Building an Artificial Star

Tanveer Gani

Last updated Feb 13 2007

### Introduction

Star testing astronomical telescopes, which also happens to be the name of an excellent book on the subject by Richard Suiter, requires an infinitely far away point source of light to be available. Real stars approximate this very well but star testing with real stars is often frustrating due to atmospheric turbulence which can prevent the Airy disk and diffraction rings from being seen at all. The limitations of testing with a real star can be removed by using an artificial one.

### Some Theory

An artificial star can obviously be only a finite distance away. A point source placed at an infinite distance results in plane wavefronts at the telescope aperture. Telescopes are designed to focus plane wavefronts at their aperture stops to points on a focal plane, but an artificial star placed a finite distance away results in spherical wavefronts at the telescope aperture. This causes apparent optical aberrations in the telescope but if the artificial star is placed far away enough, the aberrations fall below the practical detection limit because the spherical wavefronts are almost plane in shape.

Suiter gives various formulae and tables for determining the minimum distance to the artificial star, but for most mid-range amateur instruments, the distance is given as 20 times the focal length. This rule of thumb is too small for instruments of "fast" focal ratios or large apertures.

Once the distance to the artificial star is fixed, the next dimension that needs to be determined is the size of the artificial star. This depends only on the aperture size of the telescope.

The size of the Airy disk of a telescope of aperture diameter A is given by:

R = 1.22 L/A ......................(1a)

where L is a specific wavelength of light, typically chosen to be 587nm. Both L and A must be measured in the same units and R is measured in radians, which can be converted to arcseconds by multiplying with 206265.  For example, for a 200mm instrument, R = 0.00000358 radians = 0.739 arcsec. The size of the Airy disk is considered to be the resolution limit of the telescope. In other words, an object subtending a smaller angle cannot be resolved in the telescope and its image will be indistinguishable from that of a geometric point.

The angle subtended by a length S at distance D is simply S/D (in radians). Since the distance D to the artificial star is known, we can determine its maximum size S by restricting the angle subtended to be less than that of the Airy disk, i.e.,

S = D * R...............................(1b)

For a 200mm f/10 instrument, the focal length is 2m. 20 times this number is 40m, not an overly large distance. The Airy disk size, 0.739 arcseconds = 0.00000358 radians gives S = 0.000143m = 0.143mm. This is the maximum size of the artificial star for the given aperture and focal length, placed at a distance of 40m. If the artificial star is placed at a larger distance, S can be correspondingly larger.

One can physically make a small pin-hole but determining its size is problematic. A better method is to optically reduce the image of a larger and brighter source to the required dimensions.

Consider a convex lens of focal length f and place an object a distance o away on the axis of the eyepiece which forms an image of the object at a distance i from the lens. Here's the figure from the Wikipedia article on lens optics to illustrate the configuration:

In the above figure, S1 is o and S2 is i.  Using the Gaussian optics equation,

1/o + 1/i = 1/f........................(2)

which rearranges as

i = fo/(o-f)................................(2a)

The sizes of the image (H') and the object (H) are related by:

H'/i = H/o...........................(3)

which gives H' in terms of H, i and o:

H' = H i/o.............................(4)

From (2a), we can see that if an object is placed beyond the focus, say, at a distance many times the focal length, i.e., o is made much larger than f, i is almost the same as f, i.e., a fraction of the object distance. Then (4) tells us that the image size is only a fraction of the object size in the same proportion as i/o.

All this is leading to the idea that we can place a small light source behind a positive (i.e., magnifying) lens and have an image of reduced dimensions "float" in front of the lens. The reduced image can then be used as a point source.

Let M, the magnification (really the reduction) factor, be defined as

M = i/o or i = Mo.

Substituting this in (2a) we find

Mo = fo/(o-f)

which gives

M(o-f) = f

or

o = f/M + f................................(5)

This is the distance at which we require the light source to be placed to get the appropriate reduction in size.

