Transcript Deep Impact Projects (2) Mike A’Hearn Deep Impact Projects mfa - 1 Problem 1 Optical Depth & Albedo of Ejecta from DI Deep Impact Projects mfa -

Deep Impact Projects (2)
Mike A’Hearn
Deep Impact Projects
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Problem 1
Optical Depth & Albedo of Ejecta
from DI
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Optical Depth
• Well known and widely used in astronomy but
often ignored in cometary science because the
coma is usually optically thin
• I/I0 = e-t where t = optical depth
• For simple scattering and absorption
• t=Ns
• where s = extinction cross section
• and N = column density
• A typical photon travels t = 2/3 before being
absorbed or scattered
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Scattering Function
• A single particle (grain of dust or whatever) scatters
light anisotropically
• The phase function describes the distribution of light
with scattering angle, both for microscopic particles and
for large bodies like cometary nuclei and asteroids
• Phase function is often approximated by many different
simple functions of the scattering angle
• For a single particle,
– I = Fsun s p f() where p = geometric albedo and f() is
the scattering function or I = Fsun*s*A()
– Be careful of confusion between geometric albedo and
Bond albedo - geometric is backscattering and Bond is
integrated around the sphere.
– The widely used quantity Afr uses Bond albedo as A, even
though this is not well defined for microscopic particles
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Phase Functions
Phase
0
5
10
15
20
25
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Dust
Nucleus
1.00
1.0000
0.92
0.5963
0.85
0.4426
0.77
0.3429
0.70
0.2713
0.65
0.2175
0.60
0.1760
0.50
0.1177
0.45
0.0815
0.45
0.0595
0.45
0.0471
0.45
0.0409
0.45
0.0386
0.55
0.0382
0.80
0.0382
1.50
0.0376
2.50
0.0357
4.00
0.0320
7.00
0.0265
10.00
0.0191
15.00
0.0100
20.00
0
• Sample phase functions
appropriate to comets
– There is variation from
one comet to another
– Dust from Ney & Merrill,
comet West (1976 VI)
– Nucleus from LummeBowell law
• These are normalized to
unity at phase = 0°
– correct for use with
geometric albedo
– Renormalize to unity of
integral over 4p for use
with Bond albedo
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Albedo
• Solving for the albedo from t and I is a matter
of algebra left to the student
– Use the approximation that the optical depth is not
large (t < 1) so that I(column) = N*I(particle)
– Be very careful of the units!
• I has units of radiance (see the image labels for the
exact units)
• Look up solar flux for a typical wavelength (look at a
document that describes the bandpass or use the label
values for center wavelength and bandpass)
– Remember that phase = 180° - scattering angle
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Making the Measurements - 1
• Consider MRI images just before and just after
the impact (ID 9000910_050 is before, ID
9000910_080 is after)
• You may wish to register the images first
– IDL has, e.g., a cross-correlation function that you
can use to determine the offset as long as the ejecta
are not too bright
– There are numerous other ways to do this if you are
familiar with image processing
– If the images are registered, you can make the
measurements in the same pixels on all images
• Alternatively, you can choose a feature on the
surface or the limb and make sure that you
make all measurements at the same feature
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Making the Measurements - 2
• Measure the profile of brightness across the
limb of the nucleus at your “feature”
• Extrapolate the coma brightness to the
brightness just inside the limb
• Measure the height, I, of the limb above the
extrapolated coma
– Extrapolation should only be a few pixels
– Call it I0 for the pre-impact image (or averaged over
a few pre-impact images)
– Use I/I0 to determine t
– Use I and t to determine p*f() and use the table of
scattering functions to get p, the geometric albedo
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Extending the Project
• Ultimately, the interesting question is the variation of
albedo with position around the limb and with time
– This can show variations in the type of particles (ice vs.
dust and organics) both with direction of ejection and with
depth of excavation
• Optical depth is >1 after about 5-10 seconds
– The problem is much messier because one must allow both
for the attenuation of the incident sunlight from one
direction and the attenuation of the scattered light along
the line of sight
• Any other sharp boundary (a crater or a scarp) on the
nucleus can be used to measure the optical depth but
– the contrast is lower
– One must use features on the sunward side of the impact
site or the “true” brightness of the surface will change
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Problem 2
Plot the Light Curve of an Outburst
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Scientific Goal
• We do not understand what causes the natural
outbursts
• Determine how long the emission of material
continues, which is important in choosing a
mechanism for producing the outburst, and
whether the outflow of material explains the
drop in brightness after the outburst
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Outbursts on July 1 & 3
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Outburst Motion
• July 3 outburst has most
data
• Lightcurve best done
from MRI data
• Images best done from
HRI data, but requires
deconvolution
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Making the Measurements
• Select the images that include the outburst
plus a couple before and after
– NOTE!!! This analysis should start with the clear filter
and should eventually use BOTH the science data
and the navigation data.
• On each image
– Locate the nucleus of the comet - usually at the
photocenter
– Add up the intensity within a box centered on the
comet’s nucleus
• Try boxes of several sizes, e.g., 5, 9, 15, 25 pixels
• Plot the intensities vs. time to see the light curve
• A circular aperture is better than a box, but requires a
little more programming in IDL or use of procedures in
the GSFC library for IDL (probably also in IRAF)
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Interpreting the Result
• Calculate the distance (in m or km at the
comet) from the nucleus to the edge of the
aperture
• Use the time from beginning of outburst to
end of outburst to estimate the velocity of the
material
• Verify that the deduced velocity is the same
for various apertures
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Extending the Project
• The HRI images can be used to trace the material in
time - see the deconvolved images shown in a previous
slide
– Not many images during the outburst
– Requires subtracting a pre-outburst, registered image and
then doing the deconvolution for the focus problem
– Mapping the intensity variation with time can provide
further constraints on the motion of the ejecta - need a
true simulation
• Finding all the outbursts and coordinating them with
rotational phase of the nucleus
• Tracing the outbursts to specific locations on the nucleus
• Simulating a mechanism to produce the outbursts
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