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The Maury Equation, redux March 17, 2012

Posted by Maury Markowitz in solar power satellites.
Tags: ,
Mir's solar panels were heavily damaged after years in space


It seems that every time a space-based solar power supporter reads my post on The Maury Equation, they pick one point of the many and try to attack the entire argument on that. That point normally has to do with launch costs, which isn’t the real argument at all.

So, let’s make this simple:

A solar cell in space will deliver less energy to the grid than the same cell on Earth.

Energy delivery

The amount of energy delivered to the grid is easily estimated. It looks like this: E = R x I x L x T Where:

  • E is the total energy delivered over the lifetime of the system
  • R is the “nameplate rating” of the cell (or panel), the amount of power it produces under optimum conditions
  • I is the “insolation”, the amount of sunlight hitting the cell in a year
  • L is the system lifetime, in years
  • T is the loss in transmission between the supply and demand

R is simply a measure of the system’s size. In order to avoid apples-to-oranges comparisons, we want our systems to be the same size when comparing. Generally we use standard size systems, 1 kW, so this number is 1 in both equations and you can ignore it.

I, for insolation, can be measured in a number of ways. The simplest, and the one I use here, is in “hours of bright direct sunlight per year”. In space that is simply 365 x 24, but down here on Earth we have night and weather to contend with. There’s no way to guess it, the only way to get an accurate number is us an insolation calculator.

That equation is normally used for ground-based applications, but we’re talking about space. So I’m going to change it very slightly to make this part of the argument clearer. Instead of a single term for T, I’m going to use two (and R has been eliminated): E = I x L x Tg x Ts Where:

  • Tg is the loss in transmission between the supply and demand (g for ground)
  • Ts is the loss between the two antennas (s for space)

Since Tg is the same for a field of solar panels in a field or a field of rectennas in a field, it too can be eliminated from both sides of the equation.

There are some other effects, like the slightly strong sunlight in space and the slightly higher efficiency of panels when on Earth (yes, that’s correct), but they are minor considerations that have no material effect on the outcome.

Real world

So let’s put in the numbers. For solar panels on mountings facing south in the Nevada desert, the industry-standard PVWatts tools and real-world panel lifetime produces uses these input numbers:

  • I = 2300 (using a tracker)
  • L = 40 years
  • Ts = 1

So every 1 kW of panels in Nevada will produce 92,000 kWh of power over its lifetime. In space, the numbers look like this:

  • I = 8760
  • L = 12 years
  • Ts = 0.50 (see this article for reasons why)

So every 1 kW of panels in space 52,560 kWh over its lifetime. 52,560 < 92,000.

The long and short of it is that space is an extremely nasty place. Putting anything up there seriously shortens its life, and that includes humans.

If you’re wondering where those lifetime numbers come from, it’s the measured real-world 20% degradation rate. That is, it’s the measured time it takes for the panels to drop their output by 20% from their original power. The numbers for ground-mounted panels are widely available, and you can read NASA’s experienceNATO papers and this lengthy report (page 422) for the space examples.

Why that number, 20%? It’s just the number the industry uses. It’s widely measured and available, as opposed to some other number, say the time to 50% reduction which you will not easily find.

End of story

If you take a panel and put it in Nevada it will deliver some kWh. If you put that same panel in space, it will make less kWh.

What else needs to be said?

If previous history is any guide, lots. So I’ll reiterate my previous note: if you’re going to take issue with the basic gist of this article, post your math.


1. ivancho - March 17, 2012

Something got lost in transmission here. “A solar cell in space will deliver less power to the grid than the same cell on Earth.” does not follow from 31,762 kWh in Nevada and 44,225 kWh in space.

Mind you, I don’t think that changes your argument, you still need 3 orders of magnitude drop in launch costs to get them in the same ballpark

Maury Markowitz - March 17, 2012

Odd, I must have cut-n-pasted from the wrong place. The lifetime on Earth is 40 years, not 25 (25 is the warrantee). These are real-world numbers in both cases.

