I2

The expression (16) properly transforms the bandwidths. The specific intensity I can be converted with the same ratio l2/c according to the same line of reasoning.

Frequency and energy

Energy units in the reference band in use by high-energy astronomers are usually keV or MeV rather than joules. Thus one might see S (erg cm-2 s -1 keV-1). The conversion of this expression to our standard SI units S (W m-2 Hz-1) requires only numerical multiplicative factors. The conversion of erg cm-2 to J m-2 is made with the factors 10-7 J/erg and 104 cm2/m2. The conversion of keV-1 to Hz-1 requires the conversion factor 1/(1.6 x 10-16)keV/J and then the expression E = h v which provides the conversion factor h = 6.63 x 10-34J/Hz (Section 2.3). The conversion requires multiplication by the product of these four factors, or 4.14 x 10-21.

X-ray astronomers sometimes use units of I (keV s-1cm-2 keV-1 sr-1). The two keVs may seem confusing, but remember that the energy term in the numerator refers to the accumulated energy and that in the denominator to the width of the energy band in which the data are being accumulated. This awareness makes straightforward the various conversions between these units.

Spectral bin widths

Data may be taken with various frequency resolutions. A high resolution (narrow bandwidth) allows one to detect and study narrow spectral lines. Since a number must be stored or telemetered for each of the many narrow spectral bands, a high resolution can require a lot of data storage space (or telemetry from a space vehicle) if the overall bandwidth is large. If, in addition, high time resolution is required, the spectral numbers must be stored for every time interval. The required data storage can be huge.

The observation time required for a spectral measurement must be such that sufficient signal or statistics are obtained in each narrow spectral or time band (often called channel or bin). For example, N = 100 counts (photons) must be accumulated in each channel if each is to yield a 10% uncertainty in the measured value. (See Section 6.5; we assume here negligible background.) If the 2.0 to 12.0 keV band is divided into 10 channels of 1-keV width and if the accumulation rate is uniform across the channels, a total of 1000 counts must be accumulated. Improved energy resolution of 0.01 keV, again with 10% precision, requires 1000

channels accumulating a total of 105 counts. The observation time would be 100 times longer than that for the 10-channel case.

Thus, measurements with high spectral resolution carry a high price and must be used only when needed, such as for the detailed measurement of a specific spectral line. If only the broad spectral shape is desired, as in Fig. 2, then broad frequency bins will suffice.

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