## Archive for May, 2014

### Maximal Shutter Speed for Night Sky Photography without Tracking

2014/05/10

When taking night sky photography, we need to grab as much light as possible by increasing the ISO, using a larger aperture and/or using a longer shutter speed. However, larger ISO means more noise and there is a limit of aperture size. If the shutter speed is too long, star trails become noticeable.

The star trails can be avoided by attaching the camera on a equatorial mount. However, not everyone can afford one.

What’s the longest shutter speed without leaving any noticeable trail? Experienced astrophotographers have a good estimation through a trial and error approach. Let’s instead look at the mathematics behind it.

Different stars move at different speed.

Stars at the poles, e.g. the Polaris, do not move. Stars on the earth equator plane, e.g. the Orion constellation, move the fastest, i.e. about $360^{\circ}$ per day. In astronomy, declination ($-90^{\circ} \le \delta \le 90^{\circ}$) measures the angle of the star above the equator plane.

During time $t$, a star at declination $\delta$ moves for an angle of $\frac{360^{\circ} \times t \times cos \delta}{24 hours}$. If (i) a $f$mm lens focuses at infinity, (ii) a star moves for an angle of $\alpha$ and (iii) it’s at the center of the picture, then its image on the CCD sensor will move for $f \times \alpha \times \pi / 180^{\circ}$. The Polaris’ declination is about $90^{\circ}$ and the Orion constellation’s declination is around $[-10^{\circ}, 10^{\circ}]$.

Putting them together, the star’s image on the CCD will move for $t \times cos \delta \times f \times 2\pi / 24 hours$ mm. For instance, if we use 30s shutter speed and a standard 50mm lens to shoot Orion, the length of the trail will be 0.1mm on the camera CCD.

Now, the question is, whether a 0.1mm trail on the CCD is noticeable on the final photo? It’s hard to answer as it depends on several factors: size and resolution of the print, resolution of the sensor and resolving power of the lens. Fortunately, this has been well studied in the optics community. The circle of confusion (CoC) measures the largest blur spot that is indistinguishable from a point. This can be rephrased in our flavour by substituting “largest blur spot” to “longest trail”. A widely used CoC is d/1500, where d is the diagonal of the sensor. For example, for full frame (d= 43mm), CoC = 0.029mm, thus 0.1mm is quite noticeable. I found d/1500 tends to be too large for today’s high resolution camera. Moreover, I usually crop a bit. I’m more comfortable with d/2000.

We now can compute the longest shutter speed $t$:

$\frac{t \times cos \delta \times f \times 2\pi}{24 hours} = \frac{d}{2000}$

$t = \frac{d \times 24 hours}{2000 \times cos \delta \times f \times 2\pi}$

Here is a table for some configurations:

$f$ $\delta$ $d$ max $t$
14mm $0^{\circ}$ 43mm (full frame) 21s
50mm $0^{\circ}$ 43mm (full frame) 6s
500mm $0^{\circ}$ 43mm (full frame) 0.6s
50mm $0^{\circ}$ 28mm (APS-C) 4s
4.1mm $0^{\circ}$ 5mm (iPhone 5S) 6s
50mm $90^{\circ}$ 43mm (full frame) $\infty$

A general rule of thumb is $300/F$ sec where $F$ is the 35 mm equivalent focal length.