# Gravitational waves emitted by accelerating rocket

What is the gravitational wave luminosity of a rocket with mass $$m$$ (kg) accelerating at $$\Delta v$$ (m/s$$^2$$)? Intuitively, I expect the luminosity (in Watt) to depend on the mass and acceleration. Bonus question: What is the frequency of the emitted GWs?

Context: Gravitational waves (GWs) are generated by accelerated masses. The LIGO detector famously observed GWs from the inspiral (last phase of merger) of black holes and binary neutron stars. In the literature, the usual description of GW sources uses binary objects. For example, the luminosity $$L$$ in GWs is dependent on the masses of the bodies ($$m_1$$, $$m_2$$), their separation $$a$$ and natural constants such as the speed of light $$c$$ and gravity $$G$$:

$$L = \frac{32 \, G^4}{5 \, c^5 \, a^5} {\left(m_1 \, m_2\right)}^{2} \, {\left(m_1 + m_2\right)}$$

Correspondingly, the strain $$h$$ detected by LIGO and caused by these waves at an observer distance $$d$$ is

$$h = \frac{2\,G^2\,m_1\,m_2}{a\,c^4\,d}$$.

Now, there are (few) sources in accelerated but non-circular motion. An accelerating rocket comes to mind. After an extensive literature research, I could find no treatment of this linear case. I expect the GWs from a typical rocket to be VERY weak, but I'd like to understand how to quantify it.

Update:

• Changed $$v$$ to $$\Delta v$$ for acceleration.
• BTW, using $v$ for acceleration is a little confusing. "I expect the GWs from a typical rocket to be very weak" Definitely. From physics.stackexchange.com/q/537986/123208 the Earth in its orbit around the Sun emits around 200 watts of GWs. – PM 2Ring Jun 10 at 12:54
• What is the quadruple moment of a rocket and its exhaust, and how fast is it changing? – JEB Jun 10 at 13:48
• Changed $v$ to $\Delta v$ to improve notation. – Michael Jun 10 at 16:14
• You might want to have a look at the literature for the so called "C-metric", the exact solution for an accelerating black hole. – mmeent Jun 12 at 17:20