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 (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 (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
. The database you
wish to use is selected by the absindex
parameter in the relevant parameter
block of the Analysis Plans file. The keyword is simply the capitalized name of
the file without the “.par” extension. For example, to use the extindx.par
database, the plan file would include
[default.indices]
absindex = 'EXTINDX'
To provide a user-defined database, simply replace the absindex
keyword
with the name of the local file defining the database (in the format given
above). For example,
[default.indices]
absindex = '/path/to/my/local/file/my_abs_database.par'
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
. The database you wish to
use is selected by the bandhead
parameter in the relevant parameter block of
the Analysis Plans file. The keyword is simply the capitalized name of the file
without the “.par” extension. For example, to use the bhbasic.par
database, the plan file would include
[default.indices]
bandhead = 'BHBASIC'
To provide a user-defined database, simply replace the bandhead
keyword
with the name of the local file defining the database (in the format given
above). For example,
[default.indices]
bandhead = '/path/to/my/local/file/my_bhd_database.par'