Spectral Indices¶
The spectral indices to be measured by the DAP are divided into two
groups: (1) absorption-line indices that measure the equivalent width
of an absorption feature and (2) bandhead or color indices that
measure the ratio of fluxes is in two passbands. Both sets of indices
are measured using
SpectralIndices
; see
Spectral-Index Measurements.
Absorption-line Indices¶
Calculation¶
The absorption-line index calculations are performed as defined/used by Worthey et al. (1994) and Trager et al. (1998) (\({\mathcal I}_{\rm WT}\)), as well as defined/used by Burstein et al. (1984) and Faber et al. (1985) (\({\mathcal I}_{\rm BF}\)).
Specifically, let
where \({\rm d}p_i\) is the fraction of pixel \(i\) (with
width \({\rm d}\lambda_i\)) in the passband defined by
\(\lambda_1 < \lambda < \lambda_2\); the discrete sum is
performed by passband_integral()
.
Also, let \(f\) be the spectrum flux density and define a linear
continuum using two sidebands (“blue” and “red”):
where \(\lambda_{\rm blue}\) and \(\lambda_{\rm red}\) are the wavelengths at the center of the two sidebands — e.g., \(\lambda_{\rm red} = (\lambda_{1,{\rm red}} + \lambda_{2,{\rm red}})/2\) — and
When no pixels are masked in the spectrum, \(S(1) = (\lambda_2 - \lambda_1) \equiv \Delta\lambda\).
Following the Worthey et al. (1994) definition, we can then calculate the absorption-line spectral indices as follows:
The difficulty with the Worthey et al. definitions is that it makes it difficult to construct an aggregate index measurement based on individual index measurements from multiple spectra; however, this is straight-forward under definitions closer to those provided by Burstein et al. (1984) and Faber et al. (1985).
Let:
where \(C_0\) is the value of the linear (not necessarily “flat”) continuum at the center of the main passband. Given that the continuum is a linear function, \(S(C) = C_0 \Delta\lambda\); i.e., the integral of the continuum over the passband is mathematically identical to the continuum sampled at the center of the bandpass times the bandpass width.
Now, we can calculate a weighted sum of indices using the value of \(C_0\) for each index as the weight and assume that no pixels are masked such that we can replace \(S(1)\) with \(\Delta\lambda\) to find:
That is, this weighted sum of the individual indices is mathematically identical (to within the limits of how error affects the construction of the linear continuum) to the index measured for the sum (or mean) of the individual spectra. Similarly for the indices in magnitude units:
Given the ease with which one can combine indices in the latter definition, the DAP calculates both \({\mathcal I}_{\rm WT}\) and \({\mathcal I}_{\rm BF}\).
Index Parameters¶
The table below provides a compilation of absorption-line indices. Recent survey-level runs of the DAP have included all of these measurements.
