Astronomy · From collecting light to recording data

Filters and spectrographs

Follow radiation through a response-weighted band or into a spectrum, using discrete photon bookkeeping to distinguish passband selection, detector yield and spectral resolving power.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Explain why a filter measurement integrates a wavelength-dependent response.
Calculate expected electrons from declared photon-bin totals, transmission and yield.
Define a spectral resolving-power convention without confusing it with sampling or velocity precision.

A band measurement combines wavelengths

A filter transmits selected wavelength ranges more strongly than others. Its passband is a response curve or specified range, not generally one exact wavelength. Radiation transmitted through it is also affected by the rest of the optical path and detector response.

An image taken through a filter therefore measures a weighted combination of radiation across its band. The count rate depends on the source spectrum as well as the system response. A plotted filter label or a display colour does not by itself identify the source’s composition.

A spectral energy density and a photon spectral density require different weights when predicting detected electrons. At vacuum wavelength λ, dividing radiant energy by hc/λ converts it to photon number. The detector’s electron yield is then another conversion, not a second measurement of the incoming energy.

The existing photon and spectrometer Foundations records supply those quantity distinctions. The following invented bin example makes the response bookkeeping explicit without requiring integral calculus.

Equal photon inputs can give unequal outputs

Stipulate three disjoint populated windows:445–455 nm,545–555 nm and 645–655 nm. Each contains an expected 100 incident photons at the filter input during the same exposure. In this toy case there are no incident photons outside the listed windows; the totals are integrated per bin, not photons per nanometre.

The optical detector yield is constant within each window and stipulated as 0.5, 0.8 and 0.25 electron per incident detector photon respectively. All other losses are unity. A wide ideal filter passes all three windows with transmission 1; a selected ideal filter passes only 545–555 nm.

Multiply incident photon total by filter transmission and electron yield in each bin. The wide case gives 50, 80 and 25 expected electrons, totaling 155. The selected case gives 0, 80 and 0, totaling 80. The same incoming source and exposure were used in both cases.

There is no extra 10 nm factor to apply to these numbers because the 100-photon inputs already cover each whole window. An interval-width factor would be needed if the inputs instead described a density per wavelength. The diagram keeps yields and transmission separate from the expected-electron output scale.

These are synthetic expectations, not measured filter specifications or an empirical detector curve. No photon noise, background, read noise or saturation is shown. The unequal yields do not identify any chemical species in the invented source.

Original three-window model; values are bin-integrated expectations for the same exposure. Detector yield and filter transmission are constant inside each listed bin.
Window	Input photons	Electron yield	Wide output	Selected output
445–455 nm	100	0.5 electron/photon	50 electrons	0
545–555 nm	100	0.8 electron/photon	80 electrons	80 electrons
645–655 nm	100	0.25 electron/photon	25 electrons	0
Totals	300	Not a sum of yields	155 electrons	80 electrons

The same photons, two filters

Scroll sideways to see the full diagram. Its explanation is below.

Original synthetic response bookkeeping for three disjoint wavelength windows: 445–455 nm, 545–555 nm and 645–655 nm, centred at 450, 550 and 650 nm. Each window is 10 nm wide and contains an expected 100 incident photons at the filter input during the same exposure; there are no incident photons outside these listed windows in this toy. Inputs are already integrated over each window, not photon densities per nanometre. The mean detector yields are 0.5, 0.8 and 0.25 collected electron per photon arriving at the detector. Other losses are unity, and response values are constant within each window. The wide ideal filter has transmission 1 in all three populated windows. The selected ideal filter transmits only 545–555 nm: its listed transmissions are 0, 1 and 0. Multiply input photons by filter transmission and detector yield in each window. Wide-filter expectations are 50, 80 and 25 electrons, totalling 155 electrons. Selected-filter expectations are 0, 80 and 0 electrons, totalling 80 electrons. The plotted outputs share one zero-based expected-electron scale from 0 to 100. Filled blue circles and solid stems identify wide-filter outputs; hollow orange diamonds and dashed stems identify selected-filter outputs. Markers have a slight horizontal offset only to separate the two cases; horizontal groups identify the three listed windows and marker width is not a passband width. Zero outputs lie on the same zero baseline. The input, transmission and yield table is separate from the electron-output plot: yields are not output counts. No extra 10 nm factor multiplies the already integrated inputs. No continuous source spectrum, empirical filter/detector curve, noise realization, background, saturation or chemical identity is implied.

