Princeton, New Jersey

April 11, 1975

TO: | Copernicus Astronomers and Guest Investigators |

FROM: | Edward B . Jenkins |

SUBJECT: | Computations of the U1 Instrumental Profile |

Note that the brightness of the image varies along its length; this is caused by uneven illumination of the grating. Occultation by the spectrometer and secondary spider casts a shadow on the grating which produces an intensity distribution

|x/x_{max}| |
0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |

I(x) | 0.00 | 0.30 | 0.29 | 0.28 | 0.88 | 0.83 | 0.76 | 0.67 | 0.55 | 0.40 | 0.10 |

If `R` is the radius of curvature of the grating (and the
diameter of the Rowland circle; this value is 998.8 mm), the height
`L` of the illumination on the grating is `R`(`cos
i`)/20 for an *f*/20 beam with an angle of incidence
`i`. The astigmatic image height is given by

where Gamma =2

x_{max}=LGamma

where

y= -Psix^{2}/2R

Psi = [

sin i tan^{2}i-sin r+ {tan r(1- Gamma)^{2}}/cos r] Gamma^{-2}

For most wavelengths along the spectrum the segmented slit compensates for the curve. To ascertain the small smearing which results from the lack of a perfect match, we may define an offset

and

y'(x) =y(x) for |x| <= 3.0 mm

If the entrance and exit slits had infinitesimal widths, we would expect an instrumental profile whose shape follows the function

y'(x) =y(x) - 0.013(|x| - 3.0 mm) for |x| > 3.0 mm

where the sum includes all points along the line from 0 to

P(y') = Sum (dy'/dx)^{-1}I(x)

(Delta Lambda =

mR/d cos r y'= 2.0858 x 10^{3}cos r y'fory'in mm and Delta Lambda in mÅ

mR/d W_{1}cos
i = 49.82 mÅ(= FWHM _{1} at all Lambda) |

where `W`_{1} and
`W`_{2} are the entrance and exit slit
widths, respectively.

Although the face of the exit slit is always tangent to the Rowland
circle, the effective width `W`_{2} varies
slightly with wavelength since the light strikes the jaws at the
diffraction angle `r` from the normal, and the jaws are not
perfectly thin. The projected widths as seen from the grating
`W`_{2} `cos r` has been measured
by Perkin-Elmer and a functional fit to these data has been made by J. B.
Rogerson.

A convolution of `P`(Delta Lambda) with the above shown trapezoid
gives us
the net instrumental profile, which includes the slit widths and the small
lack of registration of exit slit shape with the image curvature. Some
additional smearing should be caused by spherical aberration, giving an
image diameter

A=L^{3}/(8R^{2}) (sin i tan i+sin r tan r)cos r

which may be multiplied by 2.0858 x 10^{3} `cos
r` to give the value in mÅ. The above figure, however, is
appropriate at the paraxial focus. The aberration may be substantially
reduced (by about a factor of four, I am told), by viewing the line at the
circle of least confusion, instead of the paraxial focus.

where Phi = (

r_{e}=R/Phi

- Simple trapezoids from the slit convolutions.
- Profile for a star imaged on the slit center,
- Profile for a diffuse source illuminating the whole slit.

It is of interest to compare the computed curves with actual observations
in orbit. Two sources of information are available at present. First, W.
Morton and L. Spitzer have measured stacked profiles of
H_{2} lines which are presumed to be quite narrow.
They find a good fit for the region 1025 <~ Lambda <~ 1100 Å is a
gaussian profile with a FWHM = 51 mÅ. This relation is plotted as a
series of dots on the 1050 Å graph, and it is evident that no really
significant source of additional smearing is at work in the spectrograph,
except possibly in the wings beyond | Delta Lambda | = 40 mÅ. The
other main source of observational material is the stack of geocoronal
Lyman Alpha emission scans. Here, however, the emission profile is not so
much smaller than the instrumental profile that we may ignore its effect
in producing a slightly broader profile. In connection with research on
the cometary Lyman Alpha observations, J. L. Bertaux has examined the
geocoronal data and has also theoretically estimated the true Lyman Alpha
profile shape. The observations give an approximately gaussian profile
with FWHM = 67 mÅ while the emission profile is expected to be 38
mÅ wide. We therefore would expect to find the response of our
instrument to a diffuse source of Lyman Alpha of infinitesimal width in
Delta Lambda to be roughly a gaussian with FWHM =
(67^{2}-38^{2})^{½}
= 55 mÅ. The computed curve at 1200 Å has FWHM = 60 mÅ;
perhaps the geocoronal emission width was overestimated by Bertaux or else
the difference may be attributable to the fact that the instrumental
profile is not really a pure gaussian distribution.

Samson, J. A. 1967, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York) pp. 5-19.

Welford, W. T. 1965, in Progress in Optics

4ed. by E. Wolf p. 243.