The Virtuosi (Posts about solar sails)https://thephysicsvirtuosi.com/enContents © 2019 <a href="mailto:thephysicsvirtuosi@gmail.com">The Virtuosi</a> Thu, 24 Jan 2019 15:05:02 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rss- Solar Sails III (because two just isn't enough)https://thephysicsvirtuosi.com/posts/old/solar-sails-iii-because-two-just-isn-t-enough-/Corky<p><a href="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_IKFerS7nI/AAAAAAAAACo/DPNAyMeuMaQ/s1600/leezle+pon+justice.jpg"><img alt="image" src="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_IKFerS7nI/AAAAAAAAACo/DPNAyMeuMaQ/s320/leezle+pon+justice.jpg"></a>
One thing that I've wanted to quantify since reading <em>Intelligent Life
in the Universe</em>, an outstanding book by Carl Sagan and I.S. Shklovskii,
is the idea of exogenesis. Exogenesis is the hypothesis that life formed
elsewhere in the universe and was somehow transferred to earth in the
form of some small seed or spore. Now since E.T. E. coli presumably do
not have little tiny jetpacks or other means of active transport, they
would need to traverse the cosmos in some passive way. One such way
would be solar sailing.
Way back in Solar Sails I, we derived equations describing the maximum
speeds and time-of-travel for various distances for a given solar sail.
Each of these equations was a function of the surface mass density of
the sail, which is just the amount of mass per unit cross-sectional
area. All we need to know is the cross-sectional area and mass of a
given object and we can apply these equations to just about anything!
Assume we have some spherical blob with the density of water (1g/cm^3).
The effective sigma of this blob would just be the mass divided by the
cross-sectional area. In other words,
$$ \sigma = \frac{m}{Area} = \frac{\frac{4}{3}\pi r^3 \rho}{\pi
r^2} = \frac{4}{3}\rho r .$$
Rearranging to get r in terms of the other variables, we have
$$ r = \frac{3\sigma}{4\rho} . $$
Plugging in our density of 1g/cm^3 and a suitable sigma (10^-4
g/cm^2), we get
$$ r \le 0.75 \times 10^{-4} cm = 0.75 \mu m .$$
Check out
<a href="http://learn.genetics.utah.edu/content/begin/cells/scale/">this</a> fun
site to see what kind of critters can fit in this blob.
From the previous post, we saw that for a sigma of 10^-4 g/cm^2, our
sail would get to the nearest stars on a timescale of order 10,000
years. Thus if our blob has a radius of less than about a micron, it
could spread to hundreds of stars in around 10,000-100,000 years. Even
if it would take millions of years, that would be almost no time at all
on the cosmic scale. Just based on this calculation it all seems fairly
feasible.
In making these calculations I have neglected several important aspects
of the problem. First, in no way have I actually calculated any sort of
probability of this happening. Additionally, I would have to see how
likely it is for some blob to reach planetary escape velocity
(presumably just by riding that tail of the Boltzmann distribution).
Finally, and perhaps most important of all, I have not given any sort of
motivation or mechanism by which some living body could survive hundreds
of thousands of years in the vacuum of space with constant radiation
exposure. But I have heard that some forms of life are totally
<a href="http://en.wikipedia.org/wiki/Extremophile">extreme</a> (especially if they
drink <a href="http://en.wikipedia.org/wiki/Mountain_Dew">this</a>).
Even though such a process seems possible, it certainly doesn't seem
like the easiest way to get life on earth. I prefer the much more
satisfying "amino acids + lightning + magic =
<a href="http://en.wikipedia.org/wiki/Abiogenesis">life</a>" model. But it does
offer some interesting possibilities. Suppose we as people think that
people are super awesome and therefore people should be everywhere. We
do some bio magic and put whatever DNA we want into viruses, which we
then pack into as many micron spheres as we can make. We then point them
at the nearest stars and have them disperse.
What would the probabilities be that they land somewhere habitable? Are
there any ethical considerations in doing this? Is it a galactic faux
pas?</p>do the dewexogenesisscott bakulasolar sailshttps://thephysicsvirtuosi.com/posts/old/solar-sails-iii-because-two-just-isn-t-enough-/Mon, 17 May 2010 22:24:00 GMT
- Solar Sails IIhttps://thephysicsvirtuosi.com/posts/old/solar-sails-ii/Corky<p>[NOTE: In my hurry to make up for weeks of non-posts, I managed to
almost immediately knock Nic's <a href="http://thevirtuosi.blogspot.com/2010/05/why-black-holes-from-large-hadron.html">first
post</a>
from the top of the page. It's got the LHC, black holes, and about 3
full cups of metric awesome, so make sure you check it out (after
reading this one, of course).]
Last time we did some calculations on how fast and far our solar sails
can go, but those calculations were just for the sail itself. If you are
going to do any science with it, you're going to need a payload. Let's
take it a step further and make it an actual spaceship (with people and
everything!).
<a href="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_DXvcSsDXI/AAAAAAAAACg/jK_N-B4mOME/s1600/ssradius.png"></a>
Comparing it with a typical people-carrying space hotel (the
International Space Station), let's give our payload a mass of 300,000
kg. Remembering from the last post that a sigma of less that about
10^-4 g/cm^2 gave fairly nice results, we can make an effective sigma
of our payload carrying sail as
$$ \sigma_{eff} = \frac{m_{total}}{Area} = \frac{m_s +
m_p}{Area}, $$
where m_s is the mass of the sail and m_p is the mass of our payload
(the ship). Assuming the sail has some surface density of sigma and the
sail is circular with some radius r, we have
$$ \sigma_{eff} = \frac{\pi r^2 {\sigma}<em eff>s + m_p}{\pi r^2} =
{\sigma}_s + \frac{m_p}{\pi r^2} . $$
Now we can find the radius of our sail such that
$$ \sigma</em> \le 10^{-4} \frac{g}{{cm}^2}. $$
Rearranging our equation above and solving for radius, we find that
$$ r \ge \left[\frac{m_p}{\pi \left( 10^{-4}\frac{g}{cm^2} -
\sigma_s \right)} \right]^{1/2} cm. $$
Below is a plot of sail radius (in meters) against sail surface density
(g/cm^2).
<a href="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_DXvcSsDXI/AAAAAAAAACg/jK_N-B4mOME/s1600/ssradius.png"><img alt="image" src="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_DXvcSsDXI/AAAAAAAAACg/jK_N-B4mOME/s400/ssradius.png"></a>
From this plot, we see that we will need our sail radius to be AT LEAST
10 km and the surface density of our sail must be less than 10^-4
g/cm^2. Now that's a big sail, but it's not obscenely big (depending,
of course, on your definitions of obscenity). One could certainly
imagine such a sail being built, but it would be an impressive
engineering feat.
So get to work engineers! I've already made a whole plot in Mathematica,
I can't do everything.</p>Nic shout outsscott bakulasolar sailshttps://thephysicsvirtuosi.com/posts/old/solar-sails-ii/Mon, 17 May 2010 01:15:00 GMT
- Solar Sails Ihttps://thephysicsvirtuosi.com/posts/old/solar-sails-i/Corky<p><a href="http://1.bp.blogspot.com/_fa6AZDCsHnY/S_C9Ll986gI/AAAAAAAAACY/eTykcbU6PTE/s1600/solarsail.jpg"><img alt="image" src="http://1.bp.blogspot.com/_fa6AZDCsHnY/S_C9Ll986gI/AAAAAAAAACY/eTykcbU6PTE/s200/solarsail.jpg"></a>
Solar sails are in the news again, and this time not just for <a href="http://www.cbsnews.com/stories/2005/06/22/tech/main703405.shtml">blowing
up</a>.
The Japanese space agency is
<a href="http://www.space.com/businesstechnology/japan-venus-double-mission-100516.html">launching</a>
what they hope to be the first successful solar sail tomorrow. In honor
of that, we will be discussing the physics of solar sails.
