Units
Contents
Units@
Nondimensionalization@
Becoming accustomed to the three-body problem can be challenging, and it starts immediately with the choice to work in a system of units that does not have any dimensions. That is, the CRTBP works with units that are completely normalized such that mass, distance, and time are all dimensionless.
This choice not to use standard units (kilometers and seconds) is done for a number of reasons:
Compatibility across all three-body systems
Greater numerical precision and stability
A nice property of this nondimensionalization is that \(G=1\), the universal gravitational constant. This effectively means that we will never explicitly use the universal gravitational constant in CRTBP calculations. That’s not to say that it’s not there, it still very much is - rather, we just “don’t see it”.
Mass@
Very simply, mass is normalized by the total mass in the system such that the nondimensional total mass is unity. That is, given that the primary (larger body) has mass \(m_1\) and the secondary (smaller body) has mass \(m_2\), then the mass unit \(\mathrm{MU}\) is given by
In this system of units, the primary mass becomes \(m_1/\mathrm{MU}\), and likewise the secondary mass becomes \(m_2/\mathrm{MU}\).
The nondimensional mass turns out to be rather important in the CRTBP, so important actually that it gets its own symbol. We call \(\mu\) the mass parameter, which is defined as \(m_2/\mathrm{MU}\). Fully expressed in the fundamental quantities, the mass parameter is
The primary’s nondimensional mass is \(1-\mu\)
The secondary’s nondimensional mass is \(\mu\)
Note
Sometimes the primary and secondary’s nondimensional masses are also represented as \(\mu_1 = 1-\mu\) and \(\mu_2 = \mu\), respectively
Important
If \(m'\) is a dimensional mass, then the corresponding nondimensional mass \(m\) is
Distance@
Because of the “circular” assumption in “CRTBP” the primary and secondary will always be a constant distance from each other. So like mass, distance will also be normalized such that the two bodies are unit distance apart from each other.
This means that we define the distance unit \(\mathrm{DU}\) to be the physical distance between the two bodies. In real systems, there is some debate as to what the best distance unit is to use, but this is a complication that can be avoided in the context of the standard CRTBP.
Important
If \(d'\) is a dimensional distance, then the corresponding nondimensional distance \(d\) is
Time@
Unlike mass and distance which are normalized to unity, time in the CRTBP is normalized by the time unit \(\mathrm{TU}\) which is related to the orbital period of the two bodies by a factor of \(2\pi\).
One full orbit of the primary and secondary takes \(2\pi\) time units
Important
If \(t'\) is a dimensional time, then the corresponding nondimensional time \(t\) is
Example (Sun-Jupiter)@
Consider the Sun-Jupiter system. In dimensional units, the Sun and Jupiter have the following parameters [1]:
\(\text{Body}\) |
\(GM\ (\text{km}^3/\text{s}^2)\) |
\(\text{Radius}\ (\text{km})\) |
\(a\ (10^6\ \text{km})\) |
\(\text{Orbital Speed}\ (\text{km/s})\) |
|---|---|---|---|---|
\(\text{Sun}\) |
\(132712\times 10^6\) |
\(695700\) |
– |
– |
\(\text{Jupiter}\) |
\(126.687\times 10^6\) |
\(69911\) |
\(778.479\) |
\(13.06\) |
The mass parameter is
and, supposing the distance unit is the semimajor axis, the nondimensionalization parameters that convert from standard units (kilometers and seconds) to nondimensional units are
Note that \(\mu\) was calculated using the gravitational parameters (\(GM\)). This can be done in real systems since a body’s gravitational parameter is often known to higher accuracy than its mass alone.
Using \(\mathrm{DU}\) and \(\mathrm{TU}\), we can nondimensionalize anything from kilometers and seconds to Sun-Jupiter distance and time units. For example, the Sun’s radius \(R_S\) and Jupiter’s radius \(R_J\) in Sun-Jupiter distance units become
Because \(\mathrm{DU}\) is unitless, the value is often understood to be nondimensional so Jupiter’s radius would simply be \(8.980461\times 10^{-5}.\)
Additionally, Jupiter’s orbital speed \(V_J\) in Sun-Jupiter units becomes
Similarly, the unit is already nondimensional, so the velocity is simply stated as \(1.000256\).
It shouldn’t be too surprising that the orbital velocity turned out to be approximately \(1\). In fact, this is quite generic when calculating the secondary’s speed relative to its own units.
Why did this happen?
The orbital period was purposefully nondimensionalized to be \(2\pi\) time units
The Sun and Jupiter are in (nearly) circular orbit, and the orbital distance of the two bodies is also \(1\)
This means that Jupiter has to traverse a distance \(s = r\theta\) with \(r= 1\) and \(\theta = 2\pi\) over one orbit (\(2\pi\) time units). Thus, Jupiter’s nondimensional velocity has to be approximately \(1\ \mathrm{DU}/\mathrm{TU}.\)
Try it yourself
Verify that \(G \approx 1\) in the Sun-Jupiter system.