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.

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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

\[S(y) \equiv \int_{\lambda_1}^{\lambda_2} y\ d\lambda \approx \sum_i y_i\ {\rm d}p_i\ {\rm d}\lambda_i,\]

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”):

\[C(\lambda) = (\langle f\rangle_{\rm red} - \langle f\rangle_{\rm blue})\ \frac{\lambda - \lambda_{\rm blue}}{\lambda_{\rm red}-\lambda_{\rm blue}} + \langle f\rangle_{\rm blue},\]

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

\[\langle y\rangle = S(y)/S(1).\]

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:

\[\begin{split}{\mathcal I}_{\rm WT} = \left\{ \begin{array}{ll} S(1 - f/C), & \mbox{for angstrom units} \\[3pt] -2.5\log\left[\langle f/C\rangle\right], & \mbox{for magnitude units} \end{array}\right..\end{split}\]

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:

\[\begin{split}{\mathcal I}_{\rm BF} = \left\{ \begin{array}{ll} S(1) - S(f)/C_0, & \mbox{for angstrom units} \\[3pt] -2.5\log\left[\langle f\rangle/C_0\right], & \mbox{for magnitude units} \end{array}\right.,\end{split}\]

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:

\[\begin{split}\begin{eqnarray} \frac{\sum_i C_{0,i} {\mathcal I}_{\rm BF}}{\sum_i C_{0,i}} & = & \frac{\sum_i C_{0,i} (\Delta\lambda - S(f)_i / C_{0,i})}{\sum_i C_{0,i}} \\ & = & \Delta\lambda - \frac{\sum_i S(f)_i}{\sum_i C_{0,i}} \end{eqnarray}.\end{split}\]

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:

\[\begin{split}\begin{eqnarray} -2.5\log\left[\frac{\sum_i C_{0,i} 10^{-0.4 {\mathcal I}_{\rm BF}}}{\sum_i C_{0,i}}\right] & = & -2.5\log\left[\frac{\sum_i C_{0,i} (S(f)_i / C_{0,i})} {\Delta\lambda \sum_i C_{0,i}}\right] \\ & = & -2.5\log\left[\frac{\sum_i S(f)_i}{\Delta\lambda \sum_i C_{0,i}}\right] \end{eqnarray}.\end{split}\]

Given the ease with which one can combine indices in the latter definition, the DAP calculates both \({\mathcal I}_{\rm WT}\) and \({\mathcal I}_{\rm BF}\).

See AbsorptionLineIndices.

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	[1]
CN2	4142.125 – 4177.125	4083.875 – 4096.375	4244.125 – 4284.125	air	mag	[1]
Ca4227	4222.250 – 4234.750	4211.000 – 4219.750	4241.000 – 4251.000	air	Å	[1]
G4300	4281.375 – 4316.375	4266.375 – 4282.625	4318.875 – 4335.125	air	Å	[1]
Fe4383	4369.125 – 4420.375	4359.125 – 4370.375	4442.875 – 4455.375	air	Å	[1]
Ca4455	4452.125 – 4474.625	4445.875 – 4454.625	4477.125 – 4492.125	air	Å	[1]
Fe4531	4514.250 – 4559.250	4504.250 – 4514.250	4560.500 – 4579.250	air	Å	[1]
C24668	4634.000 – 4720.250	4611.500 – 4630.250	4742.750 – 4756.500	air	Å	[1]
Hb	4847.875 – 4876.625	4827.875 – 4847.875	4876.625 – 4891.625	air	Å	[1]
Fe5015	4977.750 – 5054.000	4946.500 – 4977.750	5054.000 – 5065.250	air	Å	[1]
Mg1	5069.125 – 5134.125	4895.125 – 4957.625	5301.125 – 5366.125	air	mag	[1]
Mg2	5154.125 – 5196.625	4895.125 – 4957.625	5301.125 – 5366.125	air	mag	[1]
Mgb	5160.125 – 5192.625	5142.625 – 5161.375	5191.375 – 5206.375	air	Å	[1]
Fe5270	5245.650 – 5285.650	5233.150 – 5248.150	5285.650 – 5318.150	air	Å	[1]
Fe5335	5312.125 – 5352.125	5304.625 – 5315.875	5353.375 – 5363.375	air	Å	[1]
Fe5406	5387.500 – 5415.000	5376.250 – 5387.500	5415.000 – 5425.000	air	Å	[1]
Fe5709	5696.625 – 5720.375	5672.875 – 5696.625	5722.875 – 5736.625	air	Å	[1]
Fe5782	5776.625 – 5796.625	5765.375 – 5775.375	5797.875 – 5811.625	air	Å	[1]
NaD	5876.875 – 5909.375	5860.625 – 5875.625	5922.125 – 5948.125	air	Å	[1]
TiO1	5936.625 – 5994.125	5816.625 – 5849.125	6038.625 – 6103.625	air	mag	[1]
TiO2	6189.625 – 6272.125	6066.625 – 6141.625	6372.625 – 6415.125	air	mag	[1]
HDeltaA	4083.500 – 4122.250	4041.600 – 4079.750	4128.500 – 4161.000	air	Å	[2]
HGammaA	4319.750 – 4363.500	4283.500 – 4319.750	4367.250 – 4419.750	air	Å	[2]
HDeltaF	4091.000 – 4112.250	4057.250 – 4088.500	4114.750 – 4137.250	air	Å	[2]
HGammaF	4331.250 – 4352.250	4283.500 – 4319.750	4354.750 – 4384.750	air	Å	[2]
CaHK	3899.5 – 4003.5	3806.5 – 3833.8	4020.7 – 4052.4	air	Å	[3]
CaII1	8484.0 – 8513.0	8474.0 – 8484.0	8563.0 – 8577.0	air	Å	[4]
CaII2	8522.0 – 8562.0	8474.0 – 8484.0	8563.0 – 8577.0	air	Å	[4]
CaII3	8642.0 – 8682.0	8619.0 – 8642.0	8700.0 – 8725.0	air	Å	[4]
Pa17	8461.0 – 8474.0	8474.0 – 8484.0	8563.0 – 8577.0	air	Å	[4]
Pa14	8577.0 – 8619.0	8563.0 – 8577.0	8619.0 – 8642.0	air	Å	[4]
Pa12	8730.0 – 8772.0	8700.0 – 8725.0	8776.0 – 8792.0	air	Å	[4]
MgICvD	5165.0 – 5220.0	5125.0 – 5165.0	5220.0 – 5260.0	vac	Å	[5]
NaICvD	8177.0 – 8205.0	8170.0 – 8177.0	8205.0 – 8215.0	vac	Å	[5]
MgIIR	8801.9 – 8816.9	8777.4 – 8789.4	8847.4 – 8857.4	vac	Å	[5]
FeHCvD	9905.0 – 9935.0	9855.0 – 9880.0	9940.0 – 9970.0	vac	Å	[5]
NaI	8168.500 – 8234.125	8150.000 – 8168.400	8235.250 – 8250.000	air	Å	[6]
bTiO	4758.500 – 4800.000	4742.750 – 4756.500	4827.875 – 4847.875	air	mag	[7]
aTiO	5445.000 – 5600.000	5420.000 – 5442.000	5630.000 – 5655.000	air	mag	[7]
CaH1	6357.500 – 6401.750	6342.125 – 6356.500	6408.500 – 6429.750	air	mag	[7]
CaH2	6775.000 – 6900.000	6510.000 – 6539.250	7017.000 – 7064.000	air	mag	[7]
NaISDSS	8180.0 – 8200.0	8143.0 – 8153.0	8233.0 – 8244.0	air	Å	[8]
TiO2SDSS	6189.625 – 6272.125	6066.625 – 6141.625	6422.0 – 6455.0	air	mag	[8]

