# The Inertial Properties of the 1969 Apollo 11 Lunar Excursion Module (LEM)

How balanced was the LEM (in terms of rotational inertia)?

The CSM/LM Spacecraft Operational Data Book. Volume 3: Mass Properties⮺
(Revision 2, 20. August 1969, Manned Spacecraft Center, Houston Texas, USA) on *table 3.1-3* on page
*3.1-11* (PDF page 61) prints the raw numbers that allows us to see that
— despite its irregular appearance — the LEM was balanced very well across its three axes.

The table is titled “expected sequential mass properties”; the key rows give the numbers for each milestone in the mission. We’re interested in the “LM pre P.D.I.⮺” which is in the last row of the table.

Next to the “weight pounds”, the table features the column “C.G. inches”. Weight and the location of the
center of gravity completely specify the **translational inertia** of the vehicle, which is of no
further interest to this blog post.

The columns “inertias slug-ft2” (**rotational inertia**. The symbols

This tensor is represented by a symmetric 3x3 matrix. Let’s assemble it from the 6 values given in the book:

The choice of unit — *slug-ft²* — appears peculiar to my ‘modern’ eye; I mean, *slug square feet??*
Let’s convert this to SI units using
1 slug = 14.59390 kg⮺,
1 foot = 0.3048 m⮺,
1 pound = 0.45359237 kg⮺:

The off-diagonal elements are also called the *products of inertia*.

On the top of the table we read that the numbers are in “XE coordinates”; in Figure 2-12 on page 2-13 (PDF
page 37) we see that that is the “LM Coordinate System” (with the +X axis out the docking
interface, the +Z axis out the windows, and the +Y axis completing the right-handed coordinate
system). So, **the basis of this matrix is the LM coordinate system.**

The fact that the matrix is not purely diagonal tells us that **the LEM’s principal inertial axes
did not coincide exactly with the LEM’s coordinate system.** However, the fact that the diagonal elements
are much larger than the off-diagonal elements, and are of roughly the same magnitude, already
tells us that

**the LEM was balanced well**.

But, how well was it balanced, **exactly**? To find that out, we have to know a bit of mathematical
mechanics: The inertia matrix becomes diagonal when its *base* coincides with the *principal inertial axes*^{1}
(any symmetric matrix can be diagonalized by a *change of basis*^{2}).

This change of basis means a rotation of the coordinate system; the *eigendecomposition of a matrix*^{3} gives^{4} the following (orthogonal, unit length) base vectors of the new coordinate system, also called the *Eigenvectors* of the matrix:

Each Eigenvector has an associated *Eigenvalue*, which can be assembled to yield the diagonal *inertia matrix*:

Norming this matrix relative to the middle value (the Y axis), we can see even better how well the LEM was balanced:

Roughly speaking, rotating the LEM around its X axis (the axis from its ‘feet’ up to the docking hatch) was around 10% easier than around its other two axes.

## Footnotes

https://en.wikipedia.org/wiki/Moment_of_inertia#Principal_axes⮺ ↩

https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix⮺ ↩

Using my own interactive Eigenvector and Eigenvalue visualization . Numbers verified using independent online services 1⮺, 2⮺ and 3⮺. ↩