4 Forced precession model

We used the Sekanina's (1984, 1988) forced precession model of a rotating cometary nucleus to include the nongravitational terms into equations of comet's motion. Values of six basic parameters (four connected with the rotating comet nucleus and two describing the precession of spin-axis of the nucleus) have been determined along with the orbital elements from positional observations of the comets. Those six parameters are:

• modulus of the reactive force divided by g(r), A, where A=(A₁²+A₂²+A₃²)^1/2;
• lag angle $\eta$ of the maximum outgassing behind subsolar meridian;
• obliquity I of the orbit plane to the cometary equator;
• cometocentric longitude $\phi$ of the Sun at perihelion;
• nucleus oblateness $s= 1 - R_{\rm b}/R_{\rm a}$, where $R_{\rm a}$ and $R_{\rm b}$ are equatorial and polar radii, respectively;
• precession factor $f_{\rm p}$ which depends on a rotation period and a nucleus size.

For the rotating cometary nucleus the orbital components of nongravitational force, F_i are functions of time. Thus, F_i are related to the angular parameters $\eta , I, \phi$ of the rotating nucleus by:

$$F_i(t) = A\cdot C_i\left( \eta, I(t), v(t)+\phi (t)\right) \cdot g(r) {\rm ~~for~} i=1,2,3\,$$ (2)

where v is the true anomaly of the comet and C_idenote the direction cosines, which are time-dependent due to the orbital and precessional motion of the comet. Expressions for time variations of the angles I and $\phi$, the precession rate, and the direction cosines, C_i, of the nongravitational force are given by Królikowska et al. (1998b), (hereafter referred as KSS) together with a detailed description of the method of computations. We adopt the Marsden's water production curve, g(r), whose analytical form is useful for orbital computations.

To find satisfactory solutions we have to accept the following additional assumptions characterizing a specific physical behaviour of these comets:

• 1. Maximum of cometary activity with respect to perihelion can be asymmetric;
• 2a. The time shift parameter which describes displacement of maximum activity with respect to the perihelion could change its value during investigated apparitions; and/or
• 2b. Real activity of the comet change during the considered time interval.

The first assumption introduces an additional unknown parameter $\tau$ representing a time shift of the maximum of g(r) with respect to the perihelion time. Thus, the function g(r) is then replaced by g(r'), where $r'=r(t-\tau)$ (Sekanina 1988; Sitarski 1994a). The time shift $\tau$ can be determined together with the basic parameters from the forced precession model, but only for Comet 37P/Forbes case the model based on seven unknown parameters (basic six parameters plus $\tau$) proved to be reliable. For the remaining comets it was necessary to introduce the assumptions 2a and/or 2b. The postulated changes can be modelled by setting the discontinuities of parameters A and/or $\tau$ close to selected moments of aphelion passages ( $t_{b1},\; t_{b2}$):

$$A = \left\{ \matrix{ A ^{(1)} \mbox{~for $t<t_{b1}$~}\cr A ^{... ...t<t_{b2}$}\cr \tau _2 \mbox{~for $t\geq t_{b2}$}\cr}\right. .$$

Finally, the parameters A⁽ⁱ⁾ (i=1, 2, ...), $\eta, I_0$, $\phi _0, f_{\rm p}$, s and $\tau _k$ (k=1, 2, ...) were determined simultaneously with six orbital elements from the observational equations by an iterative least squares process; the number of parameters A⁽ⁱ⁾ and $\tau _k$depend on assumed number of discontinuities of A and $\tau$, respectively.

 

 

Table 4: Orbital elements and physical parameters of the nucleus for forced precession models linking all apparitions of each six comets. Angular elements $\omega$, $\Omega$, i are referred to Equinox J2000.0. Parameters A, A₁ and A₂ are in units of 10^-8 AU/day², the precession factor $f_{\rm p}$ is in units of 10⁷ day/AU, time shifts $\tau$, $\tau _1$, $\tau _2$and $\tau _3$ are in days, and t_b1, t_b2, t_b3 are moments of discontinuities of A or $\tau$

