Kurt,
I am *no* expert on this, but yes, unless there was something
extremely unusual about the shape of the spectrum presented,
information about the expected LCROSS spectral radiance at 545 nm
(0.545 microns) -- which I have quite arbitrarily called "SR545" in
the formula -- is sufficient to estimate an astronomer's "visual
magnitude" with high precision. And yes, the "Johnson V" photometric
system is based (as I understand it) on the number of photo-electrons
detected through a filter with a relatively broad bandpass intended to
vaguely approximate the response of the human eye. An article about
"Standard Photometric Systems" by the Australian National University's
Michael Bessell in Volume 43 of the /Annual Reviews of Astrophysics/
(2005):
www.mso.anu.edu.au/~bessell/araapaper.pdf
depicts it (Figure 1) as having a vaguely sawtooth-shaped bandpass
starting at roughly 490 nm, peaking at 530 nm, and ending at roughly
660 nm. In Table 1 he describes it as having an "effective
wavelength" of 544.8 nm and a "width" of 84 nm. In theory, one needs
to add up the responses at each individual wavelength weighted by this
curve, a process mathematicians call "integration"; but if we know
that the Sun, with SR545 = 1840 W m^-2 micron^-1 gives visual
magnitude = -26.74 in this system, then any other light source with a
similar spectrum, differing only in overall intensity, is going to
give the same result corrected only for the difference in intensity at
any one wavelength (if the spectra are the same, the ratio will be the
same at all wavelengths, but SR545 is the one most characteristic of
the V-band). I did not check Dr. Wooden's curves in detail, but since
the LCROSS signal is reflected sunlight, and the Moon is not strongly
colored, I would assume they are very Sun-like spectra, at least
between 490 and 660 nm, and probably out to where she shows "thermal"
emission beginning to overtake "scattering" (~3.7 microns).
--
Incidentally, there has been, I believe, some confusion on this forum
about the use of the word "integrated magnitude".
To the best of my admittedly limited knowledge this phrase is normally
used to mean adding up elements of varying surface brightness over a
two-dimensional area to obtain an overall magnitude of that object
treated as an unresolved point source. This is again a process of
mathematical integration, but it is unrelated to integration over
wavelength (something that is necessary only when trying to guess the
photometric response of a V-filter to a light source with a very non-
Sun-like spectrum). Surface brightness can be expressed in "W m^-2
micron^-1 arcsec^-2" (as Dr. Wooden has done), in "W m^-2 micron^-1
sr^-1" (as physicists normally do, 1 steradian = 4.25x10^10 arcsec^2),
or can be converted to mpsas using the formula you mention. Since Dr.
Wooden's "flux densities" are for 1 square arc-sec, they convert
directly to magnitudes for 1 square arc-sec (what we have been calling
"mpsas").
A very important property of surface brightnesses, known at least
subconsciously to all photographers, is that (in whatever system they
are expressed) the surface brightness of an object does *not* vary
with viewing distance. For example, if Mars has a surface brightness
of 3.9 mpsas, (a value that does not change much because its distance
from the light source -- the Sun -- is always about the same) each
square arc-sec will have that same surface brightness (and require the
same photographic exposure) whether we are viewing from a distance
where the planet as a whole subtends an angle of 4 arc-sec (near
conjunction with the Sun) or an angle of 20 arc-sec (near opposition
to the Sun). On the other hand, the surface brightness of Mars would
be expected to be lower than that of the Moon (at most phases) because
it is farther from the light source, and the planets still farther
from the Sun have still lower intrinsic surface brightnesses.
An example of integrating mpsas over area to obtain an "integrated
magnitude" would be an idealized Mars with a uniform surface
brightness of 3.9 mpsas and a diameter of 20 arc-sec. The projected
area is pi * (10)^2 = 314 sqr-arcsec, so the idealized planet as a
whole is 314 times brighter than a single sqr-arcsec. This changes
the magnitude from 3.9 (for a single sqr-arcsec) to 3.9 - 2.5*log
(314) = -2.3 (for 314 sqr-arcsec). -2.3 is the magnitude of the
object as a whole, or its "integrated magnitude", which means, if
unresolved and treated as a point source, it would be
indistinguishable (in terms of counts through the V-band detector)
from a star of that magnitude. If the calculation were repeated with
an assumed diameter of 4 arc-sec, the result would be based on an area
25x smaller, giving an integrated magnitude of 3.9 - 2.5*log(314/25) =
+1.2 near conjunction.
Relating Dr. Wooden's "flux densities" to visual mpsas is easy since
that is simply a matter of plugging them into the formula. Converting
them to "integrated magnitudes" of the plume as a whole is much more
complex since one cannot add up the intensities without knowing how
many sqr-arcsec there are at each mpsas. And here graphs do not tell
us what the surface brightnesses will be at individual points --
instead we are being instructed (I believe) to look at the grain
density charts, and where those charts say, for example, 1E7 35-micron
radius(?) particles m^-2 to assume the red curve will apply, where
they say 1E6 35-micron particles m^-2 to assume the green curve
applies, and so on.
If you asking about magnitudes in other photometric bands, such as one
of the many in the infrared, then you would need to find out what the
magnitude of the Sun (or some other reference star) is in that system,
what spectral radiance (W m^-2 micron^-1) it produces at that
wavelength, and re-determine the constant in the conversion equation
based on those two numbers.
-- Jim
P.S.: the present "flux density" curves are similar to the ones you
have found in various LCROSS presentations, except that those were
plotted in "W m^-2 micron^-1 sr^-1". Since 1 steradian = 4.25x10^10
arcsec^2, the surface brightnesses in that system are 4.25x10^10
times greater than those in "W m^-2 micron^-1 arcsec^-2" .
On Sep 12, 11:05 pm, "
canopu...@yahoo.com" <
canopu...@yahoo.com>
wrote: