From a sky cube to RGS counts: point sources, extended sources and back-tracing

0 Setting the stage: optical path, coordinates and roll

RGS is a slitless dispersive spectrometer. The grazing-incidence mirror focuses the light; on the way to focus about half of the converging beam is intercepted by the reflection grating array (RGA), which deflects it by wavelength onto a row of nine CCDs. The rest continues to EPIC-MOS. Each detected photon carries two positions and one energy:

• Dispersion angle β (position along the dispersion direction): set by the grating equation, it carries the wavelength and is the RGS spectral channel.
• Cross-dispersion coordinate x (XDSP): perpendicular to the dispersion, it essentially keeps the sky position. This project cuts three fixed strips along it.
• CCD energy PI: the coarse energy resolution of the CCD itself (of order 0.1 keV), used only to separate diffraction orders, not to measure lines.

Two coordinate frames and the roll angle

Take the target reference point as the origin and write the sky offset as (ξ, η) in arcmin. In observation o the dispersion direction has position angle φ_o on the sky, set by the spacecraft roll. The source offsets along and across the dispersion are

A different roll rotates (Δ_D, Δ_X) for the same sky point while the source spectrum stays the same. Multi-roll fusion rests on exactly this.

Why an offset along the dispersion is a wavelength shift derived

In the grating equation

the incidence angle α follows the ray direction. A source displaced by Δ_D along the dispersion rotates the whole converging beam by the same angle, so α changes by Δ_D. The pipeline converts β to wavelength for the reference position, so the photon is recorded at

For RGS α ≈ 1.58° and the groove density is about 646 lines/mm (d ≈ 15 490 Å), so in first order κ ≈ 15 490 × 0.0275 × 2.91×10⁻⁴ ≈ 0.124 Å per arcmin. At O VII 21.6 Å, a 1′ offset equals 0.124 Å, about 3.3 eV, or a line-of-sight velocity of about 1 700 km/s. The sign depends on how the DISP axis is defined; it is taken positive below.

1 A monochromatic point source at any position

Treat one voxel of the sky cube as a monochromatic point source at (Δ_D, Δ_X) with energy E. The expected counts in observation o, strip k and dispersion channel b are

T_o is the exposure in seconds inside the good time intervals (the EXPOSURE of the PHA file) and F(E) the photon flux. The algorithm page of rgsrmfgen writes the response R as a product of two kinds of factor verified:

The cross-dispersion “acceptance” and the order-selection efficiency are

The factors are explained one by one below and then combined in the interactive bench.

1a Effective area A_m(E): travels with the photon

Mirror collecting area × fraction intercepted by the gratings × grating efficiency of order m, times the time-dependent effective-area correction (dyneffareacorr, interpolated by observation date from Chebyshev parameters in the calibration files). Over a few arcmin, as for N132D, off-axis vignetting of the mirror is small, so A can be treated as a function of E alone. The CCD quantum efficiency q enters Q separately, per read-out node, because it belongs to the CCD where the photon lands.

1b The mirror PSF: split in two

The mirror PSF ψ is two-dimensional. After the gratings its two directions play completely different roles:

• The dispersion-direction component goes into the LSF. In the documentation the narrow LSF is built as: small-angle scattering ⊗ dispersion component of the mirror PSF ⊗ (optional user angular distribution) ⊗ grating bowing ⊗ defocus ⊗ finite-band broadening ⊗ misalignment. The mirror half-energy width of about 15″ corresponds to about 0.03 Å, one of the main contributors to the first-order LSF core width (about 0.06–0.07 Å FWHM).
• The cross-dispersion component ψ_X decides which strip the photon falls into. At each dispersion channel it is multiplied by the exposure map and summed over the strip, giving the “acceptance”. The broad wings from large-angle scattering use a separate empirical arctan profile.

So there is no stand-alone 2-D PSF file that “sends photons to a place on the CCD” and can be used directly: half of it is already part of the LSF.