Name |
Main Passband (Å) |
Blue Sideband (Å) |
Red Sideband (Å) |
Frame |
Units |
Ref |
---|---|---|---|---|---|---|
CN1 |
4142.125 – 4177.125 |
4080.125 – 4117.625 |
4244.125 – 4284.125 |
air |
mag |
|
CN2 |
4142.125 – 4177.125 |
4083.875 – 4096.375 |
4244.125 – 4284.125 |
air |
mag |
|
Ca4227 |
4222.250 – 4234.750 |
4211.000 – 4219.750 |
4241.000 – 4251.000 |
air |
Å |
|
G4300 |
4281.375 – 4316.375 |
4266.375 – 4282.625 |
4318.875 – 4335.125 |
air |
Å |
|
Fe4383 |
4369.125 – 4420.375 |
4359.125 – 4370.375 |
4442.875 – 4455.375 |
air |
Å |
|
Ca4455 |
4452.125 – 4474.625 |
4445.875 – 4454.625 |
4477.125 – 4492.125 |
air |
Å |
|
Fe4531 |
4514.250 – 4559.250 |
4504.250 – 4514.250 |
4560.500 – 4579.250 |
air |
Å |
|
C24668 |
4634.000 – 4720.250 |
4611.500 – 4630.250 |
4742.750 – 4756.500 |
air |
Å |
|
Hb |
4847.875 – 4876.625 |
4827.875 – 4847.875 |
4876.625 – 4891.625 |
air |
Å |
|
Fe5015 |
4977.750 – 5054.000 |
4946.500 – 4977.750 |
5054.000 – 5065.250 |
air |
Å |
|
Mg1 |
5069.125 – 5134.125 |
4895.125 – 4957.625 |
5301.125 – 5366.125 |
air |
mag |
|
Mg2 |
5154.125 – 5196.625 |
4895.125 – 4957.625 |
5301.125 – 5366.125 |
air |
mag |
|
Mgb |
5160.125 – 5192.625 |
5142.625 – 5161.375 |
5191.375 – 5206.375 |
air |
Å |
|
Fe5270 |
5245.650 – 5285.650 |
5233.150 – 5248.150 |
5285.650 – 5318.150 |
air |
Å |
|
Fe5335 |
5312.125 – 5352.125 |
5304.625 – 5315.875 |
5353.375 – 5363.375 |
air |
Å |
|
Fe5406 |
5387.500 – 5415.000 |
5376.250 – 5387.500 |
5415.000 – 5425.000 |
air |
Å |
|
Fe5709 |
5696.625 – 5720.375 |
5672.875 – 5696.625 |
5722.875 – 5736.625 |
air |
Å |
|
Fe5782 |
5776.625 – 5796.625 |
5765.375 – 5775.375 |
5797.875 – 5811.625 |
air |
Å |
|
NaD |
5876.875 – 5909.375 |
5860.625 – 5875.625 |
5922.125 – 5948.125 |
air |
Å |
|
TiO1 |
5936.625 – 5994.125 |
5816.625 – 5849.125 |
6038.625 – 6103.625 |
air |
mag |
|
TiO2 |
6189.625 – 6272.125 |
6066.625 – 6141.625 |
6372.625 – 6415.125 |
air |
mag |
|
HDeltaA |
4083.500 – 4122.250 |
4041.600 – 4079.750 |
4128.500 – 4161.000 |
air |
Å |
|
HGammaA |
4319.750 – 4363.500 |
4283.500 – 4319.750 |
4367.250 – 4419.750 |
air |
Å |
|
HDeltaF |
4091.000 – 4112.250 |
4057.250 – 4088.500 |
4114.750 – 4137.250 |
air |
Å |
|
HGammaF |
4331.250 – 4352.250 |
4283.500 – 4319.750 |
4354.750 – 4384.750 |
air |
Å |
|
CaHK |
3899.5 – 4003.5 |
3806.5 – 3833.8 |
4020.7 – 4052.4 |
air |
Å |
|
CaII1 |
8484.0 – 8513.0 |
8474.0 – 8484.0 |
8563.0 – 8577.0 |
air |
Å |
|
CaII2 |
8522.0 – 8562.0 |
8474.0 – 8484.0 |
8563.0 – 8577.0 |
air |
Å |
|
CaII3 |
8642.0 – 8682.0 |
8619.0 – 8642.0 |
8700.0 – 8725.0 |
air |
Å |
|
Pa17 |
8461.0 – 8474.0 |
8474.0 – 8484.0 |
8563.0 – 8577.0 |
air |
Å |
|
Pa14 |
8577.0 – 8619.0 |
8563.0 – 8577.0 |
8619.0 – 8642.0 |
air |
Å |
|
Pa12 |
8730.0 – 8772.0 |
8700.0 – 8725.0 |
8776.0 – 8792.0 |
air |
Å |
|
MgICvD |
5165.0 – 5220.0 |
5125.0 – 5165.0 |
5220.0 – 5260.0 |
vac |
Å |
|
NaICvD |
8177.0 – 8205.0 |
8170.0 – 8177.0 |
8205.0 – 8215.0 |
vac |
Å |
|
MgIIR |
8801.9 – 8816.9 |
8777.4 – 8789.4 |
8847.4 – 8857.4 |
vac |
Å |
|
FeHCvD |
9905.0 – 9935.0 |
9855.0 – 9880.0 |
9940.0 – 9970.0 |
vac |
Å |
|
NaI |
8168.500 – 8234.125 |
8150.000 – 8168.400 |
8235.250 – 8250.000 |
air |
Å |
|
bTiO |
4758.500 – 4800.000 |
4742.750 – 4756.500 |
4827.875 – 4847.875 |
air |
