How long does the Doctor go between visiting his companions?

Question

Concerned with the 2005 relaunch:

Do we have an estimate for how long the Doctor travels on his own between visits with his companions while he actively "has" companions?

While watching the 11th and 12th Doctor's adventures, it seems as if his companions (Amy and Clara, specifically) were left alone at times to lead a "normal" life between their trips in the TARDIS.

But given the nature of the Doctor and the TARDIS, there's no reason that he would spend the same amount of time away from his companions as his companions spend away from him (from each of their own perspectives).

So, how long is he away?

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An average is probably the only reasonable result. I'd think we'd have to take at least 2 mentions of the Doctor's age that we can take as accurate (for the sake of the question), and divide it by the number of known companion adventures. But, this may be more difficult than just getting an episode count, because we see in S8's The Caretaker that Clara is going on very many adventures not chronicled in their own full episodes, each separated by time at home without the Doctor.

This also isn't meant to be a question of "How much has the Doctor aged since the show began", because there are implied large gaps between him having companions, and those should be removed from the equation (such as the time between 10th's regen and 11th's picking of Amy as a companion, or the time between the Pond's and Clara).

Answer 1

Honestly, we have no idea how much time he spends between companions because that all happens almost entirely off-screen. However, if Eleven's behavior near the end of his run with Amy and Rory is typical, then probably "not much".

First off, it's important to note that his companions usually travel with him in the TARDIS for long periods of time. The arrangement with Clara seems unique: leaving her at home and only popping up on occasion. This was also mostly her decision: she didn't want to run off with him forever.

Usually, once he acquires a new companion, they basically move into the TARDIS. In the classic series, the companions home lives were rarely even mentioned. In the revived series, the companions friends and family have played a bigger role, but in every case: Rose, Martha, Donna, and (at first) Amy and Rory, it's clear that the companions "live with" The Doctor and "visit" Earth every so often.

Near the end of Amy and Rory's journey, The Doctor decided that he was going to have to leave them behind. This was unusual for him, because (as River explained more than once), he hates saying "goodbye". Usually he just keeps his companions until they abandon him or die. One main reason he opted to leave Amy and Rory was to avoid exactly that.

But, as we see later on with the Eleventh Doctor in "A Town Called Mercy", and even a little bit with the Tenth Doctor in "The Waters of Mars", when he's left alone without a companion to ground him, he starts to go off the rails. He begins to act less rationally, less humane than usual. Amy specifically calls him out on this:

AMY: And what then? Are you going to hunt down everyone who's made a gun or a bullet or a bomb?

DOCTOR: But they coming back, don't you see? Every time I negotiate, I try to understand. Well, not today. No. Today, I honour the victims first. His, the Master's, the Dalek's, all the people who died because of my mercy!

AMY: You see, this is what happens when you travel alone for too long. Well, listen to me, Doctor. We can't be like him. We have to be better than him. [emphasis mine]

In fact, in the classic series, it was common for the Doctor to go several incarnations without any period of "alone time"; he would often have multiple companions at once, and they'd come and go at different times. The same thing happened with Nine regenerating to Ten and Eleven regenerating into Twelve: there was no down-time for him, because Rose and Clara, respectively, were with him the whole time.