First of all, what the heck are solar sails? Solar sails are a means of
propulsion based on the simple observation that "Hey, sails work on
boats. Therefore, they should work on interplanetary spacecraft (in
space)." Boat sails work when air molecules hit into the sail and bounce
back. By conservation of momentum, this gives the boat sail an itty
bitty boost in momentum. Summing over the large number of air molecules
moving as wind, the boat gets pushed along in the water. A similar
process works with solar sails, but instead of air molecules doing the
hitting, it's photons. Since each photon of a given wavelength has some
momentum, by reflecting that photon the solar sail can gain a tiny bit
of momentum. Summing over the large number of photons coming from the
sun over a long time frame we can get a considerable boost. So let's see
how good solar sails are.
First we need to find the net force on our sail. We will certainly have
to deal with gravitational forces (which will slow us down) :
$$ F_{g} = \frac{-GM_{\odot}m}{r^2} $$
where big M is the mass of the sun and little m is the mass of the sail.
Now we need to find the radiation force on the sail. Since force is just
rate of change of momentum, we can find the change of momentum of one
photon per unit time, then find how many photons are hitting our sail.
So for one elastic collision of a photon with the sail, the change in
momentum will be
$$ \Delta p = 2 \frac{h\nu}{c} $$
and by conservation of momentum, this will also be the momentum gained
by the sail. Now we want to find the number of photons incident on a
given area in a given time. This will just be the energy flux output by
the sun ( energy/ m^2 s ) divided by the energy per photon. In other
words:
$$ f_n = \frac{L_{\odot}}{4\pi r^2}\frac{1}{h\nu} .$$
So now we can get a force by
$$ \text{Force} = \left(\frac{\Delta p}{\text{1 photon}} \right)
\times \left(\frac{\text{number of photons}}{area \times
time}\right) \times \left( Area\right) $$
which is just
$$ F_{rad} = 2 \frac{h\nu}{c} \times \frac{L_\odot}{4\pi r^2
h\nu} \times \pi R^2 = \frac{L_{\odot} R^2}{2cr^2} .$$
So combining the radiation force with the gravitation force, we have a
net force on the sail of
$$ F = \left( \frac{L_{\odot} R^2}{2c} - GM_{\odot}m \right)
\frac{1}{r^2} .$$
This can then be integrated over r to find an effective potential,
giving:
$$ U = \left( \frac{L_{\odot} R^2}{2c} -
GM_{\odot}m\right)\frac{1}{r} .$$
For simplicity, let's just write that
$$ \alpha = \frac{L_{\odot} R^2}{2c} - GM_{\odot}m $$
so
$$ U = \frac{\alpha}{r} .$$
Now we can start saying some things about this sail. The most
straightforward quantity to find would be the maximum velocity. By
conservation of energy (and starting from some r_0 at rest), we have
that
$$ v_f = \left[\frac{2\alpha}{m} \left(\frac{1}{r_0} -
\frac{1}{r_f} \right) \right]^{1/2} $$
So as r_f goes to very large values, the subtracted piece gets smaller
and smaller. In the limit that r_f goes to infinity we have that
$$ v_{max} = \left(\frac{2\alpha}{mr_0}\right)^{1/2} .$$
Plugging back in our long term for alpha and plugging in some numbers we
get:
$$ v_{max} = 42,000 m/s \left( \frac{1.5 \times 10^{-4}}{\sigma} -
1\right)^{1/2} $$
where sigma is just the surface mass density [g/cm^2] of the sail.
Below is a plot of maximum velocity ( m/s) plotted against surface mass
density (g/cm^2). For a sigma of 10^-4 g/cm^2, we get a max velocity
of about 30,000 m/s. Not bad.
<a href="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_CjbwxG-JI/AAAAAAAAAB4/PS7tTqbmLUE/s1600/maxvel.png"><img alt="image" src="http://4.bp.blogspot.com/_fa6AZDCsHnY/S_CjbwxG-JI/AAAAAAAAAB4/PS7tTqbmLUE/s400/maxvel.png"></a>
From this graph we see that there must be some maximum surface density,
above which we don't get any (forward) motion at all. This makes sense
since we want our radiation forces (which scale with area) to overcome
our gravitational forces (which scale with mass). And below this maximal
surface density we see a power law behavior. Cool.