mag |
|
aTiO |
5445.000 – 5600.000 |
5420.000 – 5442.000 |
5630.000 – 5655.000 |
air |
mag |
|
CaH1 |
6357.500 – 6401.750 |
6342.125 – 6356.500 |
6408.500 – 6429.750 |
air |
mag |
|
CaH2 |
6775.000 – 6900.000 |
6510.000 – 6539.250 |
7017.000 – 7064.000 |
air |
mag |
|
NaISDSS |
8180.0 – 8200.0 |
8143.0 – 8153.0 |
8233.0 – 8244.0 |
air |
Å |
|
TiO2SDSS |
6189.625 – 6272.125 |
6066.625 – 6141.625 |
6422.0 – 6455.0 |
air |
mag |
Input Data Format¶
The parameters that define the absorption-line-index calculations are
provided via the
AbsorptionIndexDB
object,
which is built using an SDSS-style parameter file. The core level
class that calculates the raw absorption-line indices is
AbsorptionLineIndices
.
The columns of the parameter file are:
Parameter |
Format |
Description |
---|---|---|
|
int |
Unique integer identifier of the absorption-line index. Must be unique. |
|
str |
Name of the index. Must be unique. |
|
float[2] |
A two-element vector with the starting and ending wavelength for the primary passband surrounding the absorption feature(s). |
|
float[2] |
A two-element vector with the starting and ending wavelength for a passband to the blue of the primary band. |
|
float[2] |
A two-element vector with the starting and ending wavelength for a passband to the red of the primary band. |
|
str |
The reference frame of the wavelengths; must be either ‘air’ for air or ‘vac’ for vacuum. |
|
str |
Units for the absorption index, which must be either ‘ang’ or ‘mag’. |
|
int |
Never used: Binary flag (0-false,1-true) that the index is a component of a composite index. If true (1), all components with the same NAME are added together to form the composite index. |
and an example file might look like this:
typedef struct {
int index;
char name[9];
double primary[2];
double blueside[2];
double redside[2];
char waveref[3];
char units[3];
int component;
} DAPABI;
DAPABI 1 CN1 { 4142.125 4177.125 } { 4080.125 4117.625 } { 4244.125 4284.125 } air mag 0
DAPABI 2 CN2 { 4142.125 4177.125 } { 4083.875 4096.375 } { 4244.125 4284.125 } air mag 0
Note that the functionality implied by the component
parameter has
been a notional future development for the module, but has never been
implemented. However, unfortunately, it’s still a required element of
the database file.
Changing the absorption-line index parameters¶
The absorption-line indices are measured by
SpectralIndices
; see
Spectral-Index Measurements. A set of parameter files that
define a list of absorption-line index sets are provided with the DAP
source distribution and located at
$MANGADAP_DIR/mangadap/data/absorption_indices
. There are a few
methods that you can use to change the set of absorption-line index parameters
used by SpectralIndices
:
To use one of the existing parameter databases, you can change the
absorption_indices
keyword in theSpectralIndices
configuration file. The keyword should be the capitalized root of the parameter filename. E.g., to use$MANGADAP_DIR/mangadap/data/absorption_indices/lickindx.par
, set the keyword toLICKINDX
.To use a new parameter database, write the file and save it in the
$MANGADAP_DIR/mangadap/data/absorption_indices/
directory, and then change the relevant configuration file in the same way as described above.
