Narrowband filters on fast telescopes: emission lines, cone angles and preshift

The lines narrowband filters are built for

Narrowband filters pass a few nanometers around a handful of atomic emission lines. NIST’s Atomic Spectra Database gives their positions; between 200 and 2000 nm it lists wavelengths in air, which is what a filter on a telescope sees.

Line	Wavelength in air (NIST)	Value type
H-beta	486.135 nm	observed (Ritz 486.1333)
[O III]	495.8911 and 500.6843 nm	Ritz
H-alpha	656.279 nm	observed (Ritz 656.2819)
[S II]	671.6440 and 673.0815 nm	observed

The two [S II] lines are 1.44 nm apart, less than the width of any filter discussed below; the [O III] pair is 4.79 nm apart, more than a 3.5 or 4 nm passband (computed). Either way, a filter only passes a line if its passband stays where the line is, and that is where focal ratio comes in.

Why a steep light cone shifts the passband

Narrowband filters are interference filters: stacks of thin layers whose reflections interfere to pass a chosen band. Baader’s whitepaper explains that the effect depends on the angle of incidence, and that light arriving off-normal shifts the passband to shorter wavelengths. IDEX Health & Science (Semrock) describes the same behavior for its filters, adding that s- and p-polarized light shift by different amounts. Baader approximates the shift as:

λ_θ = λ₀ × √(1 − (n₀ × sin θ ÷ n_eff)²)

Here n₀ is the index of air and n_eff the effective index of the filter coating. The whitepaper does not publish n_eff for its filters, so we do not compute shifts in nanometers.

In a telescope the filter sits in a converging cone, so each point of the image receives light over a range of angles. For an on-axis star the steepest ray comes from the edge of the aperture:

marginal ray angle = arctan(1 ÷ (2 × focal ratio))

This is plain geometry (our derivation), and it matches Baader, who writes that rays cross the filter at 0° to about 14° at f/2. A central obstruction removes the shallowest rays. Baader’s whitepaper shows it matters: the outer rings of the aperture carry the most area and the most shift, and obstruction shifts the effective passband further.

How much signal is at stake

Baader’s whitepaper plots filters centered on H-alpha. At f/4 the shift has almost no effect on transmission at the line. At f/2.5 a 6.5 nm filter keeps about 80 % at H-alpha; at f/2.3 a 3.5 nm filter about 50 %, and with a 50 % central obstruction about 30 %. The blog post adds that unshifted filters work while the focal ratio stays well above f/3.4 and the optics are unobstructed, and that cold also shifts filters to the blue, by about 0.4 nm to 1 nm at −20 °C compared with a 20 °C lab.

The makers’ answers differ. Baader preshifts by category: 6.5 nm “Highspeed” filters for f/3.4 and faster; for 3.5/4 nm, an f/3 Ultra-Highspeed for f/2.3–f/3.4 and an f/2 Ultra-Highspeed below f/2.3. Astronomik specifies ranges per coating: MFR 12 nm for f/3–f/15, 6 nm for f/4–f/15 and 4 nm for f/4.5–f/15; MaxFR 12 nm for f/1.7–f/6 and 6 nm for f/2.2–f/6, with about 10 % less at f/2. The boundaries do not line up between brands, so follow the range published for the filter you are buying.

Which telescopes in our data are fast enough to matter

Computed from published focal ratios and obstructions (obstruction as a share of the aperture diameter):

Telescope	Focal ratio	Marginal angle	Central obstruction	Baader 3.5/4 nm category
Celestron RASA 8	f/2	14.0°	45.8 %	f/2 Ultra-Highspeed
Celestron RASA 11 V2, RASA 36 cm	f/2.2	12.8°	40.9 %, 44.4 %	f/2 Ultra-Highspeed
Sharpstar 13028HNT, 15028HNT	f/2.8	10.1°	50.0 %, 46.7 %	f/3 Ultra-Highspeed
William Optics Mighty Cat 71	f/3.2	8.9°	none	f/3 Ultra-Highspeed
Askar N160, N210 with included corrector	f/3.4	8.4°	38.1 %, 33.8 %	f/3 Ultra-Highspeed, at its f/3.4 limit
Sky-Watcher Quattro 150P with included corrector	f/3.45	8.2°	42.7 %	just slower than f/3.4
Askar FRA400 with 0.7× reducer	f/3.9	7.3°	none	slower than f/3.4
Sky-Watcher Quattro 200P–300P	f/4	7.1°	33–35 %	slower than f/3.4

The obstructed fast systems are the cases Baader singles out. For the RASA 8, the 46.5 mm half-width of the obstruction over a 400 mm focal length also removes every ray shallower than about 6.6° (computed), so the light that remains is concentrated at the steep angles. The RASA and Sharpstar pages list the full specs.

Reducers make the cone steeper

A reducer shortens the focal length at the same aperture, so it speeds up the focal ratio and widens the light cone. The Askar FRA400 at f/5.6 has a 5.1° marginal angle; with its 0.7× reducer at f/3.9 the angle becomes 7.3° (computed). The reducer calculator gives the new focal ratio for any factor; check it against the filter maker’s range before buying.

Narrowband frames also collect very little sky per pixel, which changes how long subs should be: the integration time guide shows why, and the Bortle scale guide covers how sky brightness enters the choice of bandwidth.