Positronium consists of an electron and a positron. By what factor is a positronium atom bigger than a hydrogen atom?

The solution has been explained to me. The characteristic length when solving for the energy levels of the hydrogenic atom is the Bohr radius: $$a_0 \equiv \frac{4 \pi \epsilon_0 \hbar^2}{\mu e^2}$$

For positronium, the reduced mass $\mu$ is half that for hydrogen. Therefore the Bohr radius for positronium is twice that of hydrogen. However this is the distance between the two particles, rather than the center of rotation of the electron. The size of the atom is double this since the atom is symmetric, meaning that an atom of positronium is in fact the same size as a Hydrogen atom.

My question is:

Does this mean that an atom of, say, muonium will also be the same size as a hydrogen atom, or was the positronium atom a special case because the two particles have exactly the same mass?

Positronium consists of an electron and a positron. By what factor is a positronium atom bigger than a hydrogen atom?

The solution has been explained to me. The characteristic length when solving for the energy levels of the hydrogenic atom is the Bohr radius: $$a_0 \equiv \frac{4 \pi \epsilon_0 \hbar^2}{\mu e^2}$$

For positronium, we can calcuate the reduced mass, $\mu$:

\begin{equation} \mu = \frac{m_1 m_2}{m_1 + m_2} = \frac{m_e m_e}{m_e + m_e} = \frac{1}{2}m_e \end{equation}

giving a reduced mass of about half that for hydrogen. Therefore the Bohr radius for positronium is almost twice that of hydrogen.

However this is the distance between the two particles, rather than the center of rotation of the electron. The size of the atom is double this since the atom is symmetric, meaning that an atom of positronium is in fact the same size as a hydrogen atom.

My question is:

Does this mean that an atom of, say, muonium will also be the same size as a hydrogen atom, or was the positronium atom a special case because the two particles have exactly the same mass?

If we calculate $\mu$ for muonium, we get a value of

\begin{equation} \mu = \frac{m_1 m_2}{m_1 + m_2} = \frac{m_\mu m_e}{m_\mu + m_e} = 0.995 m_e \end{equation}

So the Bohr radius for a muonium atom is $\frac{1}{0.995} = 1.005$ times larger than a that of a hydrogen atom.

But this, again, is the distance between the two particles rather than the size of the atom.

So we multiply by $\frac{\mu}{m_e}$ again to get the distance of the electron to the barycenter of the system (as we did for positronium). We end up cancelling our previous factor of $\frac{1}{\mu}$, giving us the result of muonium being the same size as hydrogen.

This seems wrong!