Steve Lord's notes on MIPS Color Corrections - Aug 3, 2006
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Introduction:
We discuss corrections to be applied to MIPS reported flux densities,
when reporting with respect to different SEDs.
This work started when our group (J. Mazzarella et al.) at IPAC noticed
that our
MIPS
70 um total fluxes measured for about 200 luminous galaxies tend to be
15-20% fainter
than their corresponding IRAS 60 um fluxes. The flux densities of some
galaxies do fall off from 60 to 70 ums. Was that it? Calibration and
color
corrections were examined to try to understand what's happening. Here
are my notes made after reading the MIPS Data Handbook
v. 3.2 section 3.7.4, Pages 29-31 and studying the issues. First, I
give a few basic equations. Some relate to equations in the
Handbook. Second, I give the correction factors (in tables) that these
equations
imply. Third, I show some illustrative plots. Lastly, I give
derivations
of a few of the equations.
I) Basics:
The three MIPS (channel) filter responses as functions of wavelength
are found here.
As discussed (a little vaguely) in the Handbook, the function on the
website is called R, and when it is divided by lambda it is called
R_lam. The assertion is made that the function R has units of
electrons/photon, and the division by lambda turns this into electrons/energy.
I think of energy here as the input to the instrument at
some wavelength (through an aperture over a time, through some fiducial
wavelength interval) and "electrons" as something like "counts."
How do we get from the units of R to those of R_lam? E being proportional to
the number of photons (say this number is a constant, e.g. 1, at
a particular wavelength) times the
energy of a single photon.
E = n h c / lambda
So
(1/lambda ) (e-/E) = e- / (n h c) which is proportional to e-/n or
(e-/photon).
For some reason the fiducial wavelengths of the filters are calculated
weighted by R_lam not R (i.e. the assumpution of equal numbers of
photons at any wavelength.) I know with a continuum of emission this
statement makes little sense, but I press on regardless. The handbook
uses the term (lam R_lam)
in the integrals that follow. I instead use the more natural r_m (m for
MIPS) which is R, normalized. There
are three channels
and three functions listed, but I treat them here usually as one.
r_m= R / <R>
where < > hereon indicates an integration over lambda ("lam").
I use m, i, and a as subscripts indicate MIPS, IRAS, and Actual, while
r and o stand for "reported" and "observed." Flux densities are given
by f, SEDs (unnormalized flux densities as functions of lambda) as "S"
and MIPS and IRAS filter centers (effective wavelengths) as m0;
i0. Note that r_m is used whether we do calculations in frequency
or wavelength. While an SED has a different shape when plotted as
W/cm^2/um or Jy, the function r_m does not change since is represents
the counts in response to the energy found in in a um _or_
Hz bounded band.
The channels are assigned (by the MIPS Team) to effective wavelengths
m0 =
{23.67, 71.44, 155.9 um} which are obtained from:
m0=< lam R_lam > / < R_lam >
I use "f_mr" to indicate "MIPS-reported flux density," and "f_mo" to
indicate what I
call the "MIPS-observed flux density." Note that f_mr is
the flux density reported in the MIPS data products: the products of
the MIPS pipeline. But what
gets through
the instrument originally is f_mo....
As defined in the Handbook, f_mr is the monochromatic flux density that
a
10^4 K blackbody would have at wavelength m0, so as to have the total
flux seen in that
channel. What the MIPS instrument actually observed is (after
calibration)
proportional to f_mo, an average flux
density over a channel : f_mo = < f_a r_m> where f_a is the
actual source radiance
as a function of lambda
(as seen in the satellite frame) in [W/cm^2/um].
Note that this quantity is simply supposed here for the purposes of
calculation.
f_mo can be recovered from f_mr by:
<S_m r_m>
f_mo = f_mr
----------------
(1)
S_m|m0
where S_m is the MIPS fiducial SED, with an arbitrary leading
coefficient, of a 10^4 blackbody (thus, in our wavelength range, its
shape is
of a Rayleigh Jeans tail: S_m(lam)
= lam^-4, with arbitrary normalization) in units of [W/cm^2/um]. S_m|m0
is S_m evaluated at wavelength m0.
The ratio on the right in (1) is {1.04, 1.09, 1.04} for the 3 channels.
The derivation of (1) is given in
section IV.
