# Orbital Mechanics in Tau Station

This page describes how the stations in the Tau Station RPG game orbit their host planets.

This is a heavily simplified subset of real-world orbital mechanics.

The relative orbital positions of space stations affect local travel times and costs, as well as shipping costs between stations.

## Topology

A *Star system* in Tau land is actually a collection of stations that all orbit the same planet or dwarf planet. The star(s) that this planet orbits is not relevant for the game mechanics.

For example, the *Sol* stations all orbit around Mars, and the Sun and other
planets have no impact on the game play.

Interstellar travel happens through Jump Gates. The distances for interstellar travel are independent of station positions, and remain constant over time.

## Assumptions

Tau Station makes the following simplifying assumptions. These assumptions have been verified by building an orbital model of Sol and Alpha Centauri which worked very well, and later having staff confirm the accuracy of it.

The assumptions are:

- Stations are in circular orbits around the same planets.
- Stations all orbit in the same plane (coplanar).
- Stations all orbit in the same direction (no retrograde orbits).
- Shuttle and ship distances and travel times assume travel in a straight line, discounting
- the central planet or other stations that could cross that line
- the movement of the target station during the travel
- the possibility of other orbital maneuvers, like dropping height in favor of faster angular movement in orbital direction

- The orbits are classical two-body problems; cross-influence from other bodies are ignored.
- There are no relativistic effects.

## Static Model

The orbit of a station around the central planet is a circle, with the planet at its center.

The only static parameter of such an orbit is the radius of the orbit.

If we consider two stations, `S1`

and `S2`

, they have each have associated radius `r1`

and `r2`

. Let’s assume that `r2 > r1`

.

### Finding Orbit Radiuses

In-game, we can observe the distance of two stations over time from the *Local Shuttles* page and from the cockpit interface of a private ship.

The closest approach between two stations has the distance `dmin = r2 - r1`

, when both stations are at the same orbital angle. The farthest distance is `dmax = r2 + r1`

, when both stations are on opposite sides of the planet.

When we add those two equations, we get

```
dmin + dmax = 2 * r2
```

or

```
r2 = (dmin + dmax) / 2
```

Subtracting instead of adding the two initial equations yields

```
dmax - dmin = 2 * r1
```

or

```
r1 = (dmax - dmin) / 2
```

Or put in words, the sum and the difference of the min and max distances gives us the diameters of both station’s orbits.

Example: *Sol Jump Gate* and *Tau Station* have the closest distance of `dmin = 1937 km`

, and the farthest distance `dmax = 23997 km`

.

From this we can calculate that

```
r2 = 12967 km
r1 = 11030 km
```

We don’t know yet which is the inner and which is the outer, but repeating the calculation with another pair of stations (for example *Sol Jump Gate* and *Nouveau Limoges*) will quickly show that *Sol Jump Gate* is the outer station with radius r2 = 12967km.

You can repeat this process for each station pair, and likely get slightly different results. In this case, you can calculate averages to get the best results.

## Dynamic Model: Let’s Talk About Time

A station S1 has an orbital period of T1. That means that the station is at the same place both at times 0 and T1, or more generally, at the times t and t + T1.

If we put draw a coordinate system with the central planet in the middle, we can describe the position (x, y) of a station at time t with these equations:

```
x(t) = r * sin( 2π t/T + φ )
y(t) = r * cos( 2π t/T + φ )
```

Here `r`

is the radius and `φ`

the angle of the station at time 0 (also called the *phase*), measured in radians. `sin`

and `cos`

are the sine and cosine trigonometric functions. π is roughly 3.14159.

(This assumes that the stations are rotating clockwise in our coordinate system; but it doesn’t really matter, as long as they all rotate in the same direction, which they do).

### Relative Motion Between Two Stations

The distance between two stations as a function of time can be calculated as

```
d(t) = sqrt( (x2(t) - x1(t))² + (y2(t) - y1(t))² )
```

where `sqrt`

is the square root. The calculation is lengthy and boring, but has a pretty simple result:

```
d(t)² = r1² + (r2 -r1)² * cos( 2π t/Td + (φ2 - φ1) )
1/Td = 1/T1 - 1/T2
```

So the square of the distance varies as a cosine function, with a period of 1/(1/T1 - 1/T2). (This is easier expressed in term of *frequency* instead of periods: fd = f2 - f1).

For example, *Sol Jump Gate* has a period of T2 = 51.9 segments, and *Tau Station* has a period of 40.7 segments.

Their relative motion has a period of

```
Td = 1/(1/T1 - 1/T2) = 1/(1/40.7seg - 1/51.9seg) = 189 seg
```

So the period from shortest distance, to longest, to shortest again is 189 segments, or nearly two days.

### Relation Between Period and Radius

The farther away a station is from the central planet, the longer it needs for a full orbit.

This can be derived from the fact that in a stable orbit, the centripetal force and the gravitational force must be of equal magnitude.

Mathematically, we can describe this as

```
T = 2 π * sqrt( r³ / (G*M) )
```

where `M`

is the mass of the central planet, and `G`

is the gravitational constant, 6.674 * 10^-11 m³/(kg s²).

