A G ¼
2π f A
1 À f A
2
À
Á 3=2 tan
À1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
f A
2
À 1
s
"
#
À
2π f A
2
1 À f A
2
ð2:37Þ
where f A is the axial ratio of the spheroid. This equation is plotted for three
reasonably realistic cases in Fig. 2.9. For 67P, the present rate of angular acceleration
would suggest a lifetime against splitting of only 200 years.
As indicated by Eq. (2.34), the angular acceleration is inversely proportional to
the nucleus mass and the square of the radius so that small or unmeasurable rotation
period changes observed for comets such as 14P/Wolf, 143P/Kowal-Mrkos, and
162P/Siding Spring (Kokotanekova et al. 2018) may be indicative of larger nucleus
sizes and much longer timescales.
In the general case, a cometary nucleus can be in an excited rotational state. We
have seen that nuclei can have complex shapes. Dynamically, the most stable
rotational state is the one that requires the least amount of rotational kinetic energy
for a given total rotational angular momentum. This corresponds to rotation about its
axis of maximum moment of inertia. In this state, the moment of inertia is related to
the rotational angular momentum, L, and the rotational kinetic energy, E rot , through
I s ¼
L
2
2E rot
ð2:38Þ
where the subscript s indicates that rotation is about the short axis (assuming a
uniform density distribution within the object).
However, the external torque resulting from jet activity (Eq. 2.31) is a vector.
Thus, it may not merely modify the rotation period but can also push the comet into a
rotationally more excited state. It should also not be forgotten that major mass loss
events (e.g. splitting of a nucleus), collisions, or tidal effects (e.g. through a close
interaction with Jupiter) can produce abrupt changes in the principal moments of
inertia leading to more excited, rotational states. These types of rotation are known
as non-principal-axis rotational states (NPA rotational states) (Samarasinha and
Fig. 2.9 Critical rotation
periods for splitting as a
consequence of spin-up
following Davidsson
(2001). Solid line: Long axis
length ¼ 8 km, tensile
strength of 20 Pa and a bulk
density of 600 kg m
À3
.
Dashed line: long axis
length reduced to 2 km.
Dot-dashed line: long axis
length reduced to 2 km and
the tensile strength
increased to 200 Pa
2.4 Rotational Properties
47
2π f A
1 À f A
2
À
Á 3=2 tan
À1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
f A
2
À 1
s
"
#
À
2π f A
2
1 À f A
2
ð2:37Þ
where f A is the axial ratio of the spheroid. This equation is plotted for three
reasonably realistic cases in Fig. 2.9. For 67P, the present rate of angular acceleration
would suggest a lifetime against splitting of only 200 years.
As indicated by Eq. (2.34), the angular acceleration is inversely proportional to
the nucleus mass and the square of the radius so that small or unmeasurable rotation
period changes observed for comets such as 14P/Wolf, 143P/Kowal-Mrkos, and
162P/Siding Spring (Kokotanekova et al. 2018) may be indicative of larger nucleus
sizes and much longer timescales.
In the general case, a cometary nucleus can be in an excited rotational state. We
have seen that nuclei can have complex shapes. Dynamically, the most stable
rotational state is the one that requires the least amount of rotational kinetic energy
for a given total rotational angular momentum. This corresponds to rotation about its
axis of maximum moment of inertia. In this state, the moment of inertia is related to
the rotational angular momentum, L, and the rotational kinetic energy, E rot , through
I s ¼
L
2
2E rot
ð2:38Þ
where the subscript s indicates that rotation is about the short axis (assuming a
uniform density distribution within the object).
However, the external torque resulting from jet activity (Eq. 2.31) is a vector.
Thus, it may not merely modify the rotation period but can also push the comet into a
rotationally more excited state. It should also not be forgotten that major mass loss
events (e.g. splitting of a nucleus), collisions, or tidal effects (e.g. through a close
interaction with Jupiter) can produce abrupt changes in the principal moments of
inertia leading to more excited, rotational states. These types of rotation are known
as non-principal-axis rotational states (NPA rotational states) (Samarasinha and
Fig. 2.9 Critical rotation
periods for splitting as a
consequence of spin-up
following Davidsson
(2001). Solid line: Long axis
length ¼ 8 km, tensile
strength of 20 Pa and a bulk
density of 600 kg m
À3
.
Dashed line: long axis
length reduced to 2 km.
Dot-dashed line: long axis
length reduced to 2 km and
the tensile strength
increased to 200 Pa
2.4 Rotational Properties
47