UDC 523.66

Astronomy

D. O. MOKHNACH

ON THE PHOTODISSOCIATION OF C₂ MOLECULES IN THE ATMOSPHERE OF COMETS

(Presented by Academician B. P. Konstantinov, 24 VII 1967)

Preliminary data on the distribution of surface brightness in the heads of comets 1942g, 1959k, and others \((^{1,2})\) show that the most probable initial velocity of the molecules C₂ and CN is

\[ v_0 \sim 3 \cdot 10^5 \ \text{cm}\cdot\text{s}^{-1}. \]

At this value of \(v_0\) and at a light-pressure acceleration \(g = 0.03\ \text{cm}\cdot\text{s}^{-2}\) for CN in comet 1942g and \(g = 0.36\ \text{cm}\cdot\text{s}^{-2}\) for C₂ in comet 1959k, the isophotes retain a circular form and are concentric. For smaller values of \(v_0\), the influence of the dispersion of the initial velocities and of the dissociation of the luminous molecules noticeably distorts the circular form of the isophotes at distances of about 150,000 km from the photometric center for C₂ and 500,000 km for CN at a heliocentric distance of 1 AU.

The influence of these factors, which distort the form of the isophotes, is most noticeable in the direction of the comet’s radius vector. In this direction, when only velocity dispersion is taken into account for constant isotropic emission, the apparent density is calculated by the formula

\[ N(x,0)=\frac{2n\sqrt{\pi}}{v_0\cdot |x|} e^{-(v_1/v_0)^2}, \]

where \(v_1\) is the smallest velocity necessary for a molecule to reach the point \((x,0)\); \(n\) is the number of molecules flying out in 1 s into a steradian with velocities \(v \geq v_1\).

If \(\Delta\) is the observational error, then the form of the isophotes will be circular at a value of \(x\) for which

\[ 1-e^{-(v_1/v_0)^2}\leq \Delta. \]

Expansion of the left-hand side in a series gives the condition \((v_1/v_0)^2 = \Delta\) at the point \((x,0)\), where the influence of dispersion on the form of the isophotes is still not noticeable. Since \(v_1^2 = 2gx_1\), we obtain

\[ v_0^2 = 2gx_1/\Delta . \tag{1} \]

Formula (1) makes it possible to estimate \(v_0\), if the values of \(\Delta\) and \(x_1\) are known.

Haser’s work \((^{3})\) shows how sharp the distortion of isophotes can be when overestimated values of accelerations and underestimated velocities are specified. For example, at \(v_0 = 10^5\ \text{cm}\cdot\text{s}^{-1}\) and \(g = 3\ \text{cm}\cdot\text{s}^{-2}\), the deformation is noticeable already at a distance of 10,000 km from the emission center, and at greater distances the isophotes can in no way be called circular.

The acceleration \(g\) under resonance radiation is determined rather confidently for C₂ and CN, since for these molecules the values of the oscillator strengths \(f = 0.02\) have not yet been disproved. Consequently, \(v_0\) can be estimated if the observational error \(\Delta\) is known.

1. Study of \(\Delta\) for comet 1959k. At the author’s request, D. Miller of Frimen thoroughly investigated the observational errors in studying the distribution of surface brightness in the C₂ component of the head

comet 1959k. The results of this study were kindly made available to the author by F. Miller.

At the University of Michigan Observatory, 6 photographs were obtained with the Curtis—Schmidt instrument on Eastman-Kodak 103 a-J plates through an interference filter isolating the $\lambda 5165$ bands of $\mathrm{C}_2$. This filter transmits 7% of the radiation from the head of the $(1,1)\ R_2$ band and of the $\mathrm{CO}^+$ tail radiation.

In processing the photographs, the errors of the isophotometer, the effect of poor guiding, light scattering by the emulsion, and the influence of the error in estimating the sky background were studied. Therefore the results of the surface-brightness measurements may be regarded as practically reliable.

Table 1

The isophotes proved to be somewhat elongated in the direction away from the Sun. The shape of the isophotes approaches a circle with increasing distance from the center of the coma. If by $r_2$ we denote the radius in the direction toward the Sun, and by $r_1$ that in the opposite direction, then the deformation of the isophotes can be illustrated by Table 1.

Such a deformation can be explained by: a) the true properties of the $\mathrm{C}_2$ coma; b) the influence of continuous radiation of a type-II tail; c) the influence of $(1,1)\ \mathrm{CO}^+$ radiation, 7% of which is transmitted by the filter.

But O’Dell’s paper ($^4$) speaks of the insignificance of continuous radiation in comet 1959k; therefore one should think that the deformation is caused by $\mathrm{CO}^+$ radiation, whose maximum is displaced somewhat relative to the center of emission in the direction opposite the Sun. In any case, the dependence of the deformation of the isophotes on the distance from the photometric center given in Table 1 cannot be explained by the effect of dissociation of $\mathrm{C}_2$ molecules or by velocity dispersion at small values of $v_0$, because in this case $r_2/r_1$ would increase rapidly with distance from the center. The latter is not confirmed by the observations.

Combining his measurements with those of O’Dell ($^4$) and Males ($^5$), F. Miller gives the dependence of the surface brightness $S$ on the distance $d/2$ ($d$ is the greatest diameter of an isophote) (see Table 2).

Table 2

A preliminary study of F. Miller’s material suggests that the error is $\Delta \sim 0.1$ at $\log d/2 = 5.2$. The mean deviation of the points from the smoothed curve passing through them is $\pm 0.31$ in $\log S$.

Comparison of the data in Table 2 with the author’s calculations ($^{1,2}$) shows that, for a model with emission center $v_0 = 3 \cdot 10^5$ cm/sec, $g = -0.36$ cm/sec$^2$ and $\tau = 3 \cdot 10^4$ sec, we obtain complete agreement of the observations with the calculations for $\log d/2 \geq 4.1$, i.e., for $d/2 > 12\,000$ km. At smaller values of $d/2$, the calculated value of $S$ exceeds the measured one (Table 3).

Table 3

The fact that for $d/2 < 10\,000$ km $S_{\mathrm{calc}} > S_{\mathrm{obs}}$ is easily explained by a peculiarity of our model. At small $d/2$ we approach the region filled by parent molecules. To simplify the model we assumed this

region into a point; therefore \(S \to \infty\) as \(d/2 \to 0\). In reality this region may have a radius of several hundred kilometers. Therefore, for small values of \(d/2\), the calculated surface brightnesses should be greater than the observed ones. But it must be remembered that a decrease in \(S_{\rm obs}\) may also result from overexposure during photography. Hence one should conclude that both observers and theorists must concentrate their attention on the inner regions of the comet’s head, close to the photometric center.

2. On the possibility of photodissociation of \(\mathrm{C}_2\) by the extreme ultraviolet radiation of the Sun. The explanation of the surface-brightness distribution presented in Table 2 by a dispersion of the initial velocities is automatically ruled out after studying the measurements of F. Miller in the direction toward the Sun. A dispersion of the initial velocities at \(v_0 = 5 \cdot 10^4\) cm/sec gives \(\log S\) close to that given in Table 2, but produces a sharp asymmetry in the distribution along the radius vector, which contradicts the observations.

In studying the possibility of photodissociation of \(\mathrm{C}_2\), it is necessary to note that the surface-brightness distribution obtained by Miller, in our model, requires fulfillment of the condition

\[ v_0 \tau \sim 9 \cdot 10^9 \ \text{cm}, \]

which practically coincides with the condition derived by Yu. N. Gnedin and A. Z. Dolginov \((^6)\):

\[ v_0 \tau \sim 10^{10} \ \text{cm}, \]

where \(\tau\) is the lifetime of the \(\mathrm{C}_2\) molecule before dissociation; \(v_0\) is the most probable velocity.

For \(\mathrm{C}_2\), C. M. Poloskov \((^7)\) obtained in 1948 \(\tau = 0.5\) hour. At this value of \(\tau\), Miller’s distribution requires \(v_0 = 5 \cdot 10^6\) cm/sec. Such a value of \(v_0\) seems to us improbable. To obtain such velocities, absorption of a colossal energy during dissociation of the parent molecules is necessary. Indeed, \(\tfrac{1}{2} m v_0^2 = 312\) eV.

