This sounds a lot like Ursula K. LeGuin's The Disposessed.
You read it about 40 years ago - The Disposessed was released in 1974, so that's 40 years and you either read it soon after it came out or maybe you're rounding up.
Irreconcilable differences in political systems have led the anarchists to settle the moon of the mother planet.
From Wikipedia:
The story takes place on the fictional planet Urras and its habitable
moon Anarres. In order to forestall an anarcho-syndicalist rebellion,
the major Urrasti states gave the revolutionaries the right to live on
Anarres, along with a guarantee of non-interference, approximately two
hundred years before the events of The Dispossessed.4 Before this,
Anarres had had no permanent settlements apart from some mining.
The anarchists have a hard-scrabble existence but have learned to thrive in their environment.
Again, Wikipedia:
However, in order to insure the survival of their society in a harsh
environment, the people of Anarres are taught from childhood to put
the needs of their society ahead of their own personal desires. Shevek
and Takver, as good Odonians, take work postings away from each other,
and Shevek does hard, agricultural labor in a dusty desert instead of
working on his research, because he is needed there due to a famine.
Their non-government is loosely led by a physicist, a born leader but a modest man who prefers his lab to any seat of power.
I think you may be mis-remembering this. Their non-government is actually simply organized by syndicates, but the main character is a physicist who is critically important because his work on FTL communications technology and a broader theory of how time works.
Some catastrophe that will destroy the mother planet is impending and only the physicist can save them. He makes the controversial decision to travel to the planet and help them. The bulk of the story is about his experiences on the planet.
I'm not sure if this part matches or not - the main character does travel to the main planet and it is controversial, but I don't think it's because of a major crisis. Basically the main planet is in a major cold war and the physicist is on the verge of inventing an FTL communications device. I believe that the general idea is that he is about to invent some critical technology and he's leveraging this to get some social change and to help his people interact with the main planet.
He makes the controversial decision to travel to the planet and help them. The bulk of the story is about his experiences on the planet.
The book actually alternates chapters between his experience on the planet. The controversy of him going to the main planet is because the people who live on the moon are radical separatists and have no contact with the main planet, except trade. From the wikipedia (sub)article on the planet:
It was settled by Odonian separatists coming from Urras. Ever since,
contact to Urras has been strictly limited by a treaty, the only point
of contact being Urrasti freighters landing and exchanging cargo at
the spaceport in Abbenay.
