In "The Paradise Syndrome" a giant asteroid is only 2 month away from striking an inhabited planet and killing all life on it. Our heroes are trying to divert the asteroid.
Spock says:
Spock may mean that the asteroid has almost:
1) the mass of Earth's moon.
2) the volume of Earth's moon, its mass depending on its relative density.
3) the dimensions of Earth's moon. Since the volume varies with the cube of the dimensions, an asteroid with 0.90 the diameter of Earth's moon would have only 0.729 the volume of Earth's moon.
For the sake of being able to divert the asteroid we may hope Spock doesn't mean almost the mass of Earth's moon.
Earth's moon Luna has a mass of 7.432 times 10 to the 22nd power kilograms, which is 0.012300 of Earth's mass. It has a volume of 2.1958 times 10 to the 10th power cubic kilometers, giving it a density of 3.344 grams per cubic centimeter, about 0.606 of Earth's average density. It has an average radius of 1,737.1 kilometers (1,079.3839 miles), 0.273 of Earth's, giving it an average diameter of 3,474.2 kilometers (2,158.7678 miles).
If Spock meant that the asteroid had a diameter "almost as large as your Earth's moon", he might have meant that it was 0.75 the diameter of the moon, or 2,605.65 kilometers (1,619.075 miles). That would give it a volume 0.4218 that of the moon, and thus a mass 0.4218 of the moon if they had the same density. If the asteroid had a lower density it might have about 0.33 of the mass of the moon or something. That's a mighty big asteroid.
Astronomical bodies above a certain mass will have a gravity intense enough to overcome the resistance of their materials and pull themselves into spheroidal shapes like planets and achieve hydrostatic equilibrium.
Rhea, a moon of Saturn, was said to be the smallest body in our solar system carefully measured and probably in hydrostatic equilibrium, with a radius of 763.8 kilometers (457.82 miles) and a diameter of 1,527.6 kilometers (949.20 miles). But the asteroid Ceres, with a a radius of 473 kilometers (293.909 miles) and a diameter of 946 kilometers (587.8 miles) is said to be in hydrostatic equilibrium and considered a dwarf planet.
Vesta, the second largest asteroid, has a slightly irregular shape. It has dimensions of 572.6 kilometers (355.79 miles) by 557.2 kilometers (343.23 miles) by 446.4 kilometers (277.38 miles).
https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
Here you can see an image of the asteroid from the original version of "The Paradise Syndrome".
http://www.theviewscreen.com/the-paradise-syndrome/
Here you can see the less colorful but more realistic version of the asteroid from the remastered version of the episode.
http://www.startrek.com/database_article/paradise-syndrome-the
You can see that the asteroid seems more irregular than Vesta and thus probably a lot smaller than Vesta.
How can the asteroid be almost as large as Earth's moon and yet have such an irregular shape?
Possibly it is part of the nickel-iron core of some larger body that was struck by another body and shattered. If that was recent enough, the solar system in the episode would be full of dangerous fragments threatening the planets, and the asteroid might not yet have pulled itself into a spherical shape. I don't know how to calculate how long it would take for an irregular nickel-iron core fragment to pull itself into a spherical shape.
Many stony - not nickel-iron - asteroids are made of two or more smaller pieces loosely joined together, and rotate fast enough that their centrifugal force is almost enough to overcome their gravity and fling them apart. Thus they have weak spots where their pieces are joined together and cutting them at the junction will cause them to fly apart, possibly with enough speed to drift far enough to avoid the planet after 2 months.
So Spock's strategy would make a lot of sense with a more normal sized asteroid that is weakly held together.
Maybe Earth's entire solar system is a member of the Federation, but is subdivided into several subdivisions that are not subject to each other, such as the Martian colonies, the United Earth, and the moon government. Thus Earth's physical moon is not under Earth's jurisdiction, only that of its own government, the United Solar System, and the Federation. But maybe an asteroid, a little larger than the asteroid in "The paradise Syndrome" has been moved into Earth orbit to be mined. Maybe that asteroid is under the political jurisdiction of the United Earth Government, and is owned by the United Earth, and thus by the citizens of Earth.
There is reason to believe that during the Cardassian occupation of Bajor several asteroids were moved into orbit around Bajor and mined until they were totally destroyed, thus making the number of Bajoran moons fluctuate.
In the DS9 first season episode "The Nagus" Nog reads a text (date of composition uncertain):
And in another first season episode, "Progress" Sisko makes a log entry:
Maybe Spock was thinking of that hypothetical asteroid moved into Earth orbit as being a similar temporary second moon of Earth when he said:
That would make "The Paradise Syndrome" seem much more plausible.