1c LSF / RMF: maps to β, not to PI

L_m describes how photons of energy E are spread over dispersion channels β, peaking at β_m(E, Δ_D). When the source moves along the dispersion, the whole LSF shifts in β with almost no change of shape. At 21.6 Å a FWHM of 0.062 Å is about 1.7 eV, finer than the roughly 5 eV of XRISM Resolve, provided the source is a point.

1d Exposure map ε(β, x): fixed to the detector

The exposure map comes from rgsfilter/rgsproc. It contains the live-time fraction within the good time intervals, bad pixels and columns, the gaps between CCDs and the read-out node structure, on the detector (β, x) grid. When the source moves along the dispersion the line shifts in β but a bad column does not, so the same bad column blocks different wavelengths. This is why the ARF cannot be written as a function of E alone: the ARF produced with witharffile=yes already folds in the exposure map and strip acceptance at the reference position, and is valid for that one source position only.

Bookkeeping: T_o enters once; ε is a fraction between 0 and 1. The project's earlier probe tests already count it this way.

1e PI order selection P_m(b, E): the banana

Different orders overlap at the same β: 21.6 Å in first order and 10.8 Å in second order land at the same place. Their photon energies differ by a factor of two (0.574 keV versus 1.148 keV), which the CCD can separate. The selection region in the (β, PI) plane is curved, the “banana”; P_m is the fraction of the CCD energy redistribution ρ that falls inside it. For a selected first-order spectrum the other orders have P close to, but not exactly, zero.

1f Factor table

CCF names are those present in a standard calibration set; only files whose role is unambiguous are listed, and the grating-geometry parameter file is not broken out.

2 Extended sources: a linear sum of point sources

Photon counts are linear in source brightness. An extended source is every voxel treated as the monochromatic point source of section 1, then summed:

Discretized, this is a matrix product \(\mathbf C_{o,k}=\mathbf G_{o,k}\,\mathbf s\): s holds the amplitudes of each spectral component in each voxel, and each column of G is the count spectrum of “this component at this position” folded through the response of section 1. The project's 27-path operator is G sampled at the centres of a 3×3 grid.

2a Four consequences

1. Extent along the dispersion becomes line broadening. Each voxel's line moves by κΔ_D in λ_det; summed, the line profile is the source brightness projected on the dispersion axis convolved with the point-source LSF. N132D spans about 1.5–2′, i.e. 0.19–0.25 Å, or about 5–7 eV at O VII, three to four times the point-source LSF.
2. Position and velocity are degenerate. A line-of-sight velocity v shifts a line by λv/c. In a single roll this is identical to a change of Δ_D: \(\delta\Delta_D=\lambda v/(c\,\kappa)\approx0.58'\,(v/1000\ \mathrm{km\,s^{-1}})\) at 21.6 Å. Between rolls the position shift rotates with Δ_D and the velocity shift does not, which is what separates them.
3. Mixing across the dispersion. A strip collects not only its own patch of sky but also photons from neighbouring sky leaking through the wings of ψ_X. These are the extra paths of the 27-path operator over the 9-path one.
4. Detector features are softened. Voxels at different Δ_D place the same line at different β, so a bad column blocks only part of them; in the summed spectrum it becomes a shallow dip instead of the full notch of a point source. This requires the exposure map to be applied on the detector, not shifted with the source.

2b The standard extended-source approximation and its assumptions

rgsrmfgen with withangdist=yes (and the XSPEC convolution model rgsxsrc) provides the standard treatment: a one-dimensional angular distribution p(Δ_D) along the dispersion is convolved into the narrow LSF. According to the documentation the distribution has the same shape at all energies and is rescaled to channels by order and central energy verified. This amounts to assuming

It requires one spectrum for the whole source (no velocity field, no line-ratio variation), and evaluates the cross-dispersion acceptance at the reference position. Because angdist is convolved into the LSF while the exposure map is still applied where photons land, bad columns are handled correctly. One caution: do not convolve the mirror PSF a second time. p(Δ_D) should be the intrinsic brightness of the source, since the mirror PSF is already in the point-source LSF; switch off withmirrorpsf only if p is taken from an image that already contains the PSF.