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
index	int	Unique integer identifier of the absorption-line index. Must be unique.
name	str	Name of the index. Must be unique.
primary	float[2]	A two-element vector with the starting and ending wavelength for the primary passband surrounding the absorption feature(s).
blueside	float[2]	A two-element vector with the starting and ending wavelength for a passband to the blue of the primary band.
redside	float[2]	A two-element vector with the starting and ending wavelength for a passband to the red of the primary band.
waveref	str	The reference frame of the wavelengths; must be either ‘air’ for air or ‘vac’ for vacuum.
units	str	Units for the absorption index, which must be either ‘ang’ or ‘mag’.
component	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'

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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:

\[{\mathcal I} = \frac{\langle f\rangle_0}{\langle f\rangle_1},\]

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:

\[\frac{\sum_i \langle f_i\rangle_1 {\mathcal I}_i}{\sum_i \langle f_i\rangle_1} = \frac{\sum_i \langle f_i\rangle_0} {\sum_i \langle f_i\rangle_1} = \frac{\langle\sum_i f_i\rangle_0} {\langle\sum_i f_i\rangle_1}\]

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	[9]
Dn4000	3850 – 3950	4000 – 4100	air	\(F_\nu\)	R/B	[10]
TiOCvD	8835 – 8855	8870 – 8890	vac	\(F_\lambda\)	B/R	[5]

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
index	int	Unique integer identifier of the absorption-line index. Must be unique.
name	str	Name of the index. Must be unique.
blueside	float[2]	A two-element vector with the starting and ending wavelength for a passband to the blue of the primary band.
redside	float[2]	A two-element vector with the starting and ending wavelength for a passband to the red of the primary band.
waveref	str	The reference frame of the wavelengths; must be either ‘air’ for air or ‘vac’ for vacuum.
integrand	str	Integrand within the passband for the construction of the index, which must be either ‘fnu’ or ‘flambda’.
order	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'

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[1] (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)

Trager et al. (1998, ApJS, 116, 1)

[2] (1,2,3,4)

Worthey & Ottaviani (1997, ApJS, 111, 377)

[3]

Serven et al. (2005, ApJ, 627, 754)

[4] (1,2,3,4,5,6)

Cenarro et al. (2001, MNRAS, 326, 959); however, note that each index is considered seperately, which is not the definition provided in Table 4 of their paper.

[5] (1,2,3,4,5)

Conroy & van Dokkum (2012, ApJ, 747, 69)

[6]

Spiniello et al. (2012, ApJL, 753, 32)

[7] (1,2,3,4)

Spiniello et al. (2014, MNRAS, 438, 1483)

[8] (1,2)

La Barbera et al. (2013, MNRAS, 433, 3017)

[9]

Bruzual (1983, ApJ, 273, 105)

[10]

Balogh et al. (1999, ApJ, 527, 54)