# Planetary Society's LightSail Spacecraft's corner cube reflectors; how large, and corrected for aberration?

The Planetary Society's new 3U cubesat LightSail spacecraft (mentioned in this answer) has corner-cube retroreflectors on on the 10x10 cm +z end, facing directly away from the sail, which will be Earth-facing as the spacecraft starts to enter night time (eclipse) as it orbits the Earth.

As discussed in this Physics SE answer, light reflected by an object moving with significant velocity in a given frame will experience a small deflection because of the finite speed of light. This is often called astronomical aberration.

The deflection in radians is about $2 \ v_{\bot} / c$, so for LEO it's about 10 or 11 arcseconds when at the zenith.

If each retroreflector is 12 mm in diameter and the wavelength is 532 nm (frequency doubled Nd:YAG DPSS laser), then the spread in the reflected beam due to diffraction is smaller than the deflection, and so the signal might be weak and difficult to receive. Actually it's worse than that, see the update at the end.

Question: What is the actual diameter of the reflectors (I've just estimated a half-inch from the images below), and have the internal reflecting faces been ground and polished a few arcseconds away from orthogonal (discussed here) to spread the beam out, as it was done for LAGEOS's retroreflectors?

update 1: It looks like I may have understated the problem (for LEO at least) when I first posted. I'll do the diffraction more rigorously now:

$\theta_{ab} = 2 \ v_{\bot} / c \approx$ 2 x 7670 / 3E+08 $\approx$ 5.07E-05 $\approx$ 10.5 arcsec.

$\theta_{Airy} = 1.22 \lambda / d \approx$ 1.22 x 532E-09 / 0.0127 $\approx$ 5.11E-05 $\approx$ 10.5 arcsec also!.

$\theta_{Airy}$ is in fact the first zero of the Airy diffraction pattern, and so diffracted light is minimal at this angle.

$$I(\theta) = I_0 \left( \frac{2 J_1(x)}{x} \right)^2$$

$$x = k \ a \ sin(\theta)$$ from scipy.special import j1
import matplotlib.pyplot as plt
import numpy as np

halfpi, pi, twopi = [f*np.pi for f in (0.5, 1, 2)]
degs, rads        = 180/pi, pi/180

a   = 0.5 * 12.7E-03
lam = 532E-09
ka  = (twopi/lam) * a

arcsecs = np.arange(0, 30, 0.01)
theta   = (arcsecs/3600.) * rads
x       = ka * np.sin(theta)

I       = (2. * j1(x) / x)**2  # https://en.wikipedia.org/wiki/Airy_disk
Inorm   = I/np.nanmax(I)

plt.figure()
fs1, fs2, lw = 16, 20, 2
plt.plot(arcsecs, Inorm, linewidth=lw)
plt.xlabel('theta (arcsec)', fontsize=fs1)
plt.ylabel('I/I0 (Airy)', fontsize=fs1)
plt.title('12.7mm aperture and 532nm', fontsize=fs1)
plt.show()


below: cropped and original (but smaller size than original) image of the +z end of the Planetary Society's LightSail spacecraft from this blogpost.  below: Screen shot from this Planetary Society YouTube video. • – uhoh Jun 18 '18 at 6:34

## 1 Answer

LightSail 1

7 cubes: 4 X 10mm diameter + 3 X 12.5mm diameter https://ilrs.cddis.eosdis.nasa.gov/docs/2014/Appendix_forILRS_Form_20141217.pdf

LightSail 2

On LightSail 2, they added more corner cube reflectors in addition to the ones on the +Z end that you linked above, and standardized all of them.

The second iteration has 13 cubes in total, all of which are 12.7mm in diameter. Cubes 8 through 13 are in the cluster shown in the question, and the 1 through 7 are distributed as singles in other locations.

Here's a document from the LightSail2 project manager to NASA that has a lot of good information about the corner cubes. According to Note 2 on the CAD drawings from the optics manufacturer, the incident and reflected beams must be within 5 arcsec (0.00138889 deg) of parallel.

Therefore no, it is not likely to be corrected for aberration.

This appears to be an off-the-shelf Edmund Optics Techsphere half-inch corner cube reflector for visible light. Part number #48-606 (same as the number shown on the drawing) is the unmounted version (though the ones in the photos in the question appear to be mounted) and the drawing is downloadable from the manufacturer's web site at https://www.edmundoptics.com/document/download/418698  • Yep, I'll clean it up a bit – M.A.H. Jun 18 '18 at 4:00
• I hope you don't mind the additional edits. Feel free to adjust further. I've also added the conclusion "Therefore no, it is not likely to be corrected for aberration." Again feel free to adjust or remove. – uhoh Jun 18 '18 at 4:36
• – uhoh Jun 18 '18 at 6:34