Hmmm, I also notice some weird styling there too.

Maury Markowitz - March 19, 2012

Ugg, AND I used the wrong insolation number! The number I put into the formula was the one *after* applying the derate. It’s 2400 before. Fixed and fixed.

2. Terry M Peterson - April 16, 2012

Maury, another trivial error: I for space (the number of hours per year) is 8760 (24*365), not 8670. 🙂

Doesn’t change anything in your argument, though. I’ve long been a space-to-earth PV doubter based on the launch costs, but your point about cell lifetime differences adds a much stronger aspect.

3. Philip K. Chapman, Sc.D. - April 17, 2012

This analysis is so full of elementary mistakes that it is not worth reading, let alone correcting. If you want to be taken seriously, Maury, I suggest you start by learning the difference between power and energy. I am sure you could find some smart ten-year-old who would enlighten you.

Maury Markowitz - April 17, 2012

As I mentioned in the lead, in the past my posts on this topic have garnered some level of negative response from the supporters of SBSP. Invariably these have ignored the main point, and most could be considered to be nit picking.

But none like this! This complaint isn’t even about the topic, it’s about terminology. Way to go, Phil!

Philip is pointing out the difference between energy and power. Power is, by definition, the rate of delivering energy. Common measurements include Watts, horsepower and, for the Germans in the audience, pferdestärke.

If power is the rate at which energy is delivered, then energy is the product of power and time. And in this case, that time is in hours, the power is in kW, so then kWh is, indeed, a measure of energy.

But of course there is often the difference between technical and common usage of many terms. For instance, one’s car does not actually skid around corners due to centrifugal force, but centripetal – your car is actually trying to go straight. In spite of this terrible mis-application of the technical language, we all manage to agree on what we’re talking about.

So, consider, do you pay your power company, or your energy company? And what’s that on the outside of your house? Do you call it an energy meter, or a power meter? Do you call what comes out of a PV system solar power, or solar energy? Or both?

Well, like many cases of terminology, that depends on where you hail from. Here in North America, we tend to use “power” over “energy” in all of these cases. Inaccurate as it may be, that’s the way it is.

Another thing I’ve noted in the past is that the entire SBSP concept really has nothing to do with power – it has everything to do with building large rockets. At the recent SBSP meeting here in Toronto, for instance, there wasn’t a single person attending who had anything to do with the power industry. There were lots and lots from the space and science fiction industries though.

Philip was in the NASA astronaut corps (good on you). And he originates from Australia. So there you go.

Lets get back to the actual argument, shall we?

Philip, how do you propose to reduce the transmission losses?

How do you propose to eliminate radiation damage to the panels?

How do you propose to stop the Kessler Syndrome (which Don himself has stated makes the construction of SBSP basically unreasonable to consider)?

And then, on top of that, how do you propose to reduce launch costs to $1 a pound – which is about what it costs to ship to Nevada?

Because unless you have a solution to *all* of these problems, SBSP will, as the post demonstrates, deliver less power over its lifetime.

I’m all ears!

4. Kieran A. Carroll - April 17, 2012

Hey Maury, long time no see.

I appreciate that you think you’ve found a simple, top-level way of looking at the SBSP question, which in an easy-to-understand way makes it clear that it can’t compete with terrestrial-based photovoltaic installations. And that therefore SBSP is not worth spending any further time on.

Given the scope of what you’ve addresed, your math looks basically fine to me — I could quibble about the values of some of the numbers, but they wouldn’t swing the result by more than maybe 50%

However, I want to point out that your argument ignores a fundamental and significant factor: the intermittent nature of power delivered by terrestrial PV installations. This causes major problems for users of power from such installations, because the intermittency of the power delivered doesn’t match the time-varying demand for power. This is reflected in the need for large and expensive battery installations to store power when it’s sunny, for delivery of power when it’s not sunny, which is one solution that (for example) cottagers trying to add PVs to their cottage would face. In the wider world of Ontario Hydro, this is instead reflected in the need to add backup generation capability for when the sun don’t shine, typically natural-gas generation. Also, the way that Ontario’s energy market has been constructed, this has also resulted in significant amounts of excess power being available when it’s sunny (and also, due to windmills, when the wind is blowing), which the power system *must* purchase, and then re-sell, often at a loss.

