A 60 mm Refractor, a Fixed Tube and Two Eyepieces
An amateur astronomy club owns an achromatic objective of focal length 1200 mm and diameter 60 mm, mounted in a fixed tube whose length from objective to eyepiece holder is exactly 1200 mm and cannot be altered. Two eyepieces are available, of focal length 25 mm and 10 mm, along with a 2× Barlow lens. The club wants to split a double star whose components are 2.0 arcseconds apart, and also to view the four bright moons of Jupiter, which are faint compared with the planet. One member argues that the 10 mm eyepiece should be used for everything because it gives the greater magnification; another insists the 25 mm eyepiece is “sharper”. Observations are visual, at a wavelength of about 550 nm, and the observer's eye resolves about 1 arcminute.
1. Calculate the angular magnification with each eyepiece and the diffraction limit of the 60 mm objective at 550 nm. Decide which eyepiece, if any, can split the 2.0 arcsecond double star, and state the aperture that would be needed.
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Magnifications. M = f_o/f_e: with the 25 mm eyepiece M = 1200/25 = 48×; with the 10 mm eyepiece M = 1200/10 = 120×.
Diffraction limit. θ_min = 1.22λ/D = 1.22 × 550 × 10⁻⁹/0.060 = 1.12 × 10⁻⁵ rad. Converting: 1.12 × 10⁻⁵ × 206 265 ≈ 2.3 arcseconds.
Can it split 2.0″? No. Two stars 2.0″ apart are inside the objective's Airy disc, so no eyepiece can separate them; the 10 mm eyepiece merely enlarges two overlapping blurred discs. The limit is set by the wave nature of light and the aperture, not by magnification.
Aperture required. D = 1.22λ/θ = 1.22 × 550 × 10⁻⁹/(2.0 × 4.85 × 10⁻⁶) ≈ 0.069 m ≈ 70 mm, and in practice more (≈ 100 mm) to allow for atmospheric seeing and for the eye's own limit. Selling extra magnification cannot buy resolution - this is the point the first member has wrong.
Conclusion. Use 120× only for high-contrast detail on bright objects, never as a way to split close doubles.
2. For normal adjustment (final image at infinity) the eyepiece must be a distance f_o + f_e from the objective. Show what is wrong with the fixed 1200 mm tube for each eyepiece, and explain whether the Barlow lens rescues the situation.
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Required separations.
- 25 mm eyepiece: 1200 + 25 = 1225 mm
- 10 mm eyepiece: 1200 + 10 = 1210 mm
The tube is 1200 mm, so in both cases the eyepiece sits 25 mm and 10 mm respectively too close to the objective. The objective forms its image at 1200 mm - exactly on (or inside) the eyepiece - so the eyepiece cannot accept a real object at its focal point. The visible result is a badly defocused, blurred field which no amount of turning will fix, since there is no focusing movement.
Fix. A draw tube, helical focuser or extension ring of at least 25 mm is needed; the 10 mm eyepiece then needs only 10 mm of extension, so a 30 mm draw tube covers both. Focusing is done by moving the eyepiece, not by moving the objective.
Does the Barlow help? No. A 2× Barlow multiplies the effective focal length (to 2400 mm) and pushes the focal plane further back, so the eyepiece would need to be even further from the objective, not closer. It increases magnification and the f-ratio (a dimmer image) but does nothing for resolution and makes the tube-length problem worse. The Barlow is useful only after the focuser problem is solved.
3. Using exit pupil and the useful-magnification rule, recommend an eyepiece for the faint Jovian moons and for fine detail on the planet. Explain the competing effects and why the “sharper” claim for the 25 mm eyepiece is only half right.
Read the worked answer
Exit pupil = D/M: with 48× it is 60/48 = 1.25 mm; with 120× it is 60/120 = 0.50 mm.
Faint moons. The eye's dark-adapted pupil is about 7 mm, so a 1.25 mm exit pupil already under-fills it, and 0.50 mm is like looking through a pinhole - the image is dim and the eye's own aberrations and floaters dominate. For faint, low-contrast objects the 25 mm eyepiece (48×, 1.25 mm pupil) is clearly better, and even then the 60 mm aperture limits the faintest star visible to about magnitude 11.
Fine planetary detail. Here a small exit pupil helps: it stops down the eye's own optics and darkens the bright planet, improving apparent contrast. But the useful maximum is about 2× per mm of aperture, i.e. 120× for a 60 mm objective, and that is exactly where the 10 mm eyepiece lands. Beyond that the image is empty magnification. So 120× is the ceiling, usable only in steady air.
The “sharper” claim. The 25 mm eyepiece is not sharper in the optical sense - the objective's 2.3″ diffraction limit is unchanged. It only appears cleaner because of the larger exit pupil, lower magnification, and less obvious seeing and collimation errors. Both members are partly right: use 48× for faint things and 120× for bright, high-contrast detail, and buy a bigger objective, not a smaller eyepiece, to see more.