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3D Projectile From Spherical Planet

A projectile launched from the surface of a spinning planet under inverse-square gravity. The landing latitude and longitude are found by integrating the motion in a non-rotating frame and converting back to the planet's surface. The dashed blue path shows where the projectile would land if the planet did not rotate. Drag the globe to rotate it and use the +/− buttons to zoom.

LaunchLanding (rotating)Landing (no rotation)

Landing latitude

68.169° N

Landing longitude

1.497° E

Flight time

12.13 min

Ground range (great circle)

1855.2 km

Maximum height

502.5 km

No rotation: landing

68.202° N, 0.000° E

Shift due to rotation

62.0 km

The motion is solved in a non-rotating frame centred on the planet, with the z-axis along the spin axis. Gravity follows the inverse-square law:

\ddot{\mathbf{r}} = -\frac{GM}{|\mathbf{r}|^3}\,\mathbf{r}, \qquad GM = gR^2

The launch speed u, elevation θ and bearing α (clockwise from north) are given relative to the ground, in the local east, north and up directions. In the non-rotating frame, the ground is itself moving at ω × r:

\mathbf{v}_0 = u\left(\cos\theta\sin\alpha\,\hat{\mathbf{e}} + \cos\theta\cos\alpha\,\hat{\mathbf{n}} + \sin\theta\,\hat{\mathbf{u}}\right) + \boldsymbol{\omega}\times\mathbf{r}_0

The equations are integrated with 4th-order Runge–Kutta until |r| = R. The last step is interpolated so the impact point lies exactly on the surface. To find where it lands on the surface, the position is rotated back by the angle the planet has turned:

\mathbf{r}_{\text{planet}}(t) = R_z(-\omega t)\,\mathbf{r}(t), \qquad \omega = \frac{2\pi}{T_{\text{sidereal}}}

\phi = \sin^{-1}\!\left(\frac{z}{|\mathbf{r}|}\right), \qquad \lambda = \operatorname{atan2}(y, x) - \omega t_f

The range is the great-circle distance between the launch and landing points (haversine formula):

s = 2R\sin^{-1}\sqrt{\sin^2\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2\frac{\Delta\lambda}{2}}

If the total energy ½v² − GM/r ≥ 0, the projectile escapes. If the perigee of its orbit is above the surface, it orbits. In either case it never lands.

In the planet frame view, the globe is held still and the path bends because of the Coriolis effect. In the inertial frame view, the path is a Keplerian ellipse and the planet turns beneath it.

Planet textures: Solar System Scope, CC BY 4.0.

Launch conditions

Planet:

R = 6371 km, g = 9.81 Nkg^-1, day = 23.93 h

Launch Latitude φ (°):

Launch Longitude λ (°):

Launch Speed u ms^-1:

Elevation Angle θ °:

Bearing from North α °:

Launch Height ℎ (m):

Rotation Rate Multiplier:

Playback Speed:

View Frame:

Planet Inertial

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