Ignoring these costs weakens your argument. It may weaken it fatally.

Determining what these costs are is not easy — I’ve put some time into trying to determine the cost of using battery backup for terrestrial PV and windmills, and found that it is astoundingly high for every type of “battery” anyone has ever tried (except for storing energy by pumping water uphill during surplus times, and letting it flow through generators when needed, but almost all places where you can do that economically are already doing it). My goal in doing those calculations was actually as part of a business plan for an innovative energy-storage technology. However, my conclusion was that, when you factor in the energy-storage cost (or the cost of plants to provide backup power), you may as well just install the plants to provide backup power, and forget the PVs and windmills. Until, of course, the natural gas starts running out, and the cost of power from natural gas starts getting too expensive. And, of course, if you don’t like adding CO2 to the atmosphere, you might want to curtail such things even earlier (not to mention avoiding the use of coal to generate power).

Taking these costs into account is central to the argument for space-based solar power. If you want to claim that SBSP obviously makes no sense, you have to address this factor.

Maury Markowitz - April 17, 2012

For those of you who don’t know Kieran, he’s an actual rocket scientist, tireless popularizer of space technology, and all-round cool guy. We go back decades.

I’ll keep this short:

“Until, of course, the natural gas starts running out, and the cost of power from natural gas starts getting too expensive”

Fallacy of the excluded middle. This demands a comparison between space and terrestrial PV, as if nothing else exists.

There are literally hundreds of new power sources under development. In order for SBSP to be a winner, all of them have to fail. However, the chance that SBSP will fail is equal to any of them.

Kieran A. Carroll - April 19, 2012


You wrote, “For those of you who don’t know Kieran, he’s an actual rocket scientist, tireless popularizer of space technology, and all-round cool guy….” Aww, now I’m blushing 🙂 Actually, it’s mostly been satellites, but I actually *have* been no one launch-vehicle design team in recent years (just a study, maybe it’ll go somewhere eventually).

You may remember that Bryan Erb (ex-Avro, ex-NASA/Apollo Program, ex-CSA) gave some good papers/talks in the 1980s on the topic of the pros and cons of various large-scale sources of electrical power in the future. He reprised his main points in his talk at our SPS workshop in Toronto a couple of years ago, and I think that made it into John Mankins’ IAA study report. In those, Bryan made (I thought) a pretty good effort to be fair to all alternatives. Good reading, if you can get your hands on some of his papers (I’ll google around and see if I can find links to some downloadable versions). He definitely avoided trying to exclude the middle, and my point was drawing on his conclusions. The main points included:

– Coal, oil and natural gas put CO2 in the air, and will cause global warming, which will become bad (he was the first person I heard about global warming from, back in the 1980s).
– Nuclear is hard to grow much further, mainly due to problems with nuclear waste disposal, and proliferation of nuclear reactors to countries who’ll use them to make nuclear bombs (some of whom are crazy enough to then use those bombs).
– Geothermal is good as far as it goes, but only works well in a small number of locations.
– Wind and terrestrial PV (and other marginal things like tides) suffer from intermittency and the immensely high cost of energy storage.

Compared to those, space based solar power has no actually-known environmental problems (*maybe* there’s something we don’t yet known about health effects of microwaves, but that’s pure speculation), and can provide baseload power (i.e., it doesn’t need large-scale energy storage). It certainly is more expensive than the alternatives, especially the environmentally-awful ones. But not *that* much more expensive, if you can get get launch costs down to maybe $100/kg to LEO — which we’re finally moving towards. So, if you’re actually trying to engineer a solution to the world’s large-scale future energy problem, it seems likely that SBSP ought to be seriously considered as part of the mix.