Bandhead or Color Indices¶
Calculation¶
Bandhead, or color, indices simply measure the ratio of fluxes in two sidebands. Following the nomenclature defined for the Absorption-line Indices, a color index is:
where \(\langle y\rangle = S(y)/S(1)\) and the two sidebands are denoted with subscripts 0 and 1. The “order” of the index selects if the index is calculated as the red-to-blue flux ratio or the blue-to-red flux ratio (e.g., D4000 is defined as a red-to-blue index, whereas TiOCvD is defined as a blue-to-red index).
Given the simple definition of color indices, a combined index for the sum (or mean) of multiple spectra can be calculated by constructing the weighted-mean index, where the continuum in the denominator is used as the weight:
See BandheadIndices
.
Index Parameters¶
The table below provides a compilation of bandhead and color indices. Recent survey-level runs of the DAP have included all of these measurements.
Name |
Blue Sideband (Å) |
Red Sideband (Å) |
Frame |
Integrand |
Order |
Ref |
---|---|---|---|---|---|---|
D4000 |
3750 – 3950 |
4050 – 4250 |
air |
\(F_\nu\) |
R/B |
|
Dn4000 |
3850 – 3950 |
4000 – 4100 |
air |
\(F_\nu\) |
R/B |
|
TiOCvD |
8835 – 8855 |
8870 – 8890 |
vac |
\(F_\lambda\) |
B/R |
Input Data Format¶
The parameters that define the bandhead index calculations are provided
via the BandheadIndexDB
object,
which is built using an SDSS-style parameter file. The core level
class that calculates the raw bandhead indices is
BandheadIndices
.
The columns of the parameter file are:
Parameter |
Format |
Description |
---|---|---|
|
int |
Unique integer identifier of the absorption-line index. Must be unique. |
|
str |
Name of the index. Must be unique. |
|
float[2] |
A two-element vector with the starting and ending wavelength for a passband to the blue of the primary band. |
|
float[2] |
A two-element vector with the starting and ending wavelength for a passband to the red of the primary band. |
|
str |
The reference frame of the wavelengths; must be either ‘air’ for air or ‘vac’ for vacuum. |
|
str |
Integrand within the passband for the construction of the index, which must be either ‘fnu’ or ‘flambda’. |
|
str |
Define the order to use when constructing the index. The options are either a ratio of red-to-blue or blue-to-red, which are respectively selected using ‘r_b’ or ‘b_r’. |
and an example file might look like this:
typedef struct {
int index;
char name[9];
double blueside[2];
double redside[2];
char waveref[3];
char integrand[7];
char order[3];
} DAPBHI;
DAPBHI 1 D4000 { 3750.000 3950.000 } { 4050.000 4250.000 } air fnu r_b
DAPBHI 2 Dn4000 { 3850.000 3950.000 } { 4000.000 4100.000 } air fnu r_b
DAPBHI 3 TiOCvD { 8835.000 8855.000 } { 8870.000 8890.000 } vac flambda b_r
Changing the bandhead index parameters¶
The bandhead and color indices are measured by
SpectralIndices
; see
Spectral-Index Measurements. A set of parameter files that
define a list of bandhead index sets are provided with the DAP source
distribution and located at
$MANGADAP_DIR/mangadap/data/bandhead_indices
. There are a few
methods that you can use to change the set of bandhead-index
parameters used by
SpectralIndices
:
To use one of the existing parameter databases, you can change the
bandhead_indices
keyword in theSpectralIndices
configuration file. The keyword should be the capitalized root of the parameter filename. E.g., to use$MANGADAP_DIR/mangadap/data/bandhead_indices/bhbasic.par
, set the keyword toBHBASIC
.To use a new parameter database, write the file and save it in the
$MANGADAP_DIR/mangadap/data/bandhead_indices/
directory, and then change the relevant configuration file in the same way as described above.