(MSPCD) approach (Capanna et al. 2013) was designed to address this issue. This is
however technically and computationally challenging for high resolution data.
Cometary shapes have, in the past, been described in terms of the best-fit triaxial
ellipsoid (Table 2.2). However, describing 67P as a tri-axial ellipsoid is clearly
ridiculous but, for the sake of comparison, Jorda et al. provided the nominal figures.
From the same paper, values for the body and the head were determined independently and found to be 4.10 Â 3.52 Â 1.63 km and 2.50 Â 2.14 Â 1.64 km,
respectively with their volumes being 12.4 Æ 0.6 km
3 and 5.1 Æ 0.3 km
3
representing 66% and 27% of the total volume of the nucleus. The volume of the
neck could be derived by subtraction and was found to be 1.4 Æ 0.4 km
3 (about 7%
of the total volume) but this carries some uncertainty because of the need to define
the interfaces between the three parts of the nucleus. Thomas et al. (2013a, b)
adopted an alternate approach of giving a mean radius and values for the principal
moments derived from shape models but even this approach has difficulties if there
are large surface areas almost radial (or indeed overhanging) with respect to the
centre of figure.
It was recognized quickly after the first images of 67P were returned from Rosetta
that the neck of the nucleus would be subject to stresses as a result of self-gravity,
mechanical strength (assuming a linear elastic material) and the forces coming from
cometary activity. Self-gravity stress results from the bilobate shape of the nucleus,
while activity stress results from the surface sublimation of water ice and changes
diurnally (because of the rotation of the nucleus) and with heliocentric distance. The
resulting stresses have been modelled by S.F. Hviid (pers. comm.) using a finite
element model and are shown in Fig. 2.4. Similar calculations have been performed
by Hirabayashi et al. (2016). The stresses reach around 300 Pa. The evidence that we
may also have witnessed the effects of these stresses will be discussed later.
2.3 Mass and Density
Prior to the start of development of the instrumentation of the Rosetta mission, the
uncertainty in the bulk density of the nucleus was a major issue. Engineers designing
the landing system of what was eventually to become the Philae lander required
constraints on the density and mass of the nucleus to optimize the approach to
landing and anchoring the small spacecraft on the nucleus. At the time, the only way
to attack the problem was through analysis of the non-gravitational forces perturbing
the orbit. These analyses required numerous assumptions and values between
200 and 2400 kg m
À3 for the density were discussed at the time. However, the
dominance of water ice in the nucleus and the high porosity inferred from studies of
dust particles (e.g. arising from the first polarimetric measurements such as those of
Dollfus (1989)) led to a consensus that values <1000 kg m
À3 were highly likely and
possibly much less. Davidsson (2001) studied the rotation periods of 14 comets and
concluded that, for seven objects and assuming zero tensile strength, these nuclei
needed to have densities in the range 200 kg m
À3
ρ 530 kg m
À3 to avoid being
38
2 The Nucleus
however technically and computationally challenging for high resolution data.
Cometary shapes have, in the past, been described in terms of the best-fit triaxial
ellipsoid (Table 2.2). However, describing 67P as a tri-axial ellipsoid is clearly
ridiculous but, for the sake of comparison, Jorda et al. provided the nominal figures.
From the same paper, values for the body and the head were determined independently and found to be 4.10 Â 3.52 Â 1.63 km and 2.50 Â 2.14 Â 1.64 km,
respectively with their volumes being 12.4 Æ 0.6 km
3 and 5.1 Æ 0.3 km
3
representing 66% and 27% of the total volume of the nucleus. The volume of the
neck could be derived by subtraction and was found to be 1.4 Æ 0.4 km
3 (about 7%
of the total volume) but this carries some uncertainty because of the need to define
the interfaces between the three parts of the nucleus. Thomas et al. (2013a, b)
adopted an alternate approach of giving a mean radius and values for the principal
moments derived from shape models but even this approach has difficulties if there
are large surface areas almost radial (or indeed overhanging) with respect to the
centre of figure.
It was recognized quickly after the first images of 67P were returned from Rosetta
that the neck of the nucleus would be subject to stresses as a result of self-gravity,
mechanical strength (assuming a linear elastic material) and the forces coming from
cometary activity. Self-gravity stress results from the bilobate shape of the nucleus,
while activity stress results from the surface sublimation of water ice and changes
diurnally (because of the rotation of the nucleus) and with heliocentric distance. The
resulting stresses have been modelled by S.F. Hviid (pers. comm.) using a finite
element model and are shown in Fig. 2.4. Similar calculations have been performed
by Hirabayashi et al. (2016). The stresses reach around 300 Pa. The evidence that we
may also have witnessed the effects of these stresses will be discussed later.
2.3 Mass and Density
Prior to the start of development of the instrumentation of the Rosetta mission, the
uncertainty in the bulk density of the nucleus was a major issue. Engineers designing
the landing system of what was eventually to become the Philae lander required
constraints on the density and mass of the nucleus to optimize the approach to
landing and anchoring the small spacecraft on the nucleus. At the time, the only way
to attack the problem was through analysis of the non-gravitational forces perturbing
the orbit. These analyses required numerous assumptions and values between
200 and 2400 kg m
À3 for the density were discussed at the time. However, the
dominance of water ice in the nucleus and the high porosity inferred from studies of
dust particles (e.g. arising from the first polarimetric measurements such as those of
Dollfus (1989)) led to a consensus that values <1000 kg m
À3 were highly likely and
possibly much less. Davidsson (2001) studied the rotation periods of 14 comets and
concluded that, for seven objects and assuming zero tensile strength, these nuclei
needed to have densities in the range 200 kg m
À3
ρ 530 kg m
À3 to avoid being
38
2 The Nucleus