To “seriously consider” it would involve such normal engineering activities as building and flying small-, medium- and large-scale pilot plants, to work the bugs out and develop a proper understanding of the issues (engineering, economic, environmental, health, etc.). This will take some money to accomplish — but the amount involved is a tiny tiny fraction of the costs of the alternatives, many of which receive a huge amount of funding despite their known deficiencies.

Anyway…not trying to exclude the middle, just trying to promote a proper evaluation of the SBSP alternative.

5. Milan Vanecek - April 19, 2012

I am physicist working on thin film solar cells.
I have one comment: the highest difference comes from expected lifetime L. If you take more optimistic data for L in space
(so you should think about another cells than single crystal Si or III-V compounds) then space technology will come much more favorable.
It is well established now, that thin film Si (amorphous, nanocrystalline) and thin polycrystalline CIGS cells posses a “self-healing” effect, so damage which kills crystallne Si cells (radiation created defect=recombination centers) does not matter for them.
And so lifetime can be the same as on the Earth.
Also if you compare illumination level for Germany (PV leader)
with insolation about half of the Nevada, again your gain in space is much larger.
The real obstacle is the price od space instalation.

Maury Markowitz - April 19, 2012

“It is well established now, that thin film Si (amorphous, nanocrystalline) and thin polycrystalline CIGS cells posses a self-healing effect”

I do not believe this is “well established”.

I am aware of the effect in terms of CIGS cells, notably Rossilion’s well known paper. However, I am not aware of any suggestion that these effects can counteract the damage of solar wind or cosmic rays, nor any example of such a system being used in space applications.

Instead, what we do actually have in the market are aSi products like Unisolar, which have “well established” enormous first-light degradation problems. And bankruptcy. Again, I am unaware of any use of aSi in space.

If you have examples of such systems being used in space, and a demonstrated extended lifetime in that environment, I’m willing to change the numbers in the article to reflect that.

6. Wolfi - May 4, 2012

Hi Maury,

thanks for the interesting discussion. Yet I’d like to question whether the total amount of energy produced is the only important point.

With the numbers presented here a 1 kW solar module in space produces 3,723 kWh of energy per year, whereas the same module produces only 1,771 kWh in Arizona. It comes down to comparing investment costs and profits from selling electricity (which of course still puts earthbound PV in the lead). I’m just saying that conditions are imaginable where space PV might become a viable option, even with shorter panel lifetimes. Electricity costs would have to be very high, whereas launch costs would have to shrink considerably.


7. Arun - April 26, 2014

Excellent analysis. This sort of basic costing is all to often ignored.

In the general “space solar power” question, I’ve often wondered why photovoltaics would be used in space. I can see how PVs are useful on Earth (where insolation varies due to day/night and weather and where you have small home-scale deployments), but in space they wear out and are pretty heavy.

What are the drawbacks of using solar thermal electrical generation (i.e. concentrate light on a heat engine, convert to electricity, beam that down)? It would seem that it could be lighter (you could use very thin foil to concentrate the solar radiation at the engine), the efficiency would be higher (reflectors and heat engines should be able to get high efficiency), and the degradation issues would be less (you would lose reflectivity to micrometers and such, but probably not 20%/year). A space-based solar thermal generator could be simpler than a ground-based solar thermal system because you wouldn’t need as much thermal storage (or maybe you could add in some thermal storage and use a lower orbit?)

Of course, it still may not be competitive with ground-based solutions, and certainly wouldn’t be with today’s launch costs.

I may have missed some huge factor in solar thermal vs. photovoltaics, but it seems like a large scale space deployment solar thermal would have some big advantages.

Maury Markowitz - April 26, 2014

And of course the other question, if we’re putting mirrors in space (like the latest JAXA example), why not just shine them on existing panels? That is something I need to track down, I suspect I’m missing something obvious.