Similarly, as discussed in the Handbook, the MIPS reported flux
density may be corrected to be the monochromatic flux density f_ar at
the MIPS
effective wavelengths for an arbitrary SED, S_a, by:
<S_m r_m>
S_a|m0
f_ar = f_a|m0 = f_mr
---------------- ----------------
(2)
S_m|m0
<S_a r_m>
S_m|m0
<S_a r_m>
K = ----------------
----------------
(2a)
<S_m r_m>
S_a|m0
This is the basis of equation 3.2 in the MIPS Data Handbook,
where K is
the inverse of rightmost two terms of (2). (In the tables in
Section II here we also give the inverse of the rightmost two
terms.) The Handbook's equation 3.2 can be derived by
assuming blackbody SEDs:
S_m = B(lam, T0), S_a = B(lam, Teff)
e.g., f_a (lam,T) = C S_a = C (1/lam^5) {
1/(exp(c h/lam k T) - 1) }
with C a constant. In understanding the Handbook's eqn. 3.2 note that
the Handbook's retains (lam R_lam)
for responsivity. K is defined and tabulated in the handbook and should
be _divided_ into
the
MIPS-reported flux density to get the S_a referenced flux density at m0.
Finally - we show how to compare MIPS data with IRAS-reported flux
densities,
f_ir, given
that the source's actual SED (SED_a) shape is known, using the IRAS
filter responses found here
(use the table's last column):
Again normalize these to unit area to obtain r_i. Also, note that
IRAS-reported flux
densities are monochromatic densities at i0 with respect to a (lam
f_lam) =
constant type SED. Thus,
S_i = 1/lam. The formula to predict IRAS-reported flux densities f_ir
from the MIPS-reported flux densities f_mr (derived
below)
is:
<S_m r_m>
<S_a r_i>
S_i|i0
f_ir = f_mr x
--------------- x
---------------- x
------------------ (3)
S_m|m0
<S_a r_m>
<S_i r_i>
There are three terms on the right above. We call these T1, T2, T3.
When we report divisors K in the style of that Handbook, in tables in
(II), we are reporting the inverse of
(T1 x T2 x T3), but when we list the terms in tables individually we
give
T1, T2, T3.
K= (T1 T2 T3
)^-1
(3a)
These three terms may be understood as follows:
T1 "backs-out " MIPS-reported flux density for a T=10^4 K source to
f_mo, to obtain the
MIPS-observed flux density weighted by the bandpass response alone. T2
applies the "coupling" ratio of the IRAS and MIPS filters to the true
source SED,
and T3 puts the result into the (lam f_lam)=cont. scale of IRAS.
II. Tables:
Table 1. This is an extension of Table 3.11 in the Handbook, giving K
for more cool blackbodies from eqn. 2a. above:
lambda T= 60 K 55
K 50 K 45 K 40 K 35 K 30 K 25
K
-------
---------------------------------------------------
23.67
1.030 1.067 1.121 1.203 1.337 1.572 2.035 3.151
71.44
0.902 0.897 0.891 0.887 0.884 0.887 0.900 0.938
155.90
0.975 0.973 0.971 0.967 0.964 0.959 0.954 0.948
Table 2. This is an extension of Table 3.11 in the Handbook,
giving K for 3 actual galaxy SEDs from eqn. 2a above.
lambda
Circinus M82 NGC 4945
--------
----------------------------
23.67
1.000 0.965 1.267
71.44
0.927 0.919 0.874
155.90
1.018 1.024 0.979
Table 3. These are the conversions K to IRAS 25, 60, 100 from MIPS 24,
70, 160 for perfect blackbodies (eqn 3a above). K is
divided into the MIPS flux density to obtain the predicted IRAS flux
density.
lambda 60 K 55
K 50 K 45 K 40 K 35 K 30 K 25
K
---------------------------------------------------------
23.67
0.95 0.91 0.87 0.80 0.73 0.65
0.56 0.46
71.44
0.79 0.83 0.89 0.96 1.05 1.16
1.31 1.52
155.90 0.29
0.31 0.33 0.37 0.42 0.50 0.62 0.85
Table 3a. K= (T1 x T2 x T3) ^-1. Here we list T1, T2, and T3. Only T2
depends on S_a.