Photodissociation of \(\mathrm{C}_2\) with subsequent excitation of the C atom appears more probable. In this case V. A. Ambartsumian \((^9)\) obtains \(\tau \sim 3 \cdot 10^7\) sec, whence it is necessary to conclude that \(\mathrm{C}_2\) does not undergo dissociation during the entire time it remains in the head and tail of the comet. The reason for this conclusion is that in 1939 there was no information about the actual radiation of the Sun in the extreme ultraviolet, about the surface-brightness distribution in the coma, and about the energies of dissociation of \(\mathrm{C}_2\) and excitation of C.

According to modern data, the dissociation energy of \(\mathrm{C}_2\) is 3.6 eV \((^{10})\), and the excitation energy of C is 7.48 eV \((^{11})\); therefore, for photodissociation of \(\mathrm{C}_2\), absorption of quanta with energy \(E \geq 11.1\) eV is necessary, i.e. \(\lambda \leq 1120\) Å. According to Hinteregger \((^7)\), for \(\lambda \leq 1120\) Å

\[ \Phi = 2.7 \cdot 10^{11} \ \text{quanta}/\text{cm}^2 \cdot \text{sec}, \]

and, in order to obtain \(\tau = 3 \cdot 10^4\) sec, it is necessary that the effective cross section of photodissociation be

\[ \sigma_d = 1 / \Phi \tau = 1.2 \cdot 10^{-16} \ \text{cm}^2. \]

If the absorption coefficient is calculated by formula (8)

\[ \int_{\nu_0}^{\infty} \chi_\nu d\nu = \frac{\pi e^2}{mc} f, \]

assuming \(\chi = a/\nu^3\), then we obtain

\[ \sigma = \frac{2\pi e^2 \lambda_0}{mc^2} f = 2.1 \cdot 10^{-17} f. \]

Assuming \(f=1\), we see that the computed value of the effective cross section is 6 times smaller than the value we require.

It must be noted that the computed value \(\sigma = 2.1 \cdot 10^{-17}\) cannot be regarded as certain. The law \(\nu^{-3}\), as studies by many authors show, is not exact, nor is the adopted value \(f=1\).

O. V. Dobrovolsky gives ([(7), p. 76]) a table of ionization and dissociation probabilities for the molecules \(\mathrm{O}_2\), \(\mathrm{CH}_4\), \(\mathrm{H}_3\), \(\mathrm{H}_2\mathrm{O}\), and draws attention to the fact that the probability of dissociation for these molecules is approximately an order of magnitude greater than the probability of ionization. Therefore \(\sigma_d = 1.2 \cdot 10^{-16}\ \mathrm{cm}^2\), obtained by us for \(\mathrm{C}_2\), should be regarded as a preliminary result of an experimental determination of the effective dissociation cross section. In the future one may expect confirmation and refinement of the quantities \(\nu_0\), \(\tau\), and \(\sigma_d\), if photometry of comets is carried out no less carefully than was done by D. Miller in 1966 for Comet Burnham 1959k.

Main Astronomical Observatory
of the Academy of Sciences of the USSR

Received
13 V 1967

REFERENCES CITED

1. D. O. Mokhnach, DAN, 171, No. 4 (1966).
2. D. O. Mokhnach, DAN, 171, No. 5 (1966).
3. L. Haser, Nature et Origine des Comètes, Liège, 1966, p. 233.
4. O’Dell, Publ. Astr. Soc. Pacific, 73, No. 430, 41 (1961).
5. D. Malaise, Nature et Origine des Comètes, Liège, 1966, p. 199.
6. Yu. N. Gnedin, A. Z. Dolginov, Astr. Zhurn., 43, 1 (1966).
7. O. V. Dobrovolsky, Comets, 1966.
8. L. Unsöld, Physics of Stellar Atmospheres, 1949, p. 142.
9. V. A. Ambartsumian, Theoretical Astrophysics, 1939, pp. 210–217.
10. K. Allen, Astrophysical Quantities, 1960.
11. A. N. Zaidel et al., Tables of Spectral Lines, Moscow, 1962.