From
Scientific American: http://www.scientificamerican.com/blog/post.cfm?id=habitable-exoplanets-could-exist-at-2011-04-01
Transit Surveys for Earths in the Habitable Zones of White Dwarfs
To date the search for habitable Earth-like planets has primarily focused on nuclear burning stars. I propose that this search should be expanded to cool white dwarf stars that have expended their nuclear fuel. I define the continuously habitable zone of white dwarfs and show that it extends from ≈0.005 to 0.02 AU for white dwarfs with masses from 0.4 to 0.9 M sun, temperatures less than ≈10^4 K, and habitable durations of at least 3 Gyr. As they are similar in size to Earth, white dwarfs may be deeply eclipsed by terrestrial planets that orbit edge-on, which can easily be detected with ground-based telescopes. If planets can migrate inward or reform near white dwarfs, I show that a global robotic telescope network could carry out a transit survey of nearby white dwarfs placing interesting constraints on the presence of habitable Earths. If planets were detected, I show that the survey would favor detection of planets similar to Earth: similar size, temperature, and rotation period, and host star temperatures similar to the Sun. The Large Synoptic Survey Telescope could place even tighter constraints on the frequency of habitable Earths around white dwarfs. The confirmation and characterization of these planets might be carried out with large ground and space telescopes.
But even a white dwarf that formed over 10 billion years ago would have a relatively short "habitable window" at the present time, only a few billion years. Though a white dwarf is essentially condensed, not changing its size significantly as it cools, it can cool by a sizable amount over the age of the Universe. From http://astro1.panet.utoledo.edu/~gthomps/Presentations/Fall07ABL9.25.pdf , an approximate equation for a white dwarf's luminosity is
(L/Lsun) = 8.4*10
-4 * (M/Msun) * (time/(1 Gyr))
-7/5
1 Gyr = 10
9 years.
So a WD's planet's temperature would vary as power -7/20 of the time since it became a WD.
Let's see how some familiar white dwarfs stack up
Which | Age | Mass(S) | Rad(S) | Lum(S) | Temp | a(AU) | Per(d) | Hab. window
Sirius B | 0.2 Gyr | 0.978 | 0.0084 | 0.026 | 25200 K | 0.16 | 24 | 0.12 Gyr
Procyon B | 3 Gyr | 0.602 | 0.01234 | 0.00055 | 7740 K | 0.023 | 1.7 | 1.8 Gyr
The habitability window is the amount of time where the planet's surface temperature goes from 1.1 * Earth's to 0.9 * Earth's, and is about 0.6 * current age. The oldest white dwarfs are around 10 billion years old, giving them a present-time habitability window of 6 billion years, close to the Earth's. A habitable planet would orbit at around 0.01 AU, with a period of about half a day.
A habitable planet of a white dwarf would get tidally locked, so one side always faces its star, like the Moon with the Earth.
There is an interesting historical parallel. In the late 19th cy., various scientists had proposed that the Sun is slowly cooling down, making the Earth's surface also slowly cool down. This is what would happen to a planet of a white dwarf.
There's also the question of whether a planet could form that close to a white dwarf, or if it formed elsewhere, how it could get that close.