Shouldn't cosmic rays slow down Earth's movement around the Sun? Maybe a dumb question. Are cosmic rays slowing down earth's (or any planets) movement around the sun?
To me it sounds like an analogy of moving a mass through air and being slowed down by the impact of air-molecules with the mass (here cosmic ray particles)
 A: Let's do some Fermi back-of-the-envelope analysis.


*

*Wikipedia puts the flux of GeV particles at around $j=10^{4}\:\rm m^{-2}\:s^{-1}$.

*Assume for simplicity that the bulk of the cosmic rays at this energy are protons, for which an energy of $E=1\:\rm GeV$ corresponds to a momentum of $p=\frac1c\sqrt{E^2-m_p^2c^2}\sim 10^{-19}\:\rm kg\:m/s$.

*At the flow rate indicated that gives a momentum flow rate (a.k.a. pressure) of about $$P=jp=10^{-15}\:\rm kg\:m^{-1}\:s^{-2}.$$

*Multiplying by the cross-sectional area of the Earth, that produces a force of $$ F = P\, \pi R_\oplus^2 \sim 1 \:\rm kg\:m\:s^{-2}. $$

*Dividing that by the mass of the earth gives an acceleration of $$ a =F/M_\oplus \sim 10^{-26} \:\rm m/s^2.$$

*The orbital velocity of the Earth is $v\approx 30\:\rm km/s$, so at that rate it would take about $$T = v/a \sim 10^{22}\:\rm y$$ to slow the Earth down to a halt.


For comparison, the age of the Earth is on the ballpark of $10^{9}\:\rm y$, which puts a comfortable factor of eleven orders of magnitude between the unrealistic overestimate in the bullet points above (why is it an overestimate? because it doesn't factor in the momentum coming in from the back!) and reality. 
TL;DR: There just isn't anywhere close to enough momentum in these cosmic rays to matter on an astronomical scale.
