EarthSpeed

simple tool that calculates the rotational speed of any country on Earth

% go run . country

Some maths

Because Earth is, what we call, an 'oblate spheroid', I wanted to move from sphere-like to ellipsoid calculation of the speed at a specific latitude/longitude.

The angular velocity of Earth (≈ 7.292 × 10⁻⁵ rad s⁻¹) is fixed; the variable part is the radius of the latitude circle the point sits on. On a perfect sphere that radius is simply:

R · cos φ

... because every meridian is a great circle. On an ellipsoid that slice is not a circle of constant radius; the “horizontal” distance from the spin axis depends on the local curvature.

The WGS-84 radii we need:

symbol	meaning	WGS-84 value
a	semi-major axis (equatorial)	6 378.137 km
b	semi-minor axis (polar)	6 356.752 km
e²	first eccentricity² = (a² − b²)/a²	0.00669438

References:https://en.wikipedia.org/wiki/World_Geodetic_System

References: JSBSim-Team/jsbsim#184

Deriving the corrected radius r(φ)

1. Radius of curvature in the prime vertical

N(φ) = a / √(1 − e² sin² φ)

2. Distance from the spin axis

r(φ) = N(φ) · cos φ

// RadiusOfCurvaturePrimeVertical returns N(φ):
// N(φ) = a / sqrt(1 - e² * sin²(φ))
func RadiusOfCurvaturePrimeVertical(a, e, phi float64) float64 {
	return a / math.Sqrt(1-math.Pow(e, 2)*math.Pow(math.Sin(phi), 2))
}

// CorrectedRadius returns r(φ) = N(φ) * cos(φ)
func CorrectedRadius(a, e, phi float64) float64 {
	N := RadiusOfCurvaturePrimeVertical(a, e, phi)
	return N * math.Cos(phi)
}

Putting those together accepting a latitude (a and b are constants in main.go)

// Horizontal distance from Earth’s spin axis at latitude φ (degrees)
func radiusAtLatitude(phiDeg float64) float64 {
	phi := phiDeg * math.Pi / 180
	N := a / math.Sqrt(1-e2*math.Sin(phi)*math.Sin(phi))
	return N * math.Cos(phi) // km
}

And then, we just need to translate radius to linear speed

We need to obtain the total path side real in a whole day: C(φ) and the average linear speed v(φ). Here, C(φ) is circular:

C(φ)=2πr(φ)

So, the speed is (where T is 24h):

v(φ)= (2 * πr(φ)) / T

In go, I implemented as:

	r := radiusAtLatitude(loc.Lat)
	return math.Abs(2 * math.Pi * r / dayHours)