Documentation | Voyager Relativity

Voyager Relativity

Voyager 1 and Voyager 2 have been travelling through space since 1977. Voyager Relativity uses their trajectories to show how Einstein's theories of special and general relativity affect the rate of an ideal clock travelling with either spacecraft.

The comparison is with an ideal, non-rotating clock on the Earth's surface. A positive result means that the Voyager clock would gain time relative to that Earth clock; a negative result means that it would lose time. The displayed clock difference is a calculation from trajectory data, not a measurement from an onboard Voyager clock.

What the dashboard shows

Select Voyager 1 or Voyager 2 at the top of the main page. All values, the mission summary, the accumulated result and the graph change with the selection.

The six current-data cards show:

Distance from Earth: the straight-line distance between the current Earth and spacecraft positions.
Distance from Sun: the spacecraft's current heliocentric distance.
Heliocentric speed: the magnitude of the spacecraft's velocity relative to the Sun.
One-way signal time: The approximate time a radio signal takes to travel between Earth and Voyager, based on their current distance and the speed of light.
Solar gravity: the Sun's Newtonian gravitational acceleration at the spacecraft.
Earth orbital speed: the magnitude of the Earth's current heliocentric velocity.

The Relativistic time dilation bars separate the instantaneous rate difference into two first-order contributions:

Gravity: clocks run faster at a less negative gravitational potential. Far from the Sun, this generally favors the Voyager clock, while the Earth reference clock is also deeper in the Earth's own gravity well.
Motion: a faster-moving clock runs more slowly. Its sign depends on the difference between the spacecraft's and Earth's heliocentric speeds.
Combined: the sum of the gravity and motion terms, expressed in milliseconds gained or lost per day.

Accumulated since launch is the integrated combined clock-rate difference over the mission. The graph plots that cumulative value in seconds from launch to the present. Planetary flybys and entry into interstellar space are marked; the interstellar-space marker is a mission event and does not introduce a discontinuity into the calculation. Hover over or tap the curve to see the date, accumulated clock difference and instantaneous combined rate for that point.

Trajectory data

Current and historical positions and velocities come from the NASA/JPL Horizons system, using its documented API. The targets are Voyager 1 (-31), Voyager 2 (-32) and Earth (399). Historical calculations additionally request Jupiter, Saturn, Uranus and Neptune.

Horizons returns geometric Cartesian state vectors in kilometers and kilometers per second. This project requests Sun-centered vectors in the International Celestial Reference Frame (ICRF), with no light-time or stellar-aberration correction. Distances, speeds, signal time, gravity and relativistic rates are then derived locally from those vectors rather than being fixed representative values. For details of Horizons frames, centers and vector tables, see the Horizons manual.

What updates and when

The current state data are cached for 15 minutes. The first page request after that interval retrieves a fresh current state from Horizons. Historical state data and their derived integration result are cached for 24 hours; the first page request after expiry refreshes and reintegrates them.

The latest successful current and historical snapshots are also retained in persistent server storage. If a Horizons refresh fails, the site continues with that last-known-good data and retries the refresh after a short interval. A successful refresh then replaces the stored snapshot.

An already open page does not poll automatically, so reload it to obtain newly refreshed data. Switching between Voyager 1 and Voyager 2 is immediate: the page already contains both spacecraft datasets and does not make another Horizons request merely because the selector changes.

Historical sampling and integration

The quiet cruise portions of each mission use seven-day state-vector samples. Around every giant-planet encounter, the spacecraft trajectory is sampled every ten minutes for seven days before and seven days after closest approach. The first seven days of the mission are sampled at the same fine interval. Earth and planetary states are sampled every seven days and smoothly interpolated from their positions and velocities.

The historical gravitational potential includes the Sun, Earth, Jupiter, Saturn, Uranus and Neptune at both the spacecraft and Earth-reference locations. This ensures that the strong, rapidly changing potentials during the Jupiter, Saturn, Uranus and Neptune flybys are represented. The accumulated clock difference is obtained by trapezoidal integration of the combined fractional clock-rate difference. The graph may display a reduced set of points for responsiveness, but the integral uses the complete sampled series. Its final point is the same result shown as Accumulated since launch.

