As the universe is expanding, is the solar system also expanding?
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.
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.
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,
Gunn and Gott showed that if the over density amplitude exceeds a critical value
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.