So, T, the orbital period, increases with with the radius as T ~ r^1.5, and decreases with the square root of the mass of the central body.

We can also rearrange the formulate to read like this:

```
T² / r³ = μ
```

where the value of μ isn’t very interesting, except that it’s the same for all stations within a system.

## Finding Orbital Parameters From Observed Data

So, you are an entrepreneurial spirit, and want to find the radiuses and orbital periods for the stations within a system? Here is what you have to do:

- Visit
*Local Shuttles*regularly, and write down the distances over time (`dmax`

,`dmin`

) - For each pair of stations, find the minimum and maximum distance, and when these occurred. The is half of
`Td`

. - Now you have a systems of equations you have to solve. Assuming three stations, the equations are:

```
r3 = (dmax13 + dmin13) / 2
r3 = (dmax23 + dmin23) / 2
r2 = (dmax12 + dmin12) / 2
r2 = (dmax23 - dmin23) / 2
r1 = (dmax31 - dmin31) / 2
r1 = (dmax21 - dmin21) / 2
1 / Td13 = 1/T1 - 1/T3
1 / Td23 = 1/T2 - 1/T3
1 / Td12 = 1/T1 - 1/T2
μ = T3² / r3³
μ = T2² / r2³
μ = T1² / r1³
```

Where dmax13 is the max distance between stations 1 and 3, Td13 is the period of the relative motion of stations 1 and 3, and so on.

You have 12 equations to determine 7 unknowns (r1, r2, r3, T1, T2, T3, μ), so the equation is overdetermined.

This gives you the freedom to discard some data points that you don’t trust, and play around and try to minimize your errors.

A pragmatic approach is to first determine the radiuses, and only then approach the second half of the equations.

Finally, you can use the value of μ to determine the mass of the central body as:

```
M = (2π)² / (G * μ )
```

(Remember to convert the periods from segments and units to seconds to end up with SI units).

## Travel Times

Tau Station’s ships accelerate for the first half of the distance, and then decelerate for the second half.

The acceleration depends on the ship type, and can be obtained from this ships table.

The equation of motion for for linear acceleration in general is

```
x = 1/2 a t²
```

where `x`

is the distance traveled, `a`

the acceleration and `t`

the time.

We know that half of the distance `d`

is reached after half of the total travel time `T`

,
so we can plug in:

```
d/2 = 1/2 a (T/2)²
d/2 = 1/8 a T²
4d = a T²
T = sqrt(4d/a) = 2 * sqrt(d/a)
```

## Findings So Far

Station distances are being recorded by the Tau Tracker,
and data is submitted by volunteers through a userscript. Before the Tau Tracker, data has
been collected by storing *Local Shuttles* pages and extracting distance data afterwards.

### Sol

Data credit: JamesDragonRider, moritz.

Station | Radius / km | Period / Segments |
---|---|---|

Sol Jump Gate | 12,967 | 51.9 |

Tau Station | 11,030 | 40.7 |

Nouveau Limoges | 9,443 | 32.2 |

København | 7,238 | 21.6 |

Daedalus | 6,467 | 18.3 |

Taungoo | 3,565 | 7.5 |

Central body: Mars, mass = 6.417 × 10²³ kg, 11% of earth’s mass.

### Alpha Centauri

Data credit: JamesDragonRider, moritz.

Station | Radius / km | Period / Segments |
---|---|---|

Alpha Centauri Jump Gate | 11,870 | 503.89 |

The Ghost of Mali | 10,264 | 405.21 |

Paris Spatiale | 8,434 | 301.81 |

Moissan Station | 8,094 | 283.75 |

Yards of Gadani | 7,086 | 274.31 |

Cirque Centauri | 6,863 | 221.53 |

Bordeaux | 5,918 | 177.43 |

Spirit of Botswana | 5,621 | 164.58 |

Central body: Vercingetorix (dwarf planet), mass = 5.22 × 10²³ kg, 0.087% of earth’s mass.

### Barnard’s Star

Station | Radius / km | Period / Segments |
---|---|---|

Hopkin’s Legacy | 6013.38 | 427.51 |

The Maid of Orléans | 8168.17 | 677.69 |

Caen Stronghold | 8890.72 | 768.49 |

Barnard’s Star Jump Gate | 9028.24 | 788.73 |

Estación de Amazon | 10768.75 | 1024.25 |

### L-726 B

Station | Radius / km | Period / Segments |
---|---|---|

L-726 B Jump Gate | 9,341 | 803.85 |

Orwell Stronghold | 7,119 | 534.85 |

Spirit of Tianjin | 5,499 | 363.02 |

### YZ Ceti

Station | Radius / km | Period / Segments |
---|---|---|

YZ Ceti Jump Gate | 9,637.0 | 841.76 |

Asimov Freehold | 8,708.7 | 723.11 |

Cape Verde Stronghold | 7,322.5 | 557.51 |

Spirit of New York City | 6,243.7 | 438.98 |