Technical calculation

Reference clock and approximation

The comparison clock is idealized as non-rotating, located at the Earth's mean radius, and moving heliocentrically with the Earth's center. The model uses the first-order weak-field, slow-motion approximation to relativity. If Φ is Newtonian gravitational potential and v is coordinate speed, the clock rate is approximated by:

\frac{dτ}{dt} \approx 1 + \frac{Φ}{c^{2}} - \frac{v^{2}}{2 c^{2}}

The Newtonian potential at a position x is the sum of the relevant bodies' potentials:

Φ (x) = - \sum_{i}^{N} \frac{μ_{i}}{‖ x - r_{i} ‖}

Here μ_i = GM_i, r_i is the body's position and c is the speed of light.

For the current gravity value, the spacecraft potential uses the Sun and the reference potential uses the Sun plus the Earth's surface potential. For the historical integration, both locations additionally include Earth, Jupiter, Saturn, Uranus and Neptune as appropriate.

Instantaneous rate difference

Let subscripts V and E denote the selected Voyager and the Earth-surface reference. The fractional gravity and motion differences are:

δ_{gravity} = \frac{Φ_{V} - Φ_{E}}{c^{2}}

δ_{motion} = - \frac{v_{V}^{2} - v_{E}^{2}}{2 c^{2}}

δ_{combined} = δ_{gravity} + δ_{motion}

The instantaneous display converts a fractional rate to milliseconds per day:

rate [ms/day] = δ_{combined} \times 86400 \times 1000

Accumulation

For successive samples at times t_i−1 and t_i, the cumulative proper-time difference is integrated with the trapezoidal rule:

Δτ \approx \sum_{i}^{n} \frac{δ_{i - 1} + δ_{i}}{2} (t_{i} - t_{i - 1})

Dynamic-card formulae

Using heliocentric position and velocity vectors r and v:

D_{Earth} = ‖ r_{V} - r_{E} ‖, D_{Sun} = ‖ r_{V} ‖

t_{signal} = \frac{D_{Earth}}{c}, g_{Sun} = \frac{μ_{Sun}}{{D_{Sun}}^{2}}

Constants

Constant	Value used
Speed of light, `c`	299,792.458 km/s
Astronomical unit	149,597,870.7 km
Mean Earth radius	6,371.0088 km
Solar `GM`	132,712,440,041.27942 km³/s²
Earth `GM`	398,600.435507 km³/s²
Jupiter `GM`	126,686,531.9 km³/s²
Saturn `GM`	37,931,206.23 km³/s²
Uranus `GM`	5,793,951.3 km³/s²
Neptune `GM`	6,835,099.97 km³/s²

The gravitational parameters are consistent with the values published in JPL's astrodynamic parameters and planetary satellite physical parameters.

Assumptions and limitations

The weak-field equation keeps only the leading gravitational-potential and velocity terms. Higher-order relativistic effects are omitted.
The Earth reference clock is ideal and non-rotating. Real terrestrial clocks also depend on altitude, latitude, Earth's rotation, tides and local gravity.
The current instantaneous gravity calculation is deliberately simpler than the historical flyby model: it omits planetary potentials at Voyager and treats the distant spacecraft's Earth potential as negligible.
The historical model includes the bodies that materially affect the Voyager giant-planet encounters, but not every moon, dwarf planet, asteroid or other small Solar System body.
Weekly interpolation is used away from encounters. Fine spacecraft samples are used around flybys, where the potential changes rapidly.
Horizons begins supplying usable spacecraft states shortly after each launch. The approximately first hour is represented using the first available state.
Mission-event dates mark the graph for context. “Entered interstellar space” refers to crossing the heliopause and does not change the equations used.
Numerical precision in the interface should not be interpreted as the uncertainty of an operational navigation or time-transfer solution.

Scope

Voyager Relativity is intended for educational demonstration and exploration. It is not an operational mission-planning, navigation or precision time-transfer system.

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