	(a)   Prolate spheroid models				(b)   Oblate spheroid models
	Wolf-	Forbes	Brooks 2	Comas Solá	Schwassmann-	Giacobini-
	Harrington				Wachmann 2	Zinner
Epoch:	1997 08 20	1999 12 08	1995 02 24	1926 11 01	1995 07 04	1956 02 17
T	97 09 29.21850	99 05 04.24713	94 09 01.04396	27 03 22.19729	94 01 24.98426	53 04 16.36686
q	1.58182646	1.44616915	1.84322847	1.77244905	2.07187392	0.98755442
e	0.54394750	0.56841121	0.49075850	0.57495559	0.39993050	0.71789222
$\omega$	187 $.\!\!^\circ$13349	310 $.\!\!^\circ$74872	197 $.\!\!^\circ$98531	38 $.\!\!^\circ$50697	358 $.\!\!^\circ$66420	171 $.\!\!^\circ$82326
$\Omega$	254 $.\!\!^\circ$75651	334 $.\!\!^\circ$35354	176 $.\!\!^\circ$93159	66 $.\!\!^\circ$60484	126 $.\!\!^\circ$07531	196 $.\!\!^\circ$91409
i	18 $.\!\!^\circ$51059	7 $.\!\!^\circ$16047	5 $.\!\!^\circ$54035	13 $.\!\!^\circ$76535	3 $.\!\!^\circ$75704	30 $.\!\!^\circ$84181
A	$+.3634\pm .0061$	$+.5584\pm .0062$	$+1.1846\pm .0055$	$+.7995\pm 0.0105$	$+1.399\pm .011$	---
A(1)	---	---	---	---	---	$+.3031 \pm .0019$
tb1	---	---	---	---	---	1956 02 17
A(2)	---	---	---	---	---	$+.3944 \pm .0005$
$\eta$	6 $.\!\!^\circ$36 $\pm$ 0 $.\!\!^\circ$41	11 $.\!\!^\circ$89 $\pm$ 0 $.\!\!^\circ$11	21 $.\!\!^\circ$35 $\pm$ 0 $.\!\!^\circ$47	21 $.\!\!^\circ$70 $\pm$ 0 $.\!\!^\circ$76	13 $.\!\!^\circ$18 $\pm$ 0 $.\!\!^\circ$24	6 $.\!\!^\circ$53 $\pm$ 0 $.\!\!^\circ$02
I0	125 $.\!\!^\circ$05 $\pm$ 4 $.\!\!^\circ$40	87 $.\!\!^\circ$82 $\pm$ 0 $.\!\!^\circ$65	118 $.\!\!^\circ$22 $\pm$ 0 $.\!\!^\circ$68	50 $.\!\!^\circ$21 $\pm$ 1 $.\!\!^\circ$64	160 $.\!\!^\circ$64 $\pm$ 0 $.\!\!^\circ$50	25 $.\!\!^\circ$91 $\pm$ 0 $.\!\!^\circ$56
$\phi _0$	143 $.\!\!^\circ$47 $\pm$14 $.\!\!^\circ$80	15 $.\!\!^\circ$78 $\pm$ 4 $.\!\!^\circ$07	319 $.\!\!^\circ$49 $\pm$ 0 $.\!\!^\circ$71	85 $.\!\!^\circ$06 $\pm$ 3 $.\!\!^\circ$14	9 $.\!\!^\circ$16 $\pm$ 4 $.\!\!^\circ$31	147 $.\!\!^\circ$24 $\pm$ 0 $.\!\!^\circ$28
$f_{\rm p}$	$-.3136 \pm .0670$	$-.4623 \pm .0179$	$-.08191 \pm 0.00041$	$-.1735 \pm .0046$	$+.2865 \pm .0093$	$+.3460 \pm .0007$
s	$-.0451 \pm .0200$	$-.0589 \pm .0041$	$-.0572 \pm 0.0055$	$-.2886 \pm .0348$	$+.0923 \pm .0204$	$+.2926 \pm .0055$
$\tau$	---	$-8.3254 \pm 0.2563$	---	---	---	---
$\tau _1$	$-17.78 \pm 1.75$	---	+4.7	$-53.92 \pm 3.79$	$+3.291 \pm 1.129$	$+4.469\pm .129$
tb2	1975 01 01	---	1936 03 13	1940 01 01	1970 07 01	1956 02 17
$\tau _2$	$8.394 \pm 0.823$	---	-20.2	$+26.25 \pm 1.95$	$+61.81 \pm 1.08$	+38.75
tb3	1988 01 06	---	1949 06 13	---	1978 01 01	1969 03 01
$\tau _3$	$-11.69 \pm 0.88$	---	+1.1	---	$+1.193 \pm 0.756$	-53.06
tb4	---	---	1970 05 14	---	---	1988 12 25
$\tau _4$	---	---	+20.4	---	---	-49.38
RMS	1 $.\!\!^{\prime\prime}$39	1 $.\!\!^{\prime\prime}$47	3 $.\!\!^{\prime\prime}$07	2 $.\!\!^{\prime\prime}$01	1 $.\!\!^{\prime\prime}$67	4 $.\!\!^{\prime\prime}$17

In some cases we were able to find the best value of $\tau$ but not as a basic parameter of the mean squares procedure (these $\tau$'s are given without their mean errors in Table 4)

To show how values of the nongravitational parameters are derived, especially the values of the time shift parameter $\tau$, we take as an example the Comet 16P/Brooks 2. The procedure is as follows:

(i) Linking observations of the first four apparitions of the comet in 1889/90, 1896, 1903/04, and 1910, we find the Marsden parameters A₁,A₂,A₃ and estimate the preliminary values of $A,\eta,I_0,\phi_0$;

(ii) Joining subsequently observations of the next apparitions in 1925/26, 1932/33, and 1939/40 we are also able to determine consecutive values of the parameters $f_{\rm p}, s$, and $\tau _1$. Thus we linked all observations from the interval 1889-1940 characterized by the acceptable mean RMS residual equal to 2 $.\!\!^{\prime\prime}$7, and determine the values of six orbital elements and of seven nongravitational parameters $A,\eta,I_0,\phi_0,f_{\rm p},s,\tau_1$. However, basing on that preliminary model of the comet's motion a prediction for the next apparition failed since the predicted RMS residual for the 1946 observations amounted to 76 $.\!\!^{\prime\prime}$, thus it was impossible to obtain a reasonable solution for all the 1889-1946 observations;

(iii) To add observations of 1946 to a set of 1889-1940 for the orbit improvement we have to find (by the method of "trials and errors'' an appropriate value of $\tau _2$ and put it after the aphelion time in 1936. Thus we were able to link successfully all observations from 1889-1954 by means of eight nongravitational parameters: $A,\eta,I_0,\phi_0,f_{\rm p},s,\tau_1, \tau_2$.

(iv) To join observations from further apparitions of the comet, we have to add to a set of nongravitational parameters new values of $\tau _3$ and $\tau _4$ after 1949 and 1970, respectively, like in case of $\tau _2$.