Spock says:
Spock may mean that the asteroid has almost:
1) the mass of Earth's moon.
2) the volume of Earth's moon, its mass depending on its relative density.
3) the dimensions of Earth's moon. Since the volume varies with the cube of the dimensions, an asteroid with 0.90 the diameter of Earth's moon would have only 0.729 the volume of Earth's moon.
For the sake of being able to divert the asteroid we may hope Spock doesn't mean almost the mass of Earth's moon.
Earth's moon Luna has a mass of 7.432 times 10 to the 22nd power kilograms, which is 0.012300 of Earth's mass. It has a volume of 2.1958 times 10 to the 10th power cubic kilometers, giving it a density of 3.344 grams per cubic centimeter, about 0.606 of Earth's average density. It has an average radius of 1,737.1 kilometers (1,079.3839 miles), 0.273 of Earth's, giving it an average diameter of 3,474.2 kilometers (2,158.7678 miles).
If Spock meant that the asteroid had a diameter "almost as large as your Earth's moon", he might have meant that it was 0.75 the diameter of the moon, or 2,605.65 kilometers (1,619.075 miles). That would give it a volume 0.4218 that of the moon, and thus a mass 0.4218 of the moon if they had the same density. If the asteroid had a lower density it might have about 0.33 of the mass of the moon or something. That's a mighty big asteroid.
Astronomical bodies above a certain mass will have a gravity intense enough to overcome the resistance of their materials and pull themselves into spheroidal shapes like planets and achieve hydrostatic equilibrium.
Rhea, a moon of Saturn, was said to be the smallest body in our solar system carefully measured and probably in hydrostatic equilibrium, with a radius of 763.8 kilometers (457.82 miles) and a diameter of 1,527.6 kilometers (949.20 miles). But the asteroid Ceres, with a a radius of 473 kilometers (293.909 miles) and a diameter of 946 kilometers (587.8 miles) is said to be in hydrostatic equilibrium and considered a dwarf planet.
Vesta, the second largest asteroid, has a slightly irregular shape. It has dimensions of 572.6 kilometers (355.79 miles) by 557.2 kilometers (343.23 miles) by 446.4 kilometers (277.38 miles).
https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
Here you can see an image of the asteroid from the original version of "The Paradise Syndrome".
http://www.theviewscreen.com/the-paradise-syndrome/
Here you can see the less colorful but more realistic version of the asteroid from the remastered version of the episode.
http://www.startrek.com/database_article/paradise-syndrome-the
You can see that the asteroid seems more irregular than Vesta and thus probably a lot smaller than Vesta.
How can the asteroid be almost as large as Earth's moon and yet have such an irregular shape?
Possibly it is part of the nickel-iron core of some larger body that was struck by another body and shattered. If that was recent enough, the solar system in the episode would be full of dangerous fragments threatening the planets, and the asteroid might not yet have pulled itself into a spherical shape. I don't know how to calculate how long it would take for an irregular nickel-iron core fragment to pull itself into a spherical shape.
Many stony - not nickel-iron - asteroids are made of two or more smaller pieces loosely joined together, and rotate fast enough that their centrifugal force is almost enough to overcome their gravity and fling them apart. Thus they have weak spots where their pieces are joined together and cutting them at the junction will cause them to fly apart, possibly with enough speed to drift far enough to avoid the planet after 2 months.
So Spock's strategy would make a lot of sense with a more normal sized asteroid that is weakly held together.
Maybe Earth's entire solar system is a member of the Federation, but is subdivided into several subdivisions that are not subject to each other, such as the Martian colonies, the United Earth, and the moon government. Thus Earth's physical moon is not under Earth's jurisdiction, only that of its own government, the United Solar System, and the Federation. But maybe an asteroid, a little larger than the asteroid in "The paradise Syndrome" has been moved into Earth orbit to be mined. Maybe that asteroid is under the political jurisdiction of the United Earth Government, and is owned by the United Earth, and thus by the citizens of Earth.
There is reason to believe that during the Cardassian occupation of Bajor several asteroids were moved into orbit around Bajor and mined until they were totally destroyed, thus making the number of Bajoran moons fluctuate.
In the DS9 first season episode "The Nagus" Nog reads a text (date of composition uncertain):
And in another first season episode, "Progress" Sisko makes a log entry:
Maybe Spock was thinking of that hypothetical asteroid moved into Earth orbit as being a similar temporary second moon of Earth when he said:
That would make "The Paradise Syndrome" seem much more plausible.
