# Can a solar mirror increase an ion engine 's output?

Could an ideal size solar sail work as a parabolic mirror to increase the out put of an ion engine? Would the extra weight encumber the ion engine nullifying the increases sunlight on the ion engine? How much thrust is lost from the curvature and reflective change from the ideal shape of a solar sail? What would be the shape of it?

I would like to revisit some aspects of these questions:

At what size does a bigger solar sail stop increasing your acceleration?

Can a solar sail be added to an ion engine and work better?

• This is a really interesting question! I really like it and I'm writing an answer now, please don't change it too much, thanks! – uhoh Dec 23 '18 at 0:50
• Are you interested in combining a solar sail (generating thrust by light pressure) with an ion engine, or are you interested in using a mirror as a solar concentrator to increase electrical power available to an ion engine? – Steve Linton Dec 23 '18 at 9:03
• @SteveLinton both – Muze the good Troll. Dec 23 '18 at 14:52
• I think you'd have to be quite lucky with the geometry to use the same mirror top do both jobs at the same time – Steve Linton Dec 23 '18 at 15:48
• Thinking about this a little I realise it depends on your mission. Specifically on the total delta-V you want. At the low end, I am pretty sure the light pressure on the sail is negligible and you are operating an ion engine with a solar concentrator. At the high end, you will run out of Xenon, so you are basically operating a solar sail with a modest initial boost from an ion engine (which you probably then discard). – Steve Linton Dec 29 '18 at 0:49

This is a start, I'll add to this answer later today, or someone else is welcome to add an answer with more quantitative information.

I'll look at a solar sail in heliocentric orbit used to raise the orbital distance from the Sun. For a given mirror size and mass, it means you need roughly a 45 degree tilt to maximize the bang for the buck (delta-v per unit time per unit mass).

Let the $$\mathbf{y}$$ axis point towards the Sun and the $$\mathbf{x}$$ axis point in the direction of orbital motion (so $$\mathbf{\hat{y}}, \mathbf{\hat{x}}$$ are proxies for $$\mathbf{\hat{r}}, \mathbf{\hat{\theta}}$$ respectively) We can add scale to the problem later, so right now let's define the dimensionless parabola as

$$y=\frac{1}{2}x^2-1$$

with the focus at $$(0, 0)$$. We can already guess that the part of the mirror with the "best" thrust/mass will be somewhere around 45 degrees (at about $$(1, 0)$$) but let's see where it is.

The sail will have a mass per unit length, so we need $$ds$$:

$$ds = \sqrt{dx^2 + dy^2}$$

Also needed is the Force per unit length $$\frac{dF}{ds}$$:

$$\frac{dF}{ds} = \frac{dF}{dx} \frac{dx}{ds} = \frac{dx \sin(2 \theta)}{\sqrt{dx^2 + dy^2}}$$

The plot highlights the 95% of maximum region:

lower    peak   upper
x     0.53    0.71    0.93
θ    28.0    35.3    42.8

Optically that looks like this:

But if you used one big one, your solar collector (PV array) would be far from the sail. It's much better to have many small ones so that you don't add as much mass supporting the collectors accurately so far away.

import numpy as np
import matplotlib.pyplot as plt

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

x  = np.linspace(0, 1.2, 2001)
y  = 0.5*x**2-1

dx = x[1:]-x[:-1]
dy = y[1:]-y[:-1]
ds = np.sqrt(dx**2 + dy**2)

slope = dy/dx
theta = np.arctan2(dy, dx)

dFdx = np.sin(2.*theta)

dFds1 = dFdx * (dx/ds)
dFds = (dx * np.sin(2.*theta))/np.sqrt(dx**2 + dy**2)

nmaxi    = np.argmax(dFds)
dFdsmaxi = dFds[nmaxi]
xmaxi    = x[nmaxi]
thmax    = degs * theta[nmaxi]

n95L     = np.argmax(dFds>0.95*dFdsmaxi)
n95R     = np.argmax(dFds[nmaxi:] < 0.95*dFdsmaxi) + nmaxi

print x[n95L], x[nmaxi], x[n95R]
print degs*theta[n95L], degs*theta[nmaxi], degs*theta[n95R]

if True:
plt.figure()

plt.subplot(2, 1, 1)
plt.plot(x[:-1], dFds, '-k')
plt.plot(x[:-1][n95L:n95R+1], dFds[n95L:n95R+1], '-r', linewidth=6)
plt.xlabel('x', fontsize=16)
plt.ylabel('dF/ds', fontsize=16)

plt.subplot(2, 1, 2)
plt.plot(degs*theta, dFds, '-k')
plt.plot(degs*theta[n95L:n95R+1], dFds[n95L:n95R+1], '-r', linewidth=6)
plt.xlabel('theta (deg)', fontsize=16)
plt.ylabel('dF/ds', fontsize=16)

plt.show()

"""
lower    peak   upper
x     0.53    0.71    0.93
th   28.0    35.3    42.8
"""

xx = x[n95L:n95R+1]
yy = y[n95L:n95R+1]
xxL = np.array([xx[0], xx[0], 0])
yyL = np.array([1, yy[0], 0])
xxR = np.array([xx[-1], xx[-1], 0])
yyR = np.array([1, yy[-1], 0])
O   = np.zeros(1)

if True:
plt.figure
plt.plot(x,  y, '--k', linewidth=0.5)

dx = xx[-1] - xx[0]
dy = yy[-1] - yy[0]

for n in range(1):
plt.plot(xx  + n*dx, yy  + n*dy, '-b', linewidth=6)
plt.plot(xxL + n*dx, yyL + n*dy, '-b')
plt.plot(xxR + n*dx, yyR + n*dy, '-b')
plt.plot(O   + n*dx, O   + n*dy, 'ok')

plt.xlim(-0.1, 1.1)
plt.ylim(-1.1, 0.1)
plt.show()

if True:
plt.figure
#plt.plot(x,  y )

dx = xx[-1] - xx[0]
dy = yy[-1] - yy[0]

for n in range(5):
plt.plot(xx  + n*dx, yy  + n*dy, '-b', linewidth=6)
plt.plot(xxL + n*dx, yyL + n*dy, '-b')
plt.plot(xxR + n*dx, yyR + n*dy, '-b')
plt.plot(O   + n*dx, O   + n*dy, 'ok')

plt.xlim(-0.1, 2.1)
plt.ylim(-1.1, 1.1)
plt.show()