Gravity Train Travel Time Calculator

Introduction: why Gravity Train Travel Time Calculator matters

A gravity train calculation turns three simple inputs — planet radius, surface gravity, and the station separation angle — into four outputs that matter for planning a hypothetical tunnel: chord length, maximum depth, one-way travel time, and peak speed. This calculator is built around the uniform-density idealization, so the result is best treated as a clean comparison model rather than a full engineering design.

That makes the page useful when you want to compare one tunnel layout against another. A wider angle makes the chord longer and deeper, while the planet's radius and surface gravity set the timing through the simple harmonic model. If two scenarios share the same world but differ in angle, the calculator shows how much extra tunnel you are asking for and how fast the capsule would move at mid-chord.

The sections below explain how to enter the planetary inputs, how the formulas relate to the outputs, how to read the numbers, and which assumptions matter most if you are using the calculator as a quick first pass.

What problem does this calculator solve?

The gravity train travel-time question is really a geometry-plus-motion problem. You choose a planet, pick two surface stations, and ask how long a capsule would take to travel through a straight tunnel between them. The calculator answers that by combining the chord geometry with the ideal simple harmonic motion of a body moving through a uniform-density sphere.

In practical terms, the tool helps you answer questions such as: How deep does the tunnel go if the stations are 60°, 90°, or 180° apart? Does a bigger planet make the trip longer even if the angle stays the same? How much faster does the capsule move at the midpoint if the tunnel spans more of the planet? Once the question is framed this way, the result panel becomes a compact summary of tunnel size and transit behavior.

How to use this gravity train calculator

1. Enter Planet Radius (km): with the unit shown beside the field.
2. Enter Surface Gravity (m/s²): with the unit shown beside the field.
3. Enter Central Angle Between Stations (degrees): with the unit shown beside the field.
4. Click Calculate Gravity Train to update the chord, depth, travel-time, and peak-speed readout.
5. Check that the travel time is in minutes, the distances are in kilometers, and the peak speed changes in the direction you expect before comparing scenarios.

If you are comparing layouts, keep the angle, radius, and gravity together in a short note so you can reproduce the same gravity-train case later.

Inputs: how to choose radius, gravity, and station angle

The gravity train form asks for the three values that define the tunnel and the planet. Most mistakes come from mixing units or blending data from different worlds. Enter the values in the units shown on the page, and make sure the radius and gravity belong to the same hypothetical planet before you compare results.

• Units: confirm that radius is in kilometers, surface gravity is in meters per second squared, and the central angle is in degrees.
• Ranges: the angle should stay between 0 and 180 degrees because the model is a straight chord between two surface points.
• Defaults: the prefilled numbers are just a starting point for a familiar planetary case; replace them with your own scenario before relying on the output.
• Consistency: if two values describe the same body, they should match the same assumed planet, not a mix of unrelated sources.

Common inputs for Gravity Train Travel Time Calculator include:

• Planet Radius (km): the radius of the planet you want to model.
• Surface Gravity (m/s²): the surface gravity associated with that planet.
• Central Angle Between Stations (degrees): the angle separating the two surface endpoints of the tunnel.

If your source data use miles, different gravity units, or another angle convention, convert them before you enter the fields so the calculator can apply the tunnel model cleanly.

Formulas: how the gravity train calculator turns planet data into transit estimates

Under the calculator's uniform-density idealization, the tunnel is a straight chord and the geometry comes directly from the central angle. The chord length is L = 2R sin(θ/2), and the deepest point is d = R - R cos(θ/2). Because the interior gravity scales linearly with distance from the center, the motion becomes simple harmonic with angular frequency ω = √(g/R). From that, the half-trip time is T = π/ω = π√(R/g), and the midpoint speed is v_max = ωR sin(θ/2).

The chord formula above is the geometric part of the calculation. A wider central angle stretches the tunnel farther across the planet and also pushes the midpoint deeper below the surface.

The time formula above is the dynamic part of the model. In this idealized case, the half-trip time depends on the planet's radius and surface gravity, not on the chosen station angle.

Worked example: checking the default gravity train scenario step by step

Here is a real check using the default values on the page: radius 6371 km, surface gravity 9.81 m/s², and a 180° separation angle. Those inputs describe the longest possible straight chord through the planet, so they are a good way to verify that the outputs look sensible.

• Planet Radius (km): 6371
• Surface Gravity (m/s²): 9.81
• Central Angle Between Stations (degrees): 180

From the geometry, the chord length is 2 × 6371 × sin(90°) = 12,742 km, and the deepest point reaches the center, so the maximum depth is 6,371 km. Using ω = √(g/R), the half-trip time works out to about 42.2 minutes, and the midpoint speed is about 7.90 km/s.

That example is useful because it ties every output back to one of the inputs. If the result panel shows a very different order of magnitude, the most likely causes are a unit mix-up, a mistaken angle, or a radius that does not match the planet you intended to model.

How tunnel angle changes the gravity train result

Once you understand the formula, the result is easier to read as a set of linked outputs instead of a single number. Radius and surface gravity control the ideal half-trip time, while the station angle controls the shape of the tunnel and the midpoint speed. That means you can use the calculator to answer different questions depending on which input you vary.

• A larger planet radius makes the ideal travel time longer and also raises the midpoint speed.
• Higher surface gravity shortens the ideal travel time but increases the midpoint speed.
• A larger central angle makes the tunnel longer and deeper, and it raises the midpoint speed, but it does not change the ideal half-trip time in this model.
• A smaller central angle shortens the chord, reduces the depth, and lowers the midpoint speed.

Instead of looking for a single conservative or aggressive total, compare the result panel across a few realistic gravity-train layouts. That tells you whether the main change is geometric, dynamic, or both.

How to interpret a gravity train result

The results panel is a compact summary of the gravity train tunnel rather than a raw dump of intermediate values. The chord length tells you how far the tunnel cuts through the planet, the maximum depth shows how close the tunnel comes to the core, the travel time reports the ideal half-oscillation, and the peak speed shows how fast the capsule moves at the midpoint. Read those outputs together, not in isolation.

If you want to keep a snapshot of a scenario, use the Copy Result button and paste the text into your notes or spreadsheet. That makes it easy to compare a set of station angles or to record how the chord length and midpoint speed change when you try a different radius or gravity value.

When you compare cases, check that the distances are still in kilometers, the time is still in minutes, and the peak speed changes in the direction you expect when you alter the angle, radius, or gravity. If the units, geometry, and assumptions all line up, the estimate is a useful first pass.

Limitations and assumptions of the gravity train model

No gravity train calculator can capture every real-world complication. This page intentionally uses an idealized interior model, because that is the simplest way to see how the planet's size, gravity, and tunnel angle interact. Keep the following limits in mind if you use the output for comparison or discussion:

• Input interpretation: the station angle is the central angle between two surface points, not a map bearing or longitude difference.
• Unit conversions: convert source values into kilometers, meters per second squared, and degrees before entering them.
• Linearity: the model assumes a uniform-density sphere, which makes the interior gravity scale linearly with depth and gives the simple harmonic behavior.
• Rounding: displayed values are rounded, so tiny differences after changing inputs are normal.
• Missing factors: the calculator does not model tunnel friction, air resistance, rotation, construction limits, launch braking, or heating.

If the radius, gravity, and angle all describe the same idealized world, the output is a strong first-pass estimate. If they come from mixed sources or from an irregular planet, treat the numbers as directional rather than exact.