Issue 011 - Energy and defense - Geometry and line of sight
Are offshore wind turbines below the radar horizon?
AP reported that the Trump administration moved to halt offshore wind development while citing national-security concerns, including radar interference and vulnerability to autonomous drones. Estimate the basic geometry first: can a coastal radar see the top of a large offshore turbine before Earth curvature hides it?
The problem
Estimate how far away a large offshore wind turbine can be detected by line-of-sight coastal radar before Earth's curvature hides it.
If offshore wind turbines are 20 to 60 km from shore, are they generally below the radar horizon, near the radar horizon, or clearly visible to coastal radar?
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I had to look at the hint for which equation to use because I was not sure where to begin, but I ignored the suggested heights for radar towers and offshore wind turbines.
I assumed that the tallest radar towers would be about 20 m and that the tallest offshore windmills would be about 100 m. Then I used:
d ~= sqrt(2 x Earth radius x radar height)
+ sqrt(2 x Earth radius x windmill height)Using Earth radius as 6.4 x 10^6 m, I calculated:
radar tower term ~= sqrt(2 x 6.4 x 10^6 m x 20 m)
~= about 5 x 10^3 m in my first pass
windmill term ~= sqrt(2 x 6.4 x 10^6 m x 100 m)
~= about 3.5 x 10^4 m
total ~= 4 x 10^4 m
~= 40 kmSo my first answer was that line-of-sight detection would be around 40 km.
Calibration Score
Matt's Calibration Score: 65 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. Radar and turbine heights were underestimated.
Model: 30/30. The radar-horizon formula was the right model once supplied.
Math: 5/10. There was a square-root arithmetic slip in one term.
Result: 20/30. The final answer was about half the corrected result, but in the right order of magnitude.
Grounding facts
A compact version of the horizon rule is:
geometric horizon in km ~= 3.6 x sqrt(height in meters)That means height has diminishing returns. A 10 m radar has a horizon around 11 km, while a 100 m radar reaches about 36 km. The large turbine matters because the blade tip adds its own horizon distance.
After checking sources
Check and recalibrate
The structure was right, but two assumptions moved the answer. First, the 20 m radar term had an arithmetic slip:
radar horizon for 20 m ~= sqrt(2 x 6.4 x 10^6 m x 20 m)
~= sqrt(2.6 x 10^8 m^2)
~= 1.6 x 10^4 m
~= 16 kmThe 100 m turbine term was approximately right:
100 m turbine horizon ~= sqrt(2 x 6.4 x 10^6 m x 100 m)
~= sqrt(1.3 x 10^9 m^2)
~= 3.6 x 10^4 m
~= 36 kmSo Matt's own height assumptions imply about 52 km, not 40 km. Still, that is the same basic scale.
The bigger issue is turbine height. A large offshore turbine can reach roughly 200 to 260 m at the blade tip. Using the same geometric formula:
low radar, large turbine:
10 m radar horizon ~= 11 km
260 m turbine-tip horizon ~= 58 km
total ~= 69 km
20 m radar horizon ~= 16 km
260 m turbine-tip horizon ~= 58 km
total ~= 74 kmFor an elevated coastal radar, the range increases:
50 m radar horizon ~= 25 km
260 m turbine-tip horizon ~= 58 km
total ~= 83 km
100 m radar horizon ~= 36 km
260 m turbine-tip horizon ~= 58 km
total ~= 94 kmRadar propagation is often estimated with an effective Earth radius about 4/3 larger than the real Earth because atmospheric refraction bends radio waves slightly downward. That expands these ranges by roughly 15%, pushing the 70-to-95 km geometric range toward about 80 to 110 km under standard conditions.
A good Fermi answer is therefore: large offshore turbine blade tips are geometrically visible to many coastal radars from roughly 70 to 100 km away, depending on radar height and atmospheric assumptions. Turbines 20 to 60 km offshore are generally not hidden by Earth curvature, though 60 km can be closer to the horizon for a low radar trying to see lower parts of the turbine.
That answers only the line-of-sight question. It does not prove or disprove radar interference, military risk, sonar effects, drone vulnerability, signal clutter, mitigation technologies, or the policy claim in the AP story.
Post-check reflection
Matt's reflection
If the preferred equation had not been provided as part of the question, I would have really struggled with this one. I should remember that formula.
I also made some poor assumptions about the heights of the relevant structures. I estimated about half the actual height of the wind turbine, because those things are huge, and I underestimated how tall radar towers could be as well.
The final answer was about double what I had arrived at. That means wind turbines only 20 to 60 km from shore are still very likely to show up on radar. For that reason, we cannot necessarily dismiss the claim as a rationale against offshore wind farms.
Recommended memory peg
Remember Earth radius ~= 6.4 x 10^6 m and horizon distance ~= sqrt(2Rh). In quick metric form, horizon km ~= 3.6 x sqrt(height in meters), or about 4.1 x sqrt(height) with standard radar refraction.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.