Every amateur has at least one high power eyepiece, which is effectively a positive lens of short focal length. Suppose we use a 10mm eyepiece and a 2mm light source. To reduce the light source to a 0.14mm image, we have:

M = 0.14/2 = 0.07

and

o = 10/0.07 + 10 = 153mm.

To recapitulate, a 2mm light source placed 153mm behind an eyepiece of focal length 10mm will produce a 0.14mm image of the source in front of the eyepiece which is then used as the artificial star.

It's worth noting that the Gaussian equations (2) and (3) are for thin lenses while an eyepiece is certainly not a thin lens. However, the lens dimensions are small compared with the object distance and the equation still works to a good approximation.

Super-bright LEDs (light emitting diodes) have recently become available at very cheap prices. A good place to buy them is here. The advantages of a LED are low power consumption and a narrow spectral band. Low power consumption allows use of a compact power source -- a common 9V battery works fine. A narrow spectral band allows clean diffraction fringes to be seen because they don't smear as they would if white light were used.  The G8020 LED from the linked vendor is a good match for an artificial star since its green color (525nm) corresponds very well with the peak sensitivity of the eye.

LEDs, however, cannot be directly connected with a power source because a LED cannot sustain a current much more than its forward current. For the mentioned LED, the sustained forward current is 20mA. A LED also has a characteristic called the forward voltage which is 3.6V for the G8020 LED. This can be viewed as an opposing voltage to the external power source. Its internal resistance is negligible. This means than a resistor has to be connected in series with the LED to limit the current through the LED.  Using a 9V battery to provide suitable voltage and using the formula

V = IR ..........................(6)

where V = (battery voltage) - (forward voltage) = 9 - 3.6V = 5.4V and I = (forward current) = 0.02A, we get R = 5.4/0.02 = 270 ohms. Power consumption is VI = (5.4)(0.02) = 0.1W, so a 1/4 watt 300 ohm resistor from Radio Shack works very well. The LED is blindingly bright.

### Construction

The artificial star is constructed from these components

• A bright LED (as described above)
• A LED holder (Radio Shack sells these)
• 9V battery
• 300 ohm 1/4 watt resistor (Radio Shack sells this)
• 4 crocodile clips and wire (Radio Shack sells these)
• A piece of cardboard or plastic, cut in a 2-1/2" disc
• A suitable length of 2 to 2-1/2" ID PVC pipe (hardware store)
• End terminator for the pipe (hardware store)
• A threaded aluminium bracket for the pipe (optional, hardware store)
• 3 size 6-32 nylon screws (optional)
• Drill+tap for size 6-32 nylon screws (optional)
• 2" to 1.25" focuser eyepiece adaptor (many amateurs own one of these)

#### Step 1

From your scope size, determine the size of the artificial star using equations (1a) and (1b).  The suggested LEDs are 5mm in diameter but to get that to reduce to the required sub-mm size would require either a longer tube or an eyepiece of very short focal length. It's better to mask off the LED face to about 2mm. I used a rubber nut cover that I found lying around in my toolbox and drilled a hole though it. I later found that hardware stores sell items like this.

From the 2mm LED size and the maximum artificial star size that you calculated above, calculate M, and then using equation (5) and a suitable eyepiece's focal length, calculate the tube length as o. Cut the PVC pipe to this length or a bit longer. Remember that it's better to err on the longer side.

Since the LED is placed at the end of a relatively long tube, grazing angle reflections off the pipe walls are particularly bad and can bloat the artificial star's size. You can either paint the tube interior flat black (which is what I did) or construct a baffle just in front of the LED which prevents its light from reaching the walls (a better solution).

#### Step 2

Cut a hole through the plastic disc and insert the LED holder (with the LED in it) and screw on the nut from the other side so that the LED is held firmly in the center of the plastic disc. In the below image, I've painted the tube and plastic disc flat black, but you can see the LED in the center with the rubber cover exposing 2mm of it.

