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Dust storms

Energy and power · level 6, lesson 6 of 6

A question to start

In June 2018, a dust storm spread across the whole of Mars. Opportunity, on solar power for 14 years, went silent and never spoke again. How dark does a dust storm make the sky?

The idea

Scientists measure how dusty the sky is by its optical depth, \(\tau\) (tau). Sunlight coming straight through the air is cut down by a factor

\[\frac{I_\text{direct}}{I_\text{above}} = e^{-\tau}\]

for the Sun overhead. Each 1 of optical depth cuts the direct beam to \(e^{-1} = 37\%\).

Sky \(\tau\), about Direct sunlight getting through
Clear Mars sky 0.3 to 0.5 61 to 74%
A dusty day 1 37%
A big regional storm 3 5%
The 2018 storm over Opportunity about 11 0.002%

Not all the light is lost: dust also scatters light sideways, and some of that reaches the ground as dim, diffuse daylight. Even so, in the thickest part of the 2018 storm, only a tiny share of the usual sunlight reached Opportunity's panels. It could not charge its batteries, and its heaters could not keep it warm.

clear: τ about 0.52018 storm: τ about 11

Planet-wide storms like 2018's come every few Mars years, usually in the dusty season (southern spring and summer), and can last for months. A city on solar power alone would need to store enough energy, or keep a second source, to get through one.

Worked examples

1. A dusty day. At \(\tau = 1\), with 586 W/m² above the air and the Sun overhead, the direct beam is \(586 \times 0.37 = 216\) W/m².

2. A regional storm. At \(\tau = 3\): \(586 \times e^{-3} = 586 \times 0.050 = 29\) W/m² direct.

3. The 2018 storm. \(e^{-10.8} = 2\times10^{-5}\). Of 586 W/m², about 0.01 W/m² came straight through. Diffuse light added more, but still only a few percent of a clear day.

Try it

  1. What does optical depth measure?
  2. Find the share of direct sunlight getting through at \(\tau = 2\).
  3. A panel gives 1,000 Wh a sol in a clear sky at \(\tau = 0.5\). Roughly what might it give at \(\tau = 2.5\), counting only the direct beam?
  4. Why could a nuclear power unit keep a city going through a storm?
Answers
  1. How much the dust in the sky blocks the direct sunlight.
  2. \(e^{-2} = 0.135\), about 14%.
  3. \(1000 \times e^{-2.5}/e^{-0.5} = 1000 \times e^{-2} = 135\) Wh (in reality diffuse light adds some).
  4. It does not need sunlight: its heat comes from atomic nuclei.

Common mistake

"Twice the dust, half the light." Optical depth works by multiplying: doubling \(\tau\) from 2 to 4 cuts the direct light from 14% to 2%, not from 14% to 7%.

Where it leads