Kieran A. Carroll - April 26, 2014


Interesting to see this thread come back to life…

You wrote, “…why not just shine them on existing panels?” Clouds will block them for much of the time. Also, mirrors will tend to reflect light over a larger area than a well-designed beamed-power system would, which is wasteful (you’d need to fly much more mirror acreage due to that effect), and also the spillover reflected light will be somewhere between irritating and dangerous to people outside your PV-panel farm on the ground.

– Kieran

8. Mario - April 26, 2014

KWh is not power, but energy. You are not comparing powers, you are comparing energies! So the first paragraph of “End of story” needs revising.

Maury Markowitz - April 26, 2014

Indeed. See comment #3, above.

9. Anon - March 30, 2015

I’m not a physicist, and I’m not going to attack your calculations or the conclusions you derived from them. But I do challenge your assumptions.

In my opinion the whole idea of putting a solar panel is space is all about size, whereas you seem to assume/imply that a solar collector on earth can be of the same size as one in space. Here on earth size is limited by a number of factors some physical others social or moral. You don’t want to live in a city covered in a giant solar-collector dome; you don’t want to cause a rise in food prices by using fertile land to collect solar energy and you can’t place your collectors in a desert and then transport it back to your cities without losing a portion of the collected energy.

In space however, ignoring launch costs, you can build as big as you want/can. Nobody is going to complain any time soon about ‘sky pollution’, since it’s improbable that you’d need a solar collector that big (just intuition). Nobody will complain about the rise in food prices, and your house-with-windows-to-the-south will not decrease in value.

Obviously launch (not to mention construction) are still major obstacles, but I don’t think it’s fair to compare apples an oranges (which is actually precisely what you’re doing in my opinion despite your claims to the contrary).

Maury Markowitz - March 30, 2015

Anon appears to be forgetting about the rectenna, which is substantially the size of the original collectors.

Kieran A. Carroll - March 30, 2015


(Goodness, this discussion is still alive after quite a long while!)

Rectennas can be made to be fairly sparse, something like chicken-wire. The SPS studies in the 1970s developed designs for “rectenna farms” that could be mounted on poles above farmland, sparse enough to let most of the light get through to the crops below.

– Kieran

Maury Markowitz - March 30, 2015
Mike - March 30, 2015

Sure, the rectenna is big, but can’t it receive from multiple collectors? If there’s a 1-1 equivalence between rectenna size and collector size, I agree with the pointlessness of the whole exercise.

Maury Markowitz - March 30, 2015

It’s not 1:1, but something similar. IIRC the baseline designs put about 1.25x peak sunlight energy into the beam, around 1250 W/m2, but I’ll look that one up. Unfortunately, the current limits on environmental microwaves are around 50 W/m2, so unless they get the international laws changed, the rectenna would be 20 times the size of the array!

10. Kieran A. Carroll - March 30, 2015


Something like that, but with rather less sun blockage. Again, think chicken-wire atop poles.

– Kieran

11. Gerald Monroe - March 31, 2015

My neighbor actually works on ISS power and heat rejection systems. The 20% degradation after 12 years is thought to be caused by molecular oxygen and thruster exhaust. A power station at geosync would not be subject to as much of this. (much higher orbit, cleaner vacuum, and you could design your power station so the burns to maintain position don’t end up with exhaust impingement)

So your numbers are wrong. Is your basic conclusion wrong? Well, $1000-$10k per launched kilogram goes a long way towards saying you are still correct. SBSP is not an effective idea with any technology humanity has now or in the near term. You’d need the ability to make the panels on the Moon in self replicating factories and launch them with mass drivers or something for it to pass the pencil test, I suspect. (which is obviously technology that will take decades to centuries to develop, depending)

12. Ean - April 1, 2015

So this has come up again in reference to some article about an old man in China thinking it’s a good idea.

