- You may be looking for Short Trips: The Solar System.
The First Doctor was unfamiliar with the solar system prior to his departure on Gallifrey. However, his granddaughter Susan Foreman learned of its existence and that of Earth in her spacial cartography system. (AUDIO: The Beginning)
During the time of the Silurians and Sea Devils, a rogue planet approached the orbit of their homeworld, Earth. The Third Doctor told the Silurians that instead of crashing into the Earth, it became Earth's moon, (TV: Doctor Who and the Silurians) as did the Eleventh Doctor. (TV: Cold Blood)
Circa 12,000,000 BC the Time Lords placed Planet 5 in a time loop, in effect destroying it. The Fendahl, which came from the Fifth Planet and which the Time Lords had intended to destroy, may have destroyed life on Mars on its way to Earth. (TV: Image of the Fendahl)
- An asteroid belt somehow resulted from the time looping of the fifth planet.
In the mid-20th century, the human of Earths began to explore ther solar system via probes and, tentatively at first, with manned missions. The Moon was the first object in the solar system to be visited by humans — at least, according to the official historical record. (TV: Blink, et al)
Circa 2009, Earth was forcibly relocated to the Medusa Cascade by the Daleks. The relocation of Earth seemed to have no lasting effects on the Sol system; even Earth's Moon remained in place until the Tenth Doctor returned Earth to its proper position. (TV: The Stolen Earth/Journey's End) The end result of the incident was that the general populace of Earth were now aware of the existence of alien life (TV: Children of Earth: Day One, et al), ushering in the planet's increased presence in interstellar affairs from the 21st century onward. (TV: The Christmas Invasion, Everything Changes, et al)
Around 2089, Susie Fontana Brooke, the granddaughter of Captain Adelaide Brooke piloted the first ever lightspeed ship from Earth to Proxima Centauri. Soon after that, the human race started colonising other star systems. (TV: The Waters of Mars)
Prior to 4000, Mavic Chen assumed the position of Guardian of the Solar System. Chen's secret allies, the Daleks, at the same time proposed to conquer the solar system. (TV: Mission to the Unknown, The Daleks' Master Plan)
In the 57th segment of Time the last humans left on Earth evacuated the planet because of the increasing danger that it would fall into the Sun. They fled to planets such as Refusis II and Frontios. (TV: The Ark, Frontios) The now uninhabited Earth was preserved by the National Trust and over the billions of years following was restored to a 'classic Earth' before it was finally destroyed by the expanding sun in the year 5.5/Apple/26. (TV: The End of the World)
After a long struggle both on their own planet, Earth, and with other sentient species in the system, humans became the region's dominant species and led several different inter-planetary governments. (TV: The Daleks' Master Plan, The Monster of Peladon, Bad Wolf) Indeed, the solar system's significance was largely tied to the fate of humanity, since humans became one of the universe's longest-lived species, continuing on until the universe itself was entering its final stage of development. (TV: The End of the World, Utopia)
Behind the scenes
The term solar system can also be a simple synonym for star system in DWU fiction. However, in most instances, solar system—even though uncapitalised—refers to "the star system bound together by Sol".