Place the plastic disc with the LED inserted in the center on the end of the plastic connector with threads and screw on the end part. Then fix the connector to the PVC pipe. The rear end view then looks like this:

#### Step 3

At the front-end of the tube, tap 3 holes for 6-32 nylon screws and insert the screws. Then insert the 2"-to-1.25" adaptor and tighten the screws till the adaptor sits firm. You can also use alternate means, such as wrapping with tape, to hold the adaptor but I found the screws very convenient. Insert the eyepiece into the adaptor.

#### Step 4

Tape a 9V batter to the tube as shown above. Connect the battery with a 300 ohm resistor in series  with the LED connectors that are sticking out. Crocodile clips (available at Radio shack) make it fairly simple and soldering isn't required. Note that the LED, being a diode, works with a specific polarity. Swap the wires if the LED isn't working.

#### Step 5

Use the bracket or other means to attach the tube to a tripod or otherwise place it suitably. The final result looks like this:

(One of the nylon screws is broken off thanks to the the neighbor's kid).

This method of constructing an artificial star isn't original with me; in fact, I remember seeing something like it on the web a while back but don't have the address available.

### A Note About The Eyepiece

All eyepieces are designed to turn diverging cones of light rays, from points in the image plane formed by the objective, into parallel bundles that the eye can then focus. By symmetry, a parallel bundle of light coming in from the eye-lens side of the eyepiece will form a perfect image in front of the field lens. Because we're making the eyepiece form the image of an object at a finite distance, rather than infinitely far away, the eyepiece may suffer from spherical aberration if the eyepiece is sufficiently complex, resulting in a fuzzy or larger than ideal image of the LED. This however is not a problem in practice as the LED is far enough away from the eyepiece to consider light bundles from it to the eyepiece to be nearly parallel. It's actually more important to make sure that the eye-lens side of the eyepiece is facing the LED such that the (reduced) image of the LED is formed in front of the field lens. This is because not all eyepieces are symmetrical designs and may suffer from aberrations if used in reverse. The Plossl is an exception as its a symmetrical design so it's a good idea to use a short focal length Plossl.

What works perfectly, however, is a microscope objective. These are designed to form a large image of a small object placed close to the front of the objective. By symmetry, a large object behind the objective will be form a small image in front of the objective. This happens to be exactly what we want. By poking a hole in a film canister, which was of 1-1/4" diameter, I was able to fit in a screwed off 10x microscope objective and use it to create the artificial star with very good results.

### Testing

The only real test is to use the artificial star with a telescope but a quick sanity check can be done by piercing a piece of foil, or even paper, with an even, round hole and peering through this hole at the eyepiece end from up close, say a few inches. You should see a beautiful Airy disk with concentric rings surrounding it.

I set up the artificial star described above at a distance of 20m from an Orion Argonaut Maksutov-Newtonian (this is the same as the Intes MN-61), of aperture 150mm and focal length 900mm, and used a Philips ToUCam with Registax 3 to image the diffraction patterns.

There is some apparent misalignment in the rings but I believe this is from the offset secondary rather than miscollimation.  Note that the outer diffuse ring is the boundary of the eye-lens of a 9mm  Ortho eyepiece, imaged out of focus. Below are images of the diffraction rings inside and outside focus. Notice how remarkably similar they look even though the artificial star is only 20m away, attesting to the optical quality of the Argonaut. I tried imaging the Airy disk but could not do so due to difficulty in determining exact focus with the laptop LCD in daylight. The disk and a surrounding ring was, however, clearly visible using a 6mm eyepiece.

Outside Focus

Inside Focus

### References

1. Star Testing Astronomical Telescopes, H. R. Suiter, Willman-Bell, 1994. This is the definitive reference on star testing and gives various methods for constructing artificial stars including using a microscope objective which corresponds to the above method.

2. Wikipedia: Lens(Optics). A good introductory article on geometric lens equations.

3. SuperBrightLeds: A good source for LEDs of various colors. They also supply specs for all LEDs that they sell  by individual units. Since it's not cost effective to buy just one because of shipping costs, I suggest buying three or four to keep around in case you accidentally burn one. It might also be interesting to test your scope at various wavelengths.Buy a white LED if you want to test your telescope for chromatic aberration.