We can also find the distance traveled as a function of time. Taking the
final velocity equation above and writing v as dr/dt, we see that
$$ \frac{dr}{dt} = \left[ \frac{2\alpha}{m} \left( \frac{1}{r_0}
- \frac{1}{r_f} \right)\right]^{1/2} $$
Rearranging and integrating, we can get time (in years) as a function of
distance r (in AU):
$$ t = \frac{0.11 \left(\sqrt{(-1+r)
r}+\text{Log}\left[1+\sqrt{\frac{-1+r}{r}}\right]+\frac{\text{Log}[r]}{2}\right)}{\sqrt{-1+\frac{1.5
\times 10^{-4}}{\sigma}}}$$
A plot of t vs. r is shown below for typical solar system distances and
a sigma of 10^-4 g/cm^2. We assume that we are launching from earth (1
AU). Since Pluto is at a distance of about 40 AU, we see that our sail
could get there in less than 7 years. For comparison, the <a href="http://pluto.jhuapl.edu/">New
Horizons</a> probe will use conventional
propulsion to get to Pluto in 9.5 years (and it is the fastest
spacecraft ever made).
<a href="http://1.bp.blogspot.com/_fa6AZDCsHnY/S_Ck6enlpVI/AAAAAAAAACI/_gjIHLCm_G8/s1600/ssplutolong.png"><img alt="image" src="http://1.bp.blogspot.com/_fa6AZDCsHnY/S_Ck6enlpVI/AAAAAAAAACI/_gjIHLCm_G8/s400/ssplutolong.png"></a>
Zooming in to our starting point around 1 AU, we see that there is a
period of acceleration and then the maximum velocity is reached after a
few months. Just eyeballing it, it looks like it takes at least a month
to reach appreciable speed. That it takes so long is a result of the
very small forces involved due to radiation pressure. But even a small
acceleration amounts to a considerable speed if applied for long enough!
<a href="http://1.bp.blogspot.com/_fa6AZDCsHnY/S_CopsUAIrI/AAAAAAAAACQ/_-IqCnf8xs8/s1600/sscloseup.png"><img alt="image" src="http://1.bp.blogspot.com/_fa6AZDCsHnY/S_CopsUAIrI/AAAAAAAAACQ/_-IqCnf8xs8/s400/sscloseup.png"></a>
Now Pluto is fine I guess (it's the second largest dwarf-planet!), but
how about some interstellar flight? Well, the nearest star is Proxima
Centauri which is about a parsec away. A parsec is 3<em>10^16 m, or about
200,000 AU. From, the plot below (or plugging in to the equation above),
we see that such a trip would take of order 10,000 years. That's a long
time, but its not too shabby considering this craft uses no fuel of its
own.
<a href="http://3.bp.blogspot.com/_fa6AZDCsHnY/S_Cgqa_fqDI/AAAAAAAAABw/xxgjj5VEZww/s1600/ssTOTHESTARS.png"><img alt="image" src="http://3.bp.blogspot.com/_fa6AZDCsHnY/S_Cgqa_fqDI/AAAAAAAAABw/xxgjj5VEZww/s400/ssTOTHESTARS.png"></a>
So solar sails can do some fairly impressive things simply by harnessing
the free energy of the sun. Though this only provides a very small
acceleration, it can be taken over a long enough time to be useful.
However, since the radiation pressure of the sun falls off as 1/r^2, we
start to observe diminishing returns and the sail reaches a max
velocity. But overall the numbers seem fairly impressive. All that
remains now is whether they are feasible to construct. Right now my only
data point for feasibility was that it was in <a href="http://www.starwars.com/databank/starship/solarsailer/">Star
Wars</a>, but as I
recall that was a </em>long* time ago.</p>Geonosian solar sailerscott bakulasolar sailshttps://thephysicsvirtuosi.com/posts/old/solar-sails-i/Sun, 16 May 2010 23:29:00 GMT