To see how weak the effect of dark energy is compared to gravity, it
is useful to compute the effect of dark energy on the solar orbit of
earth and Pluto.
The acceleration on a planet orbiting the sun is G*M/R^{2},
where G is the gravitational constant, M is the solar mass, and R is
the sunplanet distance. The effect of dark energy (DE) is to give a
repulsive force; the DE acceleration between the sun and a planet is
given by +k*R, where k = 3.7*10^{36}/s^{2}.
Note that with growing distance between two objects, the effect of gravity
decreases, while the effect of dark energy increases; for example, doubling
the separation between two objects increases the DEtogravity acceleration
ratio by a factor of eight.
Now let's plug in the numbers. For earth, the acceleration from dark
energy is 9.3*10^{23} times that from the sun's
gravity. The corresponding change in the earth's orbit, compared to
an orbit without dark energy (and with the same orbital velocity), is
a tiny 140 nanometers, or about the size of a virus! For Pluto, the
distance from the sun is 40 times greater than the earthsun
distance; the corresponding change in Pluto's orbit due to dark energy
is 1 micron.
