Chirp Mass

Chirp Mass

When two compact objects spiral together and merge, they emit gravitational waves whose frequency sweeps upward — a chirp. The rate at which that frequency evolves is controlled by a single combination of the two masses called the chirp mass:

$\mathcal{M}_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}$

This is the most precisely measured parameter in gravitational-wave observations. For GW150914 (the first detection, two ~30 solar-mass black holes) LIGO measured $\mathcal{M}_c \approx 28.3\,M_\odot$ to sub-percent precision, even though individual masses were uncertain by ~10%.

Why the Chirp Mass?

The power radiated in gravitational waves depends on the masses through $\mathcal{M}_c$ at leading post-Newtonian order. Measuring the frequency sweep $\dot{f}$ directly gives $\mathcal{M}_c$ without needing to know $m_1$ and $m_2$ separately.

Related Mass Parameters

Two other combinations appear frequently:

Symmetric mass ratio (measures how equal the masses are, maximum 0.25 for equal masses): $\eta = \frac{m_1 m_2}{(m_1 + m_2)^2}$

Mass ratio (by convention $m_1 \geq m_2$, so $q \leq 1$): $q = \frac{m_2}{m_1}$

Solar Mass

One solar mass: $M_\odot = 1.989 \times 10^{30}$ kg.

Your Task

Implement three functions using the formulae above. All mass parameters must be in kg (or any consistent unit). No physical constants are needed — these are purely algebraic.