While a positive value of the lag might be explicable through some form of
thermal inertia on orbital timescales, there are exceptions that challenge this hypothesis. For example, two other comets visited by spacecraft, 81P/Wild 2 and 9P/
Tempel 1, both have negative lags—the peaks of their brightnesses are
pre-perihelion by 13 and 10 days respectively. On the other hand, the shape of the
nucleus may play a role because the surface area exposed to sunlight may change
significantly through perihelion if the rotation axis is not orthogonal to the orbital
plane. However, if this were the sole reason for the lag then one would expect lags
for a sample of comets to be equally distributed about perihelion if the rotation axes
are randomly distributed with respect to the orbital plane and, as Ferrin shows, they
are clearly not with far more positive lags than negative ones. We can see from this
that the pre-/post-perihelion asymmetries of cometary brightnesses measured from
ground remain inadequately explained.
Another aspect of Fig. 1.3 is that the brightness is greater post-perihelion than at
the same heliocentric distance pre-perihelion and that the curve is appreciably
steeper pre-perihelion than post-perihelion. Once again, this is not universally the
case and there are exceptions including comets such as C/1995 O1 Hale-Bopp and
2P/Encke. Simplistically, one might expect the brightness, m 1 (1,r h ), to be proportional to 1/r h
4
. This would arise from the decreasing solar flux with heliocentric
distance combined with the reduction in the energy being available for sublimation.
However, cometary light curves generally indicate steeper dependencies on r h ,
particularly pre-perihelion. These features would persist if further linearization of
the orbital light curves would be performed by removing the 1/r h
2 dependence of the
illumination, viz.,
m 1 1, 1
ð Þ ¼ m 1 Δ, r h
ð
ÞÀ5 log 10 Δ À 5log 10 r h
ð1:2Þ
and indeed it is common to re-write the above equation as
m 1 1, 1
ð Þ ¼ m 1 Δ, r h
ð
ÞÀ5 log 10 Δ À 2:5 n h log 10 r h
ð1:3Þ
where n h is a power-law exponent describing the decrease in brightness with r h . The
dependencies can be fit with separate power laws for pre- and post-perihelion giving
values that are useful to model non-gravitational forces on the nucleus (see below).
Finally in Fig. 1.3, at least one “outburst” or anomalous brightening can be seen
where the comet’s brightness increased by nearly 7 magnitudes (a factor of 500)
5 years after perihelion. These features are commonly seen in the orbital light curves
of comets. It is also noticeable that the short-term variability can be quite large with a
spread of measurement of the order of 2 magnitudes. While the physics of outbursts
remains a subject of considerable discussion, the spread in measured brightnesses
seen in the light curves is attributed to variations in production rate with rotational
phase and dynamical changes arising from the outgassing process itself. For further
investigation of the reasons behind these phenomena, we will need to look at the
source and the production rates of gas and dust.
1.1 Light Curves
5
thermal inertia on orbital timescales, there are exceptions that challenge this hypothesis. For example, two other comets visited by spacecraft, 81P/Wild 2 and 9P/
Tempel 1, both have negative lags—the peaks of their brightnesses are
pre-perihelion by 13 and 10 days respectively. On the other hand, the shape of the
nucleus may play a role because the surface area exposed to sunlight may change
significantly through perihelion if the rotation axis is not orthogonal to the orbital
plane. However, if this were the sole reason for the lag then one would expect lags
for a sample of comets to be equally distributed about perihelion if the rotation axes
are randomly distributed with respect to the orbital plane and, as Ferrin shows, they
are clearly not with far more positive lags than negative ones. We can see from this
that the pre-/post-perihelion asymmetries of cometary brightnesses measured from
ground remain inadequately explained.
Another aspect of Fig. 1.3 is that the brightness is greater post-perihelion than at
the same heliocentric distance pre-perihelion and that the curve is appreciably
steeper pre-perihelion than post-perihelion. Once again, this is not universally the
case and there are exceptions including comets such as C/1995 O1 Hale-Bopp and
2P/Encke. Simplistically, one might expect the brightness, m 1 (1,r h ), to be proportional to 1/r h
4
. This would arise from the decreasing solar flux with heliocentric
distance combined with the reduction in the energy being available for sublimation.
However, cometary light curves generally indicate steeper dependencies on r h ,
particularly pre-perihelion. These features would persist if further linearization of
the orbital light curves would be performed by removing the 1/r h
2 dependence of the
illumination, viz.,
m 1 1, 1
ð Þ ¼ m 1 Δ, r h
ð
ÞÀ5 log 10 Δ À 5log 10 r h
ð1:2Þ
and indeed it is common to re-write the above equation as
m 1 1, 1
ð Þ ¼ m 1 Δ, r h
ð
ÞÀ5 log 10 Δ À 2:5 n h log 10 r h
ð1:3Þ
where n h is a power-law exponent describing the decrease in brightness with r h . The
dependencies can be fit with separate power laws for pre- and post-perihelion giving
values that are useful to model non-gravitational forces on the nucleus (see below).
Finally in Fig. 1.3, at least one “outburst” or anomalous brightening can be seen
where the comet’s brightness increased by nearly 7 magnitudes (a factor of 500)
5 years after perihelion. These features are commonly seen in the orbital light curves
of comets. It is also noticeable that the short-term variability can be quite large with a
spread of measurement of the order of 2 magnitudes. While the physics of outbursts
remains a subject of considerable discussion, the spread in measured brightnesses
seen in the light curves is attributed to variations in production rate with rotational
phase and dynamical changes arising from the outgassing process itself. For further
investigation of the reasons behind these phenomena, we will need to look at the
source and the production rates of gas and dust.
1.1 Light Curves
5