THE ANNOTATED GRATINGS
2nd X-Ray Astronomy School, Berkeley Springs, WV
2002 August 19
Antonella Fruscione & Vinay Kashyap
(presented by VK)
Overview
- What is a grating?
[definition slide]
- Why use gratings?
[low-res Capella in ASCA/SIS v/s high-res Capella in HRC-S/LETG showing emission line spectra]
[low-res NGC 5548 in BeppoSAX]
[hi-res NGC 5548 in HRC-S/LETG showing absorption lines]
- How do gratings work?
- In Theory
[DISPERSION viewgraph]
- Dispersion
- Rowland Circle
- Resolving Power
- In Practice
- Current X-ray Gratings
[list of gratings on recent non-solar missions]
[photo of LETGS]
[display LETG grating facet, which is made of closely placed wires, and disperses red laser light by approx 45 deg]
* long wavelengths are dispersed to farther distances
* the points close by are due to cross-dispersion; the main first-order is displaced much farther out
[effective areas comparison]
[resolving powers comparison]
- Order Separation
[Capella ACIS-S/HETG order separation slide]
[HRC-S/LETG RSP for orders +1..+6, shown for photon of energy 0.655 keV]
* in summary: can do with CCD, but if you can't, use RSPs
- Wavelength scale
[Rowland circle v/s HRC-S plate locations]
- Line Spread Function
[LINE SPREAD FUNCTION viewgraph]
- Flux Spectra (when to and when not to)
[TO ``FLUX'' OR NOT TO ``FLUX'' A GRATING SPECTRUM?]
[EXAMPLES OF FLUXED SPECTRA: NGC 4151 AND NGC 4050]
- Response Matrices and Fitting Issues
[STRATEGIES FOR ANALYSIS OF GRATING DATA]
- Examples
[Seyfert NGC 5548 (the one with the absorption systems) from Kaastra et al.]
[Seyfert NGC 4151 (emission lines from outer regions superposed on strongly absorbed power-law spectrum from core) from Ogle et al.]
[HETG observation of Capella from Canizares et al.]
[HETG image of E0102-72]
[ACIS-S/LETG image of XTE J1118+48]
[LETG data of AD Leo compared with model spectra]
DISPERSION
- The grating equation relates the wavelength of dispersion {\lambda} at any given order {m} to the grating period {d} and the angles of incidence ({\theta_0}) and dispersion ({\theta}):
{m\lambda = d(\sin{\theta}-\sin{\theta_0})}
[Dispersion ray diagram]
- The grating focus is on the Rowland circle of diameter {D}=2{R}, so for a detector shaped to follow the Rowland circle, photons of wavelength {\lambda} come to a focus at a distance
{x = D(\theta-\theta_0) = D \Delta\theta \approx m\lambda(D/d)}
[Rowland circle geometry showing origin of design based on concave gratings that bring dispersed photons to a focus on a circle of radius half the grating radius of curvature]
* the proof of the equvivalence of the angles subtending the same arc of the inscribed circle is given in Figure 1
* same principle holds for transmission gratings; just look at the dispersion angles away from the focal point, where the rays are brought to a fo cus by the telescope mirrors.
* dispersion distance is linear in wavelength, so get used to Angstroms!
- The resolving power, {(\lambda/d\lambda) ~ ( (\delta d/d)^2 + (\delta x/x) )^(-1/2)}, and because the error in the grating period is usually small, is generally detemined by the width of telescope PSF. Also note that {\lambda/d\lambda ~ x/\delta x}, i.e., the resolving power increases for higher wavelengths.
* resolving power can be derived as an error-propagation problem
* generally limited by the PSF
- Example: Consider the LETGS, which has a Rowland diameter D=8637 mm, and a grating period d=0.99125 +- 0.000087 microns. At say 100 Angstrom in the 1st order, {\Delta\theta} ~ 0.0100883 radians ~ 37.4, and x=87.1324 mm, for a plate scale of 1.148 Angstrom/mm. The HRMA on-axis PSF is approximately 5 HRC pixels (0.032 mm) wide, so the resolving power is ~ 2700.
* example for HRC-S/LETG, back of the envelope calculation gives results close to measured values
LINE SPREAD FUNCTION
- The response of the instrument to a delta-function line is the Line Spread Function (LSF)
- Chandra LSFs usually well described by a ``Beta profile,''
I(lambda) = [ 1 + ((lambda-lambda_0)/lambda_c)^2 ]^(-beta)
(beta=1 is the Lorentzian function)
- Response matrices include multiple Gaussian and Lorentzian components
- Do you really need to use a response matrix for grating data?
* yes, if you cannot separate the orders and have significant continuum
* no, if you are analyzing individual lines and are confident of the model
vkashyap@cfa.harvard.edu
afruscione@cfa.harvard.edu