Pluto may be a planet part of the time.

From Peter Boulding comes this:

The Illinois State Senate has passed the following resolution:

SENATE RESOLUTION
WHEREAS, Clyde Tombaugh, discoverer of the planet Pluto,
was born on a farm near the Illinois community of Streator; and
WHEREAS, Dr. Tombaugh served as a researcher at the
prestigious Lowell Observatory in Flagstaff, Arizona; and
WHEREAS, Dr. Tombaugh first detected the presence of Pluto
in 1930; and
WHEREAS, Dr. Tombaugh is so far the only Illinoisan and
only American to ever discover a planet; and
WHEREAS, For more than 75 years, Pluto was considered the
ninth planet of the Solar System; and
WHEREAS, A spacecraft called New Horizons was launched in
January 2006 to explore Pluto in the year 2015; and
WHEREAS, Pluto has three moons: Charon, Nix and Hydra; and
WHEREAS, Pluto's average orbit is more than three billion
miles from the sun; and
WHEREAS, Pluto was unfairly downgraded to a "dwarf" planet
in a vote in which only 4 percent of the International
Astronomical Union's 10,000 scientists participated; and
WHEREAS, Many respected astronomers believe Pluto's full
planetary status should be restored;

therefore, be it
RESOLVED, BY THE SENATE OF THE NINETY-SIXTH GENERAL
ASSEMBLY OF THE STATE OF ILLINOIS, that as Pluto passes
overhead through Illinois' night skies, that it be
reestablished with full planetary status, and that March 13,
2009 be declared "Pluto Day" in the State of Illinois in honor
of the date its discovery was announced in 1930.

Peter notes:

(I found the link address in a Good Morning Silicon Valley story titled
"Repealing the law of gravity would probably require a two-thirds
majority".)

Acceptance of the Illinois State Senate's ability to legislate the planetary
status of a chunk of rock leads to an interesting question:

if Pluto is a planet only during the periods when it "passes overhead through Illinois'
night skies" and is re-downgraded to "minor planet" status the rest of the
time, for what proportion of the time will it once more hold its full
planetary status? What constitutes "overhead" in this context: do we assume
that it is "overhead" whenever it passes through an area bounded by lines
that start at the centre of the earth and pass through Illinois' state
boundaries? If so, how often is Pluto once more a planet?

Note that your calculation must be halved, thanks to the resolution's "night
skies" wording: daytime transits don't count.

As if Illinois state government wasn't enough of a laughing stock already, now we are claiming the right to determine the status of objects overhead.

I could not find that the Illinois House has taken any action on this.

But what about Peter's question?

So, under this resolution, what proportion of time would Pluto be a planet?

Earth has a surface area of 196,940,400 square miles
http://pages.prodigy.net/jhonig/bignum/qland2.html
Illinois has a surface area of 57,918 square miles
http://en.wikipedia.org/wiki/Illinois
Thus, Illinois covera .0294% of the surface area of the earth.

Presumably, Illinois would, at best, be able to claim 0.0294% of the sky.
As you note, this should be divided in two, to reflect the fact that it's only "night" in Illinois half the time.
This means Pluto would be a planet .0147% of the time, or about 12 seconds a day, overall.

But the above calculation assumes that Illinois was randomly located, rather than just randomly legislating. Illinois is in the northern hemisphere, about 40 degrees north latitude, and so it seems unlikely that the particular .0294% of the sky which would most logically be allocated to Illinois would ever include Pluto, although Pluto has a fairly eccentric orbit and may deviate a fair amount from an equitorial plane. The equator of the earth also tilts, which creates some further complications. These simple calculational adjustments are left to the reader as an exercise.