This should be a modestly simple application of Newton’s Laws, but I’m having trouble finding the right numbers.
Imagine that there’s a probe approaching our solar system. It has a mass of 1 tonne, plus the mass of its solar sail. Its relative velocity is 0.1c - 30,000 Km/s. It deploys a solar sail once it breaches the local heliopause, 20 Bn Km out. (Yes, I know Voyager has detected the start of it at 17.4 Bn Km…). Its target is Earth orbit. 150 M Km from the Sun. So - using v[sup]2[/sup]=u[sup]2[/sup]+2as - it needs to decelerate at about 0.1mm per second. F=ma gives us a required force of about 1 Newton (funny, that :D). And it’s at this point that I’m stuck because I can’t find decent values for the pressure of the solar wind - NASA mixing Imperial and Metric units by saying that it’s 9 Newtons per sq mile at Earth’s distance is not helpful. Does it decrease linearly? By the square, by the cube? Or is the rate of change not relevant?
How large and how massive a solar sail is needed to decelerate the probe to achieve Earth orbit?
OK, put it in one system or the other. one square mile = 2,589,988 square meters, so your pressure is 3.4749x10[sup]-6[/sup] Pa.
Things that radiate from a point source - light, sound, generally decrease in intensity with the square of distance from the source:
Ar[sup]2[/sup] = BR[sup]2[/sup]
Where:
A = solar wind pressure at earth’s orbit
r = radius of earth’s orbit, measured to sun
B = solar wind pressure at distance R from sun
The sun is not a true point source - it has a non-zero size - so that relationship is only accurate at distances that are large compared to the sun’s diameter. You’re looking at what happens at earth’s orbit and beyond (distance, 149M km, compared to sun’s diameter of 1.4M km), so you should be OK.
The units you have written here are units of velocity, not acceleration.
NASA doesn’t ever use “Imperial” units, nor do any other engineers in the U.S., because the Imperial system (and the British Empire, for that matter) came into existence decades after the independence of the U.S. from Britain. Instead, U.S. engineers use the U.S. Customary System. There are differences in the units between the two systems.
Also, while it’s not incorrect to refer to the other system of units as “metric,” the actual system of units is the SI system.
Two comments in addition to those already out there: I think you may have miscalculated the acceleration required—I get a number different than yours by around 5 to 6 orders of magnitude; 0.1 c is fast—and the weight of the sail is going to be far in excess of the weight of the rest of the spacecraft (unless we’re assuming unobtainium).
I used silica aerogel’s density (~1mg/cc or 1kg/cubic meter if you prefer), the thickness of 750 gauge Mylar (no idea how thick you’d need to make a solar sail made from aerogel), and treated the solar wind as the constant you supplied (9N per 2.59 x 10^6 m^2). (Too lazy to integrate the pressure over the distance between the heliopause and Earth, using the relationship that Machine Elf kindly supplied)
With all that, I got a figure for the sail weight necessary to have sufficient area to only decelerate the spacecraft (so, neglecting decelerating the sail’s own weight) around a thousand times that of the rest of the spacecraft. Of course, the sail is likely to be made of denser materials than the least dense material currently known and the solar pressure will be less around Pluto than that 9N per area, so your sail’s going to have to be even bigger than that.
It seems then that it would be impossible to construct a spacecraft+sail assembly that would provide the OP’s desired deceleration. If a sail weighing 1000 tonnes provides only enough force to decelerate a 1-tonne spacecraft at the desired rate, then even if the spacecraft is removed, the sail can only decelerate its own mass at 1/1000th the desired rate.
Moreover, the gravitational attraction of the sun will be fighting you the whole way.
More moreover, for most of the probe’s journey, the pressure of the solar wind will be far, far less than the value at earth’s orbital radius; at 20 billion km out - 134.4 times earth’s orbital radius - the pressure will be 1/18,066 part of the value Gray Ghost used in his calculation. The average pressure over the whole journey will somewhere between the two extremes; given the inverse square relationship, the average should be somewhat less than half the value at earth’s orbital radius.
Either the sail has to be made a lot lighter for the same area (such that it weighs a small fraction of a tonne), or the OP has to settle for a much, much smaller deceleration. Probably both.
A solar sail can operate on two different sources, both using momentum transfer; radiation pressure, the intensity of which decreases as a square factor with distance from the star, and the charged particles ejected from the sun that make up the heliosphere, which varies with both distance and orientation. The effect of the former is easy to calculate, based upon the albedo of the reflector and distance from the sun, along with standard solar intensity values. The latter is more difficult; there are some average values for the density and speed of the interplanetary medium but they vary from the mean. In general, solar sail propulsion is not going to provide an effective amount of active thrust at the distance of Jupiter or beyond, as solar intensity and momentum flux of the solar wind is just too small to be significant, although at velocities of 0.1 c the drag from even a static medium will provide some thrust. You would also want to utilize swingbys of one or more of the massive outer planets in order to waste excess energy and make direction changes (as we do with outgoing interplanetary probes).
However, entering the solar system at 0.1 c it is probably not feasible to slow a massive craft via solar sail regardless of how large the area feasibly could be. As Gray Ghost and Machine Elf have indicated, the mass of the sail will dominate and ultimately limit the effectiveness of such deceleration. I haven’t run even a back-of-envelope calculation on the o.p.'s scenario, but I’ve previously studied solar sail propulsion and what he proposes is just way outside of the level of effectiveness of such systems. However, if you use a magnetic sail (similar to a Bussard ramjet setup, but instead of funneling particles into the engine you use the drag to slow it) and start decelerating a few light years away, you could feasibly get to 1% or less of c; still probably not slow enough to stop by solar sail, but at least it gets you within the realm of feasibility for some really energetic propulsion system. So now you just need a really powerful superconducting web (preferably using magnetic monopoles for fine control), a powerful and efficient fusion rocket for insystem deceleration and maneuvering, and some way to get all of this up to relativistic speeds. If you can do all this and manage to get yourself beatified, you’ll be a candidate for canonization.
