2.5 Centripetal Accelerations
The masses of cometary nuclei are sufficiently small that the rotational velocity at the
surface can approach the escape velocity. By manipulating the equations for the
rotational velocity and the escape velocity at the equator, one can arrive at the
equation
v r
v esc
¼
Ω N r
3=2
N
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2GM N
p
ð2:56Þ
which gives the ratio of the rotational velocity to the escape velocity and shows the
dependencies on three variables, namely, angular velocity, nucleus radius and
nucleus mass, for a spherical object.
In the case of rotational instability, the centripetal acceleration must approach the
surface gravitational acceleration. For a spherical body without tensile strength, the
force balance shows that the critical rotation period is
P crit ¼ 2π
ffiffiffiffiffiffiffiffiffiffi ffi
r
3
N
GM N
s
¼
ffiffiffiffiffiffiffiffiffi
3π
Gρ N
r
ð2:57Þ
For nucleus parameters appropriate for 67P, the ratio of v r /v esc would be around
0.3 indicating a significant opposing force to gravity at the equator. Thus, while the
nucleus of 67P is not yet close to rotational instability, the centripetal acceleration
does lower, locally, the effective gravitational potential and the escape velocity.
2.6 Surface Reflectance
At this point, we have addressed, for unresolved nuclei, the geometric albedo, p, and
have introduced φ as the value of the (unresolved) phase function at the phase angle,
α (Eq. 2.1). The directional-hemispherical albedo, A H , has also been introduced as an
important component in the determination of the heat balance at a surface element
(Eq. 2.4). In resolved surface photometry, we must look at the variation in the
surface reflectance as a function of the three photometric angles, i.e. the angle of
incidence (i), the angle of emission (e) and the phase angle. These angles are shown
in Fig. 2.14.
There is a vast body of literature about the photometry of surfaces including
outside planetary research. Much of this is beyond the scope of this text (see for
example, Li et al. 2015) but is interesting in itself as it has been driven by requirements for gaming software that seeks to be as realistic as possible. The nomenclature
is often different from that used in planetary physics but, even in this field, a large
54
2 The Nucleus
The masses of cometary nuclei are sufficiently small that the rotational velocity at the
surface can approach the escape velocity. By manipulating the equations for the
rotational velocity and the escape velocity at the equator, one can arrive at the
equation
v r
v esc
¼
Ω N r
3=2
N
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2GM N
p
ð2:56Þ
which gives the ratio of the rotational velocity to the escape velocity and shows the
dependencies on three variables, namely, angular velocity, nucleus radius and
nucleus mass, for a spherical object.
In the case of rotational instability, the centripetal acceleration must approach the
surface gravitational acceleration. For a spherical body without tensile strength, the
force balance shows that the critical rotation period is
P crit ¼ 2π
ffiffiffiffiffiffiffiffiffiffi ffi
r
3
N
GM N
s
¼
ffiffiffiffiffiffiffiffiffi
3π
Gρ N
r
ð2:57Þ
For nucleus parameters appropriate for 67P, the ratio of v r /v esc would be around
0.3 indicating a significant opposing force to gravity at the equator. Thus, while the
nucleus of 67P is not yet close to rotational instability, the centripetal acceleration
does lower, locally, the effective gravitational potential and the escape velocity.
2.6 Surface Reflectance
At this point, we have addressed, for unresolved nuclei, the geometric albedo, p, and
have introduced φ as the value of the (unresolved) phase function at the phase angle,
α (Eq. 2.1). The directional-hemispherical albedo, A H , has also been introduced as an
important component in the determination of the heat balance at a surface element
(Eq. 2.4). In resolved surface photometry, we must look at the variation in the
surface reflectance as a function of the three photometric angles, i.e. the angle of
incidence (i), the angle of emission (e) and the phase angle. These angles are shown
in Fig. 2.14.
There is a vast body of literature about the photometry of surfaces including
outside planetary research. Much of this is beyond the scope of this text (see for
example, Li et al. 2015) but is interesting in itself as it has been driven by requirements for gaming software that seeks to be as realistic as possible. The nomenclature
is often different from that used in planetary physics but, even in this field, a large
54
2 The Nucleus