Stellar Continuum Spectrum

In the current version, the stellar continuum spectrum is modeled using the spec1d.StellarContinuum class. This module supports multiple methods for constructing the intrinsic spectral shape, and two distinct modes for flux calibration.

Overview

The stellar spectrum is constructed from single stellar population (SSP) templates. We adopt the E-MILES library (Vazdekis et al. 2016), which provides a broad wavelength range from \(1680\ \mathring{\mathrm{A}}\) to \(50000\ \mathring{\mathrm{A}}\), and dense sampling in age and metallicity.

Two options for constructing the spectral shape:

  • Composite Stellar Population (CSP): Assemble from a star formation history (SFH) and chemical enrichment history (CEH).

  • Single SSP: Directly use one SSP template with given age and metallicity.

Two modes for flux calibration:

  • Magnitude-calibrated mode: Use SDSS-r magnitude (mag) to normalize output spectrum (required for single SSP).

  • SFR-calibrated mode: Use physically normalized SFH to produce absolute flux (only available for CSP).

Spectral Construction

In CSP mode, the stellar continuum is built from a linear combination of SSP templates:

\[S(\lambda) = \sum_i w_i \cdot \mathrm{SSP}_i(\lambda)\]

where \(w_i\) is the weight from SFH at lookback time \(t_i\), and \(\mathrm{SSP}_i(\lambda)\) is the template spectrum for that age and metallicity.

The input SFH and CEH must be arrays with lookback time grids:

  • SFH input: array of shape (nage, 2), with columns = [lookback time, relative SFR].

  • CEH input: array of shape (nage, 2), with columns = [lookback time, [Fe/H]].

Only the relative SFR distribution is used in magnitude-calibrated mode; the absolute mass normalization is applied in SFR-calibrated mode.

In single SSP mode, the user provides age and feh, and the closest matching template is selected.

Velocity Dispersion Broadening

The E-MILES templates include intrinsic instrumental broadening, which varies with wavelength:

  • UV band (\(\lambda < 3541\ \mathring{\mathrm{A}}\)): \(\sigma \approx 200\ \mathrm{km\ s^{-1}}\)

  • Optical/NIR bands: \(\sigma < 100\ \mathrm{km\ s^{-1}}\)

If the input stellar velocity dispersion (vdisp) exceeds the intrinsic value, additional broadening is applied using Gaussian convolution in log-wavelength space, following PPXF.

Dust Attenuation

Dust attenuation is applied using the Calzetti et al. (2000) law:

\[S_{\rm dust}(\lambda) = S_{\rm nodust}(\lambda) \cdot 10^{-0.4 \cdot E(B-V) \cdot k(\lambda)}\]

where \(k(\lambda)\) is the attenuation curve and \(E(B-V)\) is the reddening input parameter (ebv).

Redshift and Peculiar Velocity

The observed-frame wavelength is computed using the cosmological redshift (z) and the line-of-sight peculiar velocity (vpec), as:

\[v_{\rm los} = (1 + z)\cdot v_{\rm pec} + z \cdot c\]
\[\lambda_{\rm obs} = \lambda_{\rm rest} \cdot \left(1 + \frac{v_{\rm los}}{c}\right)\]

This relativistic correction is consistent with the treatment used for gas emission lines.

Flux Calibration

Two options are supported:

  • Magnitude-calibrated mode:

    • Input: SDSS-r apparent magnitude (mag)

    • Output flux is scaled to match mag after redshift and attenuation.

    • Available for both single SSP and CSP inputs.

  • SFR-calibrated mode:

    • Input: normalized SFH + z

    • Output reflects physically motivated absolute flux assuming total stellar mass.

    • Only available when using SFH + CEH (not for single SSP).

Output

The final output is a 1D NumPy array representing the observed-frame spectrum, with:

  • Spectral line broadening,

  • Dust attenuation,

  • Redshift and peculiar velocity correction,

  • Proper flux normalization based on the chosen calibration mode.

Unit: \(10^{-17}\ \mathrm{erg\ s^{-1}\ cm^{-2}\ \mathring{A}^{-1}}\)