Question

As the universe is expanding, is the solar system also expanding?

Answer 1

No! Not only the solar system but the entire local group of galaxies have decoupled themselves from expansion and no longer participate in the expansion. The reason for this is explained below.

Explanation:

The problem of what happens to mass distributions in an expanding Universe has been studied carefully for an ideal case of spherical distribution of matter in a flat - matter dominated expanding Universe (called Einstein-deSitter Universe ) by Gunn and Gott in 1971.

Reference:
Title : On the Infall of Matter Into Clusters of Galaxies and Some Effects on Their Evolution
Authors : Gunn, James E.; Gott, J. Richard, III
Publication : Astrophysical Journal, vol. 176, p.1

Summary of their Conclusion:
If matter is distributed uniformly in the Universe there will be no gravitational collapse but the expansion will lead to dilution of density with time.

If the matter is distributed non-uniformly in a static Universe then the regions that are over-dense will become denser assisted by gravitation. But in an expanding Universe gravity is not the only force. Caught in the background expansion, these over-dense regions expand slowly compared to the surrounding under-dense regions.

Their dynamics of expansion and their eventual fate is determined by a parameter called the over density amplitude, `\delta` (defined below).

`\delta\equiv \frac{\rho-\bar{\rho}}{\bar{\rho}}`, where `\rho` is the matter density in the over dense region and `\bar{\rho}` is the mean density of the Universe.
`\delta > 0` : overdense regions
`\delta < 0` : underdense regions

Gunn and Gott showed that if the over density amplitude exceeds a critical value `\delta_c = 1.06` (`2.06` times greater then the background density) then such regions eventually stop expanding, turn around and collapse gravitationally. Nothing can stop them now. Gravitational collapse amplifies the density further and further until the whole system virializes. When the system reaches a stable configuration through the process called virialization, its density is 150-200 times higher than the background density.

This is how structures like galaxies, galaxy clusters etc form. Once a system collapses this way it totally decouples itself from the expansion and will not participate in it anymore. So structures like galaxies and local group of galaxies are stable and do not expand anymore. But some massive galaxy clusters may still be expanding but they too would eventually turn around and collapse, while the rest of the Universe expand.