Does an accelerating mass radiate energy? This question is actually 2 parts


*

*Just like binary star system or the 2 black wholes (which generated the recently discovered gravitational waves) shouldn't the Earth also radiate giving off energy. Reading the Wikipedia article confirmed my belief that Earth should radiate and give off gravitational waves. Now, since the Earth is 'radiating' shouldn't it lose energy (howsoever small it may be) resulting in shrinking of its orbit (howsoever ever small the shrink may be)? Radiating means losing energy after all.

*All mass (except few like neutrinos) is made up of charges and accelerating charges radiate. So normal matter, though neutral, must radiate in the sense that each of its individual charge should radiate. Shouldn't it? And there should be a net loss of energy?
 A: Not all masses, only those that have an asymmetric mass generate gravitational waves. A perfect sphere will not.

Unlike charge, which exists in two polarities, masses always come with the same sign. This is why the lowest order asymmetry producing electromagnetic radiation is the dipole moment of the charge distribution, whereas for gravitational waves it is a change in the quadrupole moment of the mass distribution. Hence those gravitational effects that are spherically symmetric will not give rise to gravitational radiation. A perfectly symmetrical collapse of a supernova will produce no waves, while a non-spherical one will emit gravitational radiation. A binary system will always radiate. Gravitational waves distort spacetime: in other words, they change the distances between free macroscopic bodies. 

Italics mine.
So for 1:  the radiation from the Earth , which is almost spherical, will be very small in any case.
For 2: The elementary particles are point particles, so symmetric and they will  not radiate gravitational waves in acceleration. Molecules and atoms accelerated may, if the outside orbitals are not S orbitals. Even in the last case as can be seen here it will be   very small, as there is G and divisions by power of c involved.
The energy has to be provided by the source that is accelerating the objects.
The amplitude of the gravitational wave, formula 2.34 :


where εE_kin(with 0≤ε≤1), is the fraction of kinetic energy of the source that is able to produce gravitational waves. The factor ε is a measure of the asymmetry of the source and implies that only a time varying quadrupole moment will emit gravitational waves. For example, even if a huge amount of kinetic energy is involved in a given explosion and/or implosion, if the event takes place in a spherically symmetric manner, there will be no gravitational radiation

A: The earth-sun system absolutely emits gravitational waves, and they will tend to reduce the radius of the earth's orbit. But the effect will be very small (I think negligible) because the accelerations involved are small in relativistic terms.
