Do planets orbiting stars emit gravitational waves? I have heard it said that charged planets could not orbit a massless (low mass) oppositely-charged star based on electromagnetic attraction the same way they can with gravitational attraction, because Maxwell's laws dictate that accelerating (orbiting) charges produce electromagnetic waves and therefore lose energy which would lead to the planets slowdown and eventually crash. But it occurred to me that something similar would seem likely with gravitational waves in real-life gravity-based orbits.
Is it true than that planets' orbits are decaying slowly and turning that energy into gravitational waves?  If not, how can that be, given we know gravitational waves exist and surely expend energy in the same way the production of electromagnetic waves do?
 A: As noted in @G.Smith's answer, Wikipedia gives a figure of $\sim 200 \, \mathrm{W}$ for Earth/Sol orbital radiation.
Wikipedia didn't clearly cite the source, but this PDF is cited not long thereafter and may be it.  That PDF claims that the radiated energy for a non-relativistic binary system is about
$$
\frac{\mathrm{d}E}{\mathrm{d}t}
~=~
- \frac{32 \, G^4}{ 5 \, c^5 \, r^5}
{\left(m_1 \, m_2\right)}^{2}
\,
{\left(m_1 + m_2\right)}
\tag{24}
\,,
$$
where the masses $m_1$ and $m_2$ are separated by a radius $r .$  The numbers seem to sync up, so I'm guessing that it may be the source.
For the solar system, WolframAlpha calculates:$
{\def\Calc{~~{{\color{darkblue}{\Large{}}} \!\!}
}}
{\def\RowHeaderPrefix{
\textbf{Mercury}
}}
{\def\RowHeader{
{\phantom{\RowHeaderPrefix{\Calc}\textbf{:}~~}}
}}
{\def\EnergyColumn{
\phantom{0 {,}\, 000 {.}\, 000 {,}\, 000 {,}\, 000 {,}\, 000 {,}\, 00}
}}
{\def\PlanetEntry#1#2{
\rlap{
{\RowHeader}
{\llap{\textbf{#1} \phantom{\Calc} \textbf{:}~}}
{\rlap{~~#2}}
}}
{\def\CalculationLink{
\rlap{
\phantom{\RowHeaderPrefix}
{\Calc}
}}}}
{\def\Placeholders#1{{
\color{lightgrey}{#1}
}}}
$$
{\rlap{\begin{array}{c}
{\smash{\RowHeader}} \\[-25px]
{\underline{\textbf{Planet}}}\phantom{:}
\end{array}}}
{\rlap{\RowHeader\begin{array}{c}
{\smash{\EnergyColumn}} \\[-25px]
{\underline{\textbf{Radiation}~\left(\mathrm{W}\right)}}
\end{array}}}
$$\PlanetEntry{Mercury}{\phantom{0{,}\,0} 69 {.}\,}$$\CalculationLink$$\PlanetEntry{Venus}{\phantom{0{,}\,} 658 {.}\,}$$\CalculationLink$
$\PlanetEntry{Earth}{\phantom{0{,}\,} 196 {.}\,}$$\CalculationLink$
$\PlanetEntry{Mars}{\phantom{0{,}\,00} {\Placeholders{0 {.}\,}} 276}$$\CalculationLink$
$\PlanetEntry{Jupiter}{\phantom{} 5{,}\,200 {.}\,}$$\CalculationLink$
$\PlanetEntry{Saturn}{\phantom{0{,}\,0} 22 {.}\, 54}$$\CalculationLink$
$\PlanetEntry{Uranus}{\phantom{0{,}\,00} {\Placeholders{0 {.}\, 0}}15 {,}\, 93}$$\CalculationLink$
$\PlanetEntry{Neptune}{\phantom{0{,}\,00} {\Placeholders{0 {.}\, 00}}2 {,}\, 349}$$\CalculationLink$
$\PlanetEntry{Pluto}{\phantom{0 {,}\, 00} {\Placeholders{0 {.}\, 000 {,}\, 000 {,}\, 000 {,}\, 00}}9 {,}\, 83}$$\CalculationLink$
To note it, these figures are theoretical; it remains to be seen if current theories work in contexts like this.
A: Yes, but undetectably. The Earth-Sun system radiates a continuous power average of about 200 watts as gravitational radiation. As Wikipedia explains, “At this rate, it would take the Earth approximately $1\times 10^{13}$ times more than the current age of the Universe to spiral onto the Sun.”
The Hulse-Taylor binary (two neutron stars, one a pulsar) was the first system in which the gravitational decay rate was measurable. It radiates $7.35\times 10^{24}$ watts as gravitational radiation, about 1.9% of the power radiated as light by the Sun.
A: Yes, two bodies orbiting each other like this will indeed emit gravitational waves, regardless of whether or not they're compact objects like neutron stars or black holes. Obviously, most exoplanets will not emit strongly; a planet-star system generally involves large separations and non-relativistic speeds. Therefore, as G. Smith noted, while all such systems emit gravitational waves, the radiation is largely insignificant.
It's been proposed (Cunha et al. 2018) that some exoplanets with extremely small semi-major axes ($a\sim0.01$ AU) could be sources of gravitational waves that would detectable in the near future. As in most of these cases $a$ is large compared to the sources LIGO has observed so far (compact objects in the process of merging), these waves would be relatively low-frequency ($f\sim10^{-4}$ Hz) and would fall in the regime of long-baseline space-based interferometers like LISA, not ground-based interferometers like LIGO. Some exoplanets could reach peak strains of $h\sim10^{-22}$, which is indeed above LISA's sensitivity curve at those frequencies. (Compare this to the binary systems LIGO has observed so far, with $f\sim10^2\mathrm{-}10^3$ and $h\sim10^{-22}\mathrm{-}10^{-21}$ at peak.)
The authors note that in these systems, orbital decay is indeed occurring, but at lower rates than, say, famous orbiting compact objects like the Hulse-Taylor binary pulsar. Over long timescales, this decay should be detectable. In a few systems, the period decay are comparable to the Hulse-Taylor binary, within a factor of a few, although the gravitational wave luminosities remain lower by a couple orders of magnitude or more.
A: G.Smith and HDE 226868 gave good answers.
I would add that, in the Solar system case, the gravitational waves are clearly not the dominant factor in changing the (Keplerian parameters of) orbits. Momentum exchange between planets, solar radiation pressure, solar wind effects, tidal effects - every one of these (and probably more that I cannot recall right now) are orders of magnitude stronger than the orbit decay because of gravitational waves radiation.
A: If you had an interferometer that was accurate enough, you would be constantly in an ocean of gravitational waves. The frequencies of the planet's waves would be very low frequencies, about 1 period per year! Jupiter going to aphelion would vary in amplitude every 12 years. For the moment, 20Hz is the record for low frequency gravitational wave detection. 