The same accounting can be written as an integral

For an optional continuous model, let Fλ be spectral energy flux density at the entrance in W m⁻² m⁻¹, A the collecting area in m², and t the exposure in seconds. Let T(λ) be total transmission before the detector and q(λ) its mean collected-electron yield per incident photon. Use wavelength in metres consistently.

Under linear response, expected electrons are Ne = tA ∫ Fλ(λ) T(λ) q(λ) λ/(hc) dλ across the chosen band. The factor λ/(hc) converts energy into photon number; T selects the transmitted fraction; q converts detector arrivals into expected collected electrons.

If the supplied source distribution is already photon spectral flux density, use that density directly and omit the energy-to-photon factor. Do not count detector efficiency twice by including q inside both T and a separate multiplier.

The integral is the continuous counterpart of summing the bin contributions above. It predicts a mean under the stated source, response and exposure assumptions. It does not supply the uncertainty model or convert a broad-band count into a unique monochromatic source flux without additional assumptions.

A spectrograph separates wavelength information

A spectrograph uses a dispersing system, such as a prism or diffraction grating, to map wavelength onto a measurable coordinate. A detector then records the resulting distribution. Its calibration relates pixel position and response to wavelength and signal.

Unlike one broad-band image value, a spectrum retains more information about how the signal is distributed across wavelength. The instrument’s line-spread response, throughput, source placement and sampling affect which features can be distinguished and how their profiles are measured.

Use a declared convention for spectral resolving power. Here R = λ/Δλ, with Δλ taken as the instrumental line-spread FWHM at that wavelength. In a synthetic example at 500 nm, R =5000 gives Δλ =0.10 nm; R =10000 gives 0.05 nm.

That width is not the pixel spacing or the statistical uncertainty of a fitted line centre. A well-calibrated feature can sometimes be located to a fraction of a resolution element, depending on its shape, signal, sampling and systematics. R alone does not promise a Doppler-velocity precision.

Dividing a fixed wavelength range into narrower bins can reduce expected photons per bin for a smooth source when throughput and exposure are otherwise held fixed. The gain in spectral detail and the uncertainty cost must be assessed together. The following radio and tradeoff lessons retain the same distinction between measuring more detail and collecting more signal.

Worked examples

Sum the outputs in their actual units

In the synthetic wide-filter case, expected photon totals are 100 in each window and yields 0.5, 0.8 and 0.25 electron/photon. What is the total expected electron signal?

Compute the three outputs:100 ×0.5 =50, 100 ×0.8 =80 and 100 ×0.25 =25 electrons.
Sum those outputs, rather than summing yields alone.
50 +80 +25 =155 electrons.

155 expected electrons. No bin-width multiplication is needed for the already integrated inputs.

Interpret one resolving-power convention

At 500 nm, an invented instrument has R =10000 using the stated FWHM convention. What width follows?

Use Δλ = λ/R.
500 nm/10000 =0.05 nm.
Retain that this is an instrument-response width, not an automatic line-centre error.

An instrumental FWHM of 0.05 nm at 500 nm under the declared convention.

Try the reasoning

Does a filter name imply that only one exact wavelength reaches the detector?
Show the answer and reasoning
No.
A passband transmits a range with wavelength-dependent response.
Should the 100 photons already integrated within a 10 nm toy bin be multiplied by 10 nm again?
Show the answer and reasoning
No.
That would count the interval width twice. A density per nm would be a different input quantity.
If the source input is already photon spectral flux density, should λ/(hc) be applied again to turn it into photon number?
Show the answer and reasoning
No.
The factor converts energy to photons; it is omitted when the input already counts photons.
Does resolving power alone specify the uncertainty of a measured radial velocity?
Show the answer and reasoning
No.
Calibration, signal, profile shape, sampling, blends and systematic effects also affect the velocity inference.

What to carry forward

A band count combines source spectrum, transmission and detector yield over a range.
Keep photon-bin totals, spectral densities and output electrons distinct in the arithmetic.
Spectral resolving width, sample spacing and parameter precision are separate quantities.

References

Teaching resources and research supporting this lesson’s explanations and examples.