The voxel-by-voxel sum (the integral at the start of this section) makes none of these assumptions: every voxel can have its own spectrum, velocity and line ratios, and the cross-dispersion acceptance follows its true position. The price is that G gains a column per voxel and component, and each column needs one point-source response.

3 Looking back from the CCD: where do the photons of this channel come from?

Read the sum of section 2 backwards. For observation o, strip k and channel b, the fraction contributed by sky position (ξ, η) and energy E is

For a line of rest wavelength λ_ℓ, only sky positions that satisfy

can send photons into channel λ_b. On the sky this is a narrow band perpendicular to the dispersion direction, about 0.5′ wide (0.062 Å / 0.124 Å per arcmin). Each line has its own band; the bands are parallel and offset by (λ_i−λ_r)/κ ≈ 1.65′ and (λ_f−λ_r)/κ ≈ 4.0′. Across the dispersion the bands are cut by the strip (plus the wings of ψ_X). A velocity field bends the bands, because v changes with position. The continuum comes from all positions, each contributing its continuum at λ_b−κΔ_D.

Three more sources of counts lie outside the sky cube and need separate bookkeeping:

• Other diffraction orders: second- and third-order photons at the same β, mostly removed by the PI selection, with the remainder counted through P_m.
• Large-angle scattering: broad wings scatter photons from distant wavelengths in, a low-level floor.
• Background: particle and soft-proton events that do not come from the sky cube and need their own detector-space model.

The bench below draws two rolls for the same channel. The bands have different directions in the two rolls; only where they intersect are the position and velocity consistent with both observations. That is the geometric meaning of multi-roll fusion.

4 Back to the project

What the current 27-path operator does

The 27-path operator of the reference observation 0210681301 samples R_o,k at the nine centres of a common 3×3 sky grid: the source position given to rgsrmfgen is moved through a private source list while the three extraction strips stay fixed, giving 9 positions × 3 strips verified (accepted probes of the reference observation). It is not a “point sources only” approximation. Building an extended source from point-source responses is exactly the linear sum; the limits are that each cell is represented by its centre (within-cell integration is not yet validated) and that every position needs its own rgsrmfgen run (about one minute and about 29 MB per file on our machine). With several rolls, the same nine sky cells map through each observation's φ_o to their own (Δ_D, Δ_X): 3 observations × 3 strips × 9 cells = 81 paths (not yet run).

Refining the cube without paying N times

The factor table of section 1 points to a shortcut: only A·L moves with the source, while ε, q, the ψ_X acceptance and P stay on the detector. In principle, then, no new response is needed along Δ_D: shift the LSF and multiply by the detector factors; only a few positions along Δ_X are needed. This rests on three acceptance tests proposed, not yet run:

1. At fixed Δ_X and several Δ_D, compare “shift plus detector factors” with direct rgsrmfgen output, and measure the shift invariance and the deviations at bad columns and gaps.
2. The relative L1 error at held-out positions not used to build the emulator, against the threshold of 0.001 used in the earlier probe tests.
3. The standard extended-source response from withangdist, compared with the emulator's integral in the single-spectrum case.

Once these pass, a cube of any resolution and any roll can use the same emulator for mocks and fits. Reproducing the pipeline does not remove calibration uncertainty: the effective area (a few per cent), the LSF wings and, above all, the pointing and wavelength zero point of each observation remain nuisance parameters. The last matters most for velocities: a 1″ pointing error is about 2 mÅ, or about 30 km/s.

What “XRISM level” means here

At O VII 21.6 Å the first-order RGS LSF for a point source is about 1.7 eV FWHM, already finer than the roughly 5 eV of Resolve. The 1.5–2′ extent of N132D broadens it to about 5–7 eV, the Resolve scale, and within one roll position and velocity are degenerate. The aim of multi-roll fusion is to use the fact that the position term rotates between rolls while the velocity term does not, to decompose the extended-source broadening back into a spatial distribution and a velocity field and approach point-source resolution. Also, to our knowledge the Resolve gate valve has not been opened, so Resolve has almost no effective area below about 1.7 keV and the O VII band still relies on RGS.

What on this page is established