Tuesday, October 6, 2009

Think. Quantitatively.

Every year I tell my students something like: ‘When you finish any calculation in physical chemistry, have a look at the number you have ended up with and see if it is a reasonable number. Is it about the size you would expect? If it isn’t, go back and find the mistake. If you are in an exam and pressed for time, write: ‘I know this number is the wrong size, but I can’t find where I went wrong.’ Or, just change the order of magnitude to the order of magnitude it ought to be, and hope I don’t read your working carefully.’

Every year, I get more cautionary tales to tell my students when I tell them this- assignments I have received with chemical bonds blithely reported as longer than the distance from here to Alpha Centauri, molecules heavier than the Sun, energies for chemical reactions greater than the annual output of all the power plants in Europe.

The chapter ‘The Spur of the Moment’ of Kim Stanley Robinson's book 'Green Mars' is about a project to build a sea in Hellas Planitia in the southern hemisphere of Mars, a sea that is quoted as being ‘1000 by 300 km’. It order to give the project verisimilitude, numbers for amounts of water are quoted throughout the chapter.


It was a running river, in an obviously riverine valley, placid in some places, agitated in others, with gravel fords, sandbars, braided sections, crumbling lemniscate islands, there a big deep lazy oxbow, frequent rapids, and far upstream, a couple of small falls. Under the tallest waterfall they could see the pink foam turn almost white, and patches of white were then carried downstream, to catch on boulders and snags sticking out from the bank.

‘Dao River,’ Diana said. ‘Also called the Ruby River by the people who live there.’

‘How many are there?’

‘A few thousand. … Upstream there are family homesteads and the like. And of course then the aquifer station at the head of the canyon, where a few hundred of them work.’

‘It’s one of the biggest aquifers?’

‘Yes. About three million cubic metres of water. So we’re pumping it out at a flow rate- well, you see it there. About a hundred thousand cubic metres a year.’

105 m3 per year = 274 m3 per day = 3 litres per second

To put this number in perspective:
“Older firehoses with 2.5 inch diameter equipped with a 1.5 inch nozzle can typically deliver 20-40 litres of water per second” (Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World, Benny Lautrup)

I suppose the description of the river *could* apply to something that one could easily step over, but I do not think this was the author’s intention.


One day at the office, news came in from the Hellespontus. They had discovered a new aquifer, very deep compared to the others, very far away from the basin, and very big. Diana speculated that earlier glacial ages had run west off the Hellespontus range, and come to rest out there, underground- some twelve million cubic meters, more than any other aquifer, raising the amount of located water from 80 % to 120% of the amount needed to fill the basin to the -1-kilometer contour.

Let’s see, the Hellas Planitia basin is quoted as being 1000 by 300 km. That’s an area of 106 m × 3 × 105 m, or 300 billion square metres.

Twelve million cubic metres would cover 300 billion square metres to a depth of

1.2 × 107 m3 / 3 × 1011 m2

= 4 × 10-5 m.

40 microns.

So if this raises the amount of located water from 80 % to 120 %, at a first approximation the Hellas Sea will be 0.12 mm deep.

I confess myself unimpressed by this feat of areological engineering.

I expect KSR made the common undergraduate mistake of assuming
1000 cubic metres = 1 cubic kilometre. I wonder if the numbers have been corrected in later printings?