be more precize, in its continuation). The third, intermediate sequence is continued by diffuse and "core plus halo'' nebulae. All three sequences join naturally in the region of objects' sizes $1{}^{\prime\prime}\ -\ 2\hbox{$.\!\!^{\prime\prime}$ }5$.

The apparent separation of the objects in Fig. 2 provides grounds to break up all the studied objects on the basis of their morphological features in H$\alpha$ into sequences: stars, bubble nebulae and intermediate diffuse (or complex) nebulae. The objects having low fluxes and small sizes are common to all three sequences. We label these 4 groups by s, d, b and c, respectively. It should be emphasized that the division can be done only on the average. Some of the objects fill space between the sequences, which is caused not only by photometric errors, but by the real complex morphology of the nebulae.

We made approximations of each of the isolated sequences in Fig.2 with the inclusion of the objects of known types (bold lines) and only for the objects under study (thin lines). The objects of type c were added, when approximating, to all three sequences. For both versions of the stellar sequence ${F= b + F_0\cdot10^{\rm ad}}$ turned out optimum. By equalling the flux to zero, we find an estimate of the seeing size d₀in a given image. In particular, in Fig.2 ${ d_0} =1\hbox{$.\!\!^{\prime\prime}$ }8$ and $1\hbox{$.\!\!^{\prime\prime}$ }6$ for all stars and for the catalogue stars, respectively.

The bubble sequence must satisfy ${ F\propto d^2}$. A formal search for the best curve fitting this data with the power function ${ F \propto d^n}$ yields $n=\,1.95-2.05$, i.e. the square power does fit this type of objects. Approximations of two samples -- with inclusion of the known bubbles and without them by ${F= a + F_0 \cdot d^2}$ yields a good fit, a formal seeing estimate is $1\hbox{$.\!\!^{\prime\prime}$ }7$.

Approximations of the intermediate type (diffuse) nebulae are also presented in Fig. 2. One may propose that the appearance of the diffuse sequence may be due to inaccuracies in determining the parameters of objects of complex structure. However, the fact that the bright diffuse nebulae fit well the intermediate sequence is a forcible argument in favour of reality of the intermediate objects as a separate diffuse sequence. Approximation of this sequence by ${ F \propto d^n}$ yields n=3.2$\pm$0.3, i.e. the cubic power describes well the diffuse branch (the formal seeing estimate there ${ d_0=2\hbox{$.\!\!^{\prime\prime}$ }0}$).

It should be noted that the approximation results prove in themselves that we are working on a nonlinear part of the effective characteristic curve. We have found that this is valid at least in F < 7000, and practically all the objects under our study are faint enough to be underexposed. The images of central region of the galaxy, however, are characterized by a stronger background and their position on the effective characteristic curve could be shifted a little to the linear region, where ${ F\propto \lg I}$. However the fact that the both fits (bold and thin lines in Fig. 2) are very similar gives us a proof that we may use the approximation ${ F \propto I}$ for all objects under study. When examining the rest of the images, we applied quite similar techniques. The conclusions drawn from a single H$\alpha$ image were confirmed for the remaining 8. The seeing estimate for all the photographs is $1\hbox{$.\!\!^{\prime\prime}$ }8 \pm0\hbox{$.\!\!^{\prime\prime}$ }1$, which is in agreement with the real seeing value during the observations (Courtes et al. 1987).

Figure 3: Relationship between H$\alpha$ surface brightness and a size for objects from image N2. Stars -- circles, diffuse nebulae -- diamonds, bubbles -- squares, common point-like objects -- triangles. Approximations of the isolated sequences are shown by dashed lines and those with allowance for the photographic effect by solid lines

A relationship between a size d and surface brightness SB=F/d²has been also studied for the selected candidates. Such a plot for one of the central images is presented in Fig.3. As was to be expected, the objects broke up into three sequences according to these described above. In this representation, some of the objects of type c are easier to identify and refer to one of the sequences s, d or b. We made the approximations in Fig.3 (dotted lines) using the objects broken up into sequences s, d, b separately and also those c-type objects, which are close to the each sequence:

$${ SB} = \left \{ \begin{array}{rcr} { (a + c 10^{\alpha {d}})... ...(a + cd^3)/d^2} \\ { (a + cd^2)/d^2}, \\ \end{array}\right.$$

where the coefficients a, c and $\alpha$ were determined for each sequence and in each image separately.

The curves in Fig.3 fit well the sequences (on average), especially the star sequence -- the upper curve in the figure. Nevertheless it is quite possible to improve the fits for the extended objects. For this purpose the photographic effect of a star size growing with star brightness has to be taken into account. This is most essential for nebulae of small sizes. From the analysis of Fig.3 it can be concluded that with nebulae sizes of $2{}^{\prime\prime}\!\!-4{}^{\prime\prime}$, the curves (two lower dotted lines) overestimate the value of SB, i.e. the data themselves are located, on average, below the curves. To allow for this effect, it has to be added to the formula of the nebulae flux ( ${ F(d)=A+F_0\,d^n}$) a term, which describes the behaviour of a point-like source, ${ F(d)=(A+B\,d^{2n}+ C\cdot10^{\beta d})^{1/2}}$. Taking ${ F_0\,d^n}$ out of brackets and expanding the rest into a series with d=d₀, one can find that the allowance for the photographic effect can be made by adding to the flux expression, a linear term ${ F(d)=A+F_0\,d^n+kd}$. The solid lines in Fig.3 show the b and d-type object approximations with the linear term allowed for. Agreement between the curves and the data for the nebulae is seen to considerably improved. With the aid of the diagram SB-d we managed to isolate a part of the objects from the group c and refer them to one of the three morphological sequences.