Nondimensionalization (Normalized Units)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Nondimensionalization rescales the variables of a physical equation of motion by characteristic quantities of the same dimension so that the new variables carry no units. In the restricted three-body problem this procedure strips the equations of all dimensional constants and leaves a single dimensionless parameter — the mass ratio $\mu = m_2/(m_1 + m_2)$ . The scaled quantities are referred to as normalized (or canonical, dimensionless) units.

Choosing the characteristic quantities

The CR3BP convention (Szebehely 1967, §1.5) is:

Characteristic mass: $m^* = m_1 + m_2$ (the sum of the two primaries' masses), so $\mu = m_2/m^*$ .
Characteristic length: $l^* = $ the (mean) distance between the two primaries.
Characteristic time: chosen so that the mean motion of the secondary about the primary equals one, i.e. $t^* = 1/n$ , where $n = \sqrt{G(m_1 + m_2)/(l^*)^3}$ is the dimensional mean motion. Kepler's third law then gives $G(m_1 + m_2) t^{*2}/l^{*3} = 1$ , and the gravitational constant becomes unity in the new system.

With these choices the primaries sit at $(-\mu, 0, 0)$ and $(1-\mu, 0, 0)$ , the angular rate of the synodic frame is unity, and the equations of motion collapse to the canonical CR3BP form containing only $\mu$ (see Equation of Motion).

Normalized units for the Earth–Moon system

Using the mean Earth–Moon distance $l^* \approx 384{,}400$ km and $\mu \approx 0.0121506$ :

unit length $l^* \approx 3.844 \times 10^5$ km;
unit time $t^* = 1/n \approx 3.752 \times 10^5$ s $\approx 4.342$ d (so one full revolution of the primaries takes $2\pi$ time units);
unit velocity $v^* = l^*/t^* \approx 1.025$ km/s.

Values vary slightly between references depending on whether the semi-major axis or the instantaneous distance is used for $l^*$ , and on the adopted Earth–Moon mass ratio; modern DE440-based values of $\mu$ are around 0.012150585.

Why it matters

Nondimensionalization serves three purposes. (i) Generality: a single dimensionless parameter $\mu$ describes every Earth–Moon-like, Sun–Earth-like, or Sun–Jupiter-like system — results carry over by scaling. (ii) Numerical conditioning: with $l^*$ , $t^*$ as the working units, position, velocity, and time are all $O(1)$ , which limits floating-point cancellation in long integrations. (iii) Conceptual clarity: the relative importance of terms in the equations (Coriolis vs. gravity vs. centrifugal) is read off directly from the dimensionless form.

A practical caveat: the dimensionless equations do not assume the actual inter-primary distance is one physical unit (Szebehely 1967, §1.5). They hold for any physical system after scaling; recovering dimensional outputs requires multiplying by $l^*$ , $t^*$ , $v^*$ appropriately.

References

Szebehely, 1967, Theory of Orbits, §§1.2–1.5 — dimensional-to-dimensionless derivation and the warning against reading dimensionless units as physical units.
Vallado, 2022, Fundamentals of Astrodynamics and Applications, §12.3 — non-dimensional CR3BP equations and the mass parameter.
李星明等, 2024, 地月周期轨道对地月 L1 与 L2 附近 Halo 轨道的可见性分析——典型 Earth–Moon canonical-unit numerical setup.
张晨, 2024, 北京航空航天大学学报——PBCR4BP normalization using Earth–Moon total mass, mean distance, and mean angular rate.