lambda 60 K 55
K 50 K 45 K 40 K 35 K 30 K 25
K
--------------------------------------------------------
T1
--------------------------------------------------------
23.67
1.04 1.04 1.04 1.04 1.04 1.04
1.04 1.04
71.44
1.09 1.09 1.09 1.09 1.09 1.09
1.09 1.09
155.90 1.04
1.04 1.04 1.04 1.04 1.04 1.04 1.04
--------------------------------------------------------
T2
--------------------------------------------------------
23.67
1.08 1.12 1.19 1.28 1.41 1.59
1.85 2.24
71.44
1.18 1.11 1.04 0.97 0.89 0.80
0.71 0.61
155.90 3.31
3.10 2.86 2.58 2.27 1.92 1.53 1.12
--------------------------------------------------------
T3
--------------------------------------------------------
23.67
0.93 0.93 0.93 0.93 0.93 0.93
0.93 0.93
71.44
0.99 0.99 0.99 0.99 0.99 0.99
0.99 0.99
155.90 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00
--------------------------------------------------------
Table 4. These are
the conversions K given three galaxy SEDs to IRAS 25, 60, 100 from MIPS
24, 70, 160 using eqn. 3a above. K is
divided into the MIPS flux density to obtain the predicted IRAS flux
density.
lambda Circinus M82 N4945
------------------------------
23.67 1.00
1.05 0.99
71.44
0.77 0.75 1.00
155.90 0.26
0.26 0.42
Table 4a. The breakdown of Table 4 into terms T1, T2, T3,
respectively.
lambda Circinus
M82 N4945
--------------------------------
T1
--------------------------------
23.67
1.04 1.04 1.04
71.44 1.09
1.09 1.09
155.90
1.04 1.04 1.04
--------------------------------
T2
--------------------------------
23.67 1.03
0.98 1.03
71.44
1.21 1.23 0.93
155.90
3.67 3.65 2.27
--------------------------------
T3
--------------------------------
23.67 0.93
0.93 0.93
71.44
0.99 0.99 0.99
155.90
1.00 1.00 1.00
--------------------------------
III. Plots
The MIPS and IRAS filter response curves are taken from the web
links above. Blackbody curves are computed from f(lam,T) = C (1/lam)^5
{1/ ( exp(hc/lam k T) -1) }.
Galaxy SEDs (S_a) are
obtained by stitching together the LWS01 HPDP products from the ISO
archive with the SWS data
served from the IPAC IRSA Atlas server. SWS and LWS (taken in different
apertures, 70", 26")
were multiplicatively scaled to match at 40um.
Fig 1. We see that MIPS 24 is narrower and centered within IRAS 25. We
see the blackbodies are no
match for Circinus SED. (There is a rich literature on the source SED.)
We see that for the Circinus SED MIPS -> IRAS requires only small
correction in _observed_
average flux density.
Figure 2. We show the three galaxy SEDs for which we found complete
coverage (longward of 40um)
and good S/N. They are relatively flat in the bandpasses.
Figure 3. Here again simple blackbodies do not match the Circinus SED,
although over the bandpass
of IRAS 60 um, the 1/lambda IRAS SED in fact does fairly well. We see
here why T2 in the correction for MIPS to IRAS dominates - the short
side of the IRAS
bandpass sees the rising spectrum.
Figure 4. Here we see the variety of galaxy SEDs possible. NGC 4945 is
the SED most similar to a blackbody.
IV. Derivations:
Derivation of Equation (1) above:
By the definition of f_rm, the average flux density MIPS receives is:
f_mo = C0 < S_m r_m>
We don't know C0 (constant) except through the reported monochromatic
flux density
f_mr: f_mr = C0 S_m(mo)
f_mr
C0 = -----------
S_m(mo)
So
<S_m r_m>
f_mo = f_mr ---------------
S_m|m0
QED
Derivation of Equation (2) above:
<S_m r_m>
f_mo = f_mr --------------- = C1 < S_a r_m>
S_m|m0
<S_m r_m>
C1 = f_mr ----------------------
<S_a r_m> S_m|m0
note: f_a = C1 S_a (we have pinned-down our source spectrum)
f_ar = f_a|m0 = C1 S_a|m0
<S_m r_m>
S_a|m0
f_ar = f_mr x
---------------- ----------------
S_m|m0
<S_a r_m>
QED
Derivation of Equation (3) above:
f_oi = C1 <S_a r_i>
but, in terms of the IRAS SED:
f_oi = C2 <S_i r_i>
C2 = C1 <S_a r_i > / <S_i r_i>
f_mr
<S_m r_m> <S_a r_i>
C2 = ------------------------------------
S_m|m0 <S_a
r_m> <S_i r_i>
and since f_ir = C2 S_i|i0
we have
<S_m r_m>
<S_a r_i>
S_i|i0
f_ir = f_mr x
--------------- x
---------------- x
------------------
S_m|m0
<S_a r_m>
<S_i r_i>
QED