If an Earth-like planet was tidally locked to a star similar to Sirius A, at which distance from the star would the night side be able to have liquid water, possibly with an average temperature of about 10ºC in the night side?
A fascinating presentation on the SETI Seminar Series goes over tidally locked worlds and lays to rest the presumption that the day and night sides are extreme and inhospitable.
The situation can be substantially complicated by
- odd half multiples of spin-orbit synchronization, like Mercury
- inclination of orbital plane
- axial tilt
In short, locking does not mean an unmoving sun; situations like Mercury are in fact the norm, and the presence of other giant planets can drive a close rocky world into a state with both high inclination and eccentricity, and eccentricity causes odd-half multiple rotations to be preferred.
In extreme cases, the overall insolation can be averaged out to the same degree as a non-locked world like ours. And it takes far less to average out the temperature to a reasonable degree.
Sirius A is 25 times more luminous than the sun but less than twice the diameter, and is only 250 million years old.
A planet close in would be baked indeed. Unlike the case of Mercury and the trend of figuring out how such planets could be more uniformly lit, you really do want it locked with a permanent night side.
Note that even Mercury has ice in permanent shadows. The key is not having an atmosphere. That means not having liquid water either.
Consider an underground aquifer. If the planet has a permanent day and night side, the place for liquid water does not have to be exactly at the anti-sol point. It can be frozen at the midnight point, and have a "just right" belt some distance between the hot and cold.
The existence of such temperature gradients depends on the composition of the rock, and how it conducts heat. So, you can considerable flexibility in making it (plausibly) come out just right.
I'm more worried that a rocky world would not have had time yet to cool down and form a solid crust. That's more of a hard math physics question.
When discussing the habitability of a tidally locked planet, temperatures are only ONE of the vast array of variables, each of which has a critical role in determining whether long term life sustenance is possible on the planet or not. Here is a list:
As presented in your query, temperature is one of the key factors determining the habitability of a planet (tidally locked or not). The temperature on the night side of the planet would depend on:
a. Composition and thickness of the atmosphere. By "thickness" I mean how far the atmosphere extends above the crust of the planet. By composition, I specifically mean the presence of greenhouse gases. You could say that temperature on the night side = density of atmosphere x ratio of greenhouse gases
b. Presence of active volcanoes on the night side. This should be self explanatory. Volcanic activity does a long term job of heating up the crust of a planet.
c. Presence of large water bodies (oceans) on the planet. Large water bodies do a magnificent job in reducing the difference of temperatures between two zones. If there is a huge ocean that stretches both sides (day and night) on the planet, the temperature difference on these sides would be much lesser than if it were a desert planet (no water).
2- Perpetual Wind Speed
Seasonal storms and gales are one thing, but when you have a large temperature difference between two sides of a PLANET, things get terribly messy. We are talking about vortexes, cyclones and dunes in speeds excess of 1000 km/h. For the sake of reference, remember that a hurricane/cyclone with wind speed exceeding 350 km/h would splatter normal concrete houses to bits. Storms exceeding 600 km/h would bend and crush metal structures (we are talking steel here). I don't want to discuss wind speeds exceeding 800 km/h here. The very thought gives me shudders of horror. You should read about the giant red spot on jupiter to learn about what I mean. All these horrible vortexes are generated due to temperature differences.
3- Renewable Food Source
The night side of a planet means no direct sunlight. No direct sunlight means no photosynthesis. No photosynthesis means no green plants. No green plants means no herbivores. No herbivores mean no mutton, beef or chicken. No photosynthesis also means no wheat, rice, maize, barley, pizza, pasta, chocolate cake ... You will need to develop a food chain base. A food chain base is something that generates food for others and makes it food by itself. On our planet earth, it is green plants making magic happen with photosynthesis.
5- Building Material Availability
What are you going to make buildings with, on the dark side of the planet? No photosynthesis means that "wood" is out of question.
a. Is there a system of natural caves where people can live in safely?
b. Is there an abundance of easily available building materials other than wood? Iron, hydrocarbons (plastics family), construction grade rocks (granite, hard layered limestone/marble etc). If your people aren't living in caves, you'd certainly need some of these things to build houses for them.
6- Perpetual Light Source(s)
Because, you know, you are living on the night side of the planet and your characters would be groping in perpetual dark unless ... you give them sort of giant planet-wide lamp or something.
7- And yet ...
The above mentioned 6 variables are the very basic things you cannot even THINK surviving without, on the night side of a tidally locked planet. Having all these does NOT mean you can host an intelligent civilization on that planet. Think about these little things:
a. In the absence of photosynthesis, how are you going to get all the vitamins required for healthy growth? You can't survive on carbohydrates or proteins alone. Where are you going to get the "fiber" in your food?
b. Sunlight is also necessary for skin, eyes and several other things in our body. If you live in an underground well-lit shelter for 2 weeks, you are going to quickly lose your skin freshness and your muscle strength. Thats what sunlight deprivation does to you!
c. For a civilization living in perpetual dark side of a planet, evolution rule states that they will lose the sense of vision rapidly as there will be no use for it. Creatures living in dark caves don't have eyes. You ought to provide a natural light source for your people. It's mandatory if you want them "seeing" things!
d. Besides energy requirements of food, you also need water. Fresh, clean, pathogen free water. At least for complex life on earth, which we are used to seeing and being.
These are the very basic factors determining the habitability of a planet. There are several other "subtle" variables too, which I haven't included here for the sake of refraining from dragging on. For a start, think about gravity, metabolism type (oxygen I mean for earthly life), sensory organs, modes of locomotion, moon systems, cosmic rays prevention system (ozone layer, in case you were wondering) etc etc etc