My question is this: What about the exposure time over a day? It would vary slightly for the space collector and feasibly send some energy back to earth while the ground collector were in darkness. Depending on how far the geostationary orbit is from earth that might be a large angle or it might be a small one. Don’t know off the top of my head.

I have the feeling this would be not nearly enough to make up for the immense cost of this venture.

Maury Markowitz - April 1, 2015

That’s what the I figure takes care of. If it was just night/day it would be 1/2 as much insolation as in space. But the I figure also includes the effects of filtering in the air, the angle between the panels and the sun, reflection off snow, dirt on the panels, clouds, etc. It’s an all in one “this is what you’re going to get” number. And this is a measured value too, not theoretical. I know my panels in Toronto slightly outperform the estimates from PVWatts, but not statistically significant.

13. Kassandra21 - November 1, 2016

One little mistake here:
“hours of bright direct sunlight per year”. In space that is simply 365 x 24 x 7

Ermm..no. It’s just days times hours per day, so I guess the “7” is in the wrong place here. *goes back to lurking*

Maury Markowitz - November 1, 2016

Well caught! I wonder how that got in there, I didn’t actually use it in the calc.

Kassandra21 - November 1, 2016

Guess you’re typing fingers thaught something like “24/7” 😉

Just read your rebuttal on Cassandra’s Legacy. I wasn’t aware the peer-review-process is that broken 😦

hkeithhenson - November 2, 2016

There are two parts to the economics, continuing operations and startup. Electric power from coal is around 4 cents per kWh, so the target for power from space was set at 3 cents per kWh. That means the capital cost can’t be more than $2400/kW. (See levelized cost of electricity.) $2400/kW is $2.4 B/GW. For microwave optics reasons, they need to be about 5 GW. (See Seth Potters presentation from 2009.) Rough analysis of the rectenna gives a cost of about a billion dollars, or one part in 12. So if we figure $200/kW for the rectenna and around $900/kW for the power satellite parts, we are left with $1300/kW to spend on transport.

I make a case here: http://spacejournal.ohio.edu/issue18/thermalpower.html for around 6.5 kg/kW. That permits transport cost to GEO of no more than $200/kg. That’s a 100 to one reduction below the current cost of lifting comm sats to GEO, but given beamed power and arcjets along with Skylon or some other cheap way to get to LEO, it looks like it might be possible.

The startup problem is really complicated, with runways, LH2 plants, propulsion power satellites and the like. The current project is to figure out how to keep a lot of construction workers in space long term.

Your point about cost is really critical. A really cheap storage method might allow ground solar to undercut solar from space.

The most optimistic ground solar that includes storage is StratoSolar. Few years ago I worked on the previous version until it got more complicated than power satellites. Ed Kelly, the main guy working on it thinks base load power can be made for around 5 cents a kWh. The storage method is lifting mass 20 km into the sky when there is excess energy available and lowering it to get the energy back when you need it.

People have not been talking about it yet, but there is a heating problem with ground solar. PV is black, reflects much less light than the deserts it would be put over. Going from 70% reflected to maybe 5% for the kind of energy civilization needs is adding a lot of heat to the Earth.

Maury Markowitz - November 2, 2016

As I pointed out in my email to you two years ago, the numbers in your estimate are ridiculously wrong. Your price on the rectenna is less than the cost of just an inverter alone. PV systems today are going in for $1/W, so unless you can build it for $1.25/W, no one will.

hkeithhenson - November 2, 2016

If I had gone back and looked at my email, I would not have replied at all.

14. mspaldingecon - December 12, 2016

Thorium reactors eliminate all of the problems of solar and space based solar and they are less technologically and economically risky than space based solar. Also, when you count fabrication with today’s energy sources, solar produces a lot of atmospheric carbon.

Maury Markowitz - December 12, 2016

And that’s true for a system that runs on ground up pandas as well. Both are just as likely to be used commercially.

mspaldingecon - December 13, 2016

Sad but true. Thanks for your detailed calculations of ground vs space solar.

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