Space Math: Planet Kepler-10b: A Matter of Gravity
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Space Math: Planet Kepler-10b: A Matter of Gravity

Topic:

Body:

Mission:

Science Education Standards:

Physical Science:

Motions and Forces

• Objects change their motion only when a net force is applied. Laws of motion are used to calculate precisely the effects of forces on the motion of objects. The magnitude of the change in motion can be calculated using the relationship F = ma, which is independent of the nature of the force. Whenever one object exerts force on another, a force equal in magnitude and opposite in direction is exerted on the first object.
• Gravitation is a universal force that each mass exerts on any other mass. The strength of the gravitational attractive force between two masses is proportional to the masses and inversely proportional to the square of the distance between them.

Short Description: Students use the measured properties of the Earth-like planet Kepler 10b such as its size and density, and by solving Newton's formula for gravity, they determine the weight of a 100 kg human standing on the planet's surface.

Source: Space Math (GSFC)

 NASA Artist rendition of the sizzling-hot, Earth-like world: Kepler 10b.

The Kepler Space Observatory recently detected an Earth-sized planet orbiting the star Kepler-10. The more than 8 billion year old star, located in the constellation Draco, is 560 light years from Earth. The planet orbits its star at a distance of 2.5 million km with a period of 20 hours, so that its surface temperature exceeds 2,500 F.

Careful studies of the transit of this planet across the face of its star indicates a diameter 1.4 times that of Earth, and an estimated average density of 8.8 grams/cc, which is about that of solid iron, and 3-times the density of Earth's silicaterich surface rocks.

Problem 1 -Assume that Kepler-10b is a spherical planet, and that the radius of Earth is 6,378 kilometers. What is the total mass of this planet if its density is 8800 kg/meter3?

Problem 2 - The acceleration of gravity on a planet's surface is given by the Newton's formula

a = 6.67x10-11M/R2 meters/sec2

Where R is distance from the surface of the planet to the planet's center in meters, and M is the mass of the planet in kilograms. What is the acceleration of gravity at the surface of Kepler-10b?

Problem 3 - The acceleration of gravity at Earth's surface is 9.8 meters/sec2. If this acceleration causes a 68 kg human to have a weight of 150 pounds, how much will the same 68 kg human weigh on the surface of Kepler-10b if the weight in pounds is directly proportional to surface acceleration?

Problem 1 -Assume that Kepler-10b is a spherical planet, and that the radius of Earth is 6,378 kilometers. What is the total mass of this planet if its density is 8800 kg/meter3?

Answer: The planet is 1.4 times the radius of Earth, so its radius is 1.4 x 6,378 km = 8,929 kilometers. Since we need to use units in terms of meters because we are given the density in cubic meters, the radius of the planet becomes 8,929,000 meters.

Volume = 4/3πR3

so V = 1.33 x (3.141) x (8,929,000 meters)3
V = 2.98 x 1021meter3

Mass = Density x Volume
= 8,800 x 2.98 x 1021
= 2.6x1025 kilograms

Problem 2 - The acceleration of gravity on a planet's surface is given by the Newton's formula

a = 6.67x10-11M/R2 meters/sec2

Where R is distance from the surface of the planet to the planet's center in meters, and M is the mass of the planet in kilograms. What is the acceleration of gravity at the surface of Kepler-10b?

Answer: a = 6.67 x 10-11 (2.6 x 1025)/(8.929x106)2
= 21.8 meters/sec2

Problem 3 - The acceleration of gravity at Earth's surface is 9.8 meters/sec2. If this acceleration causes a 68 kg human to have a weight of 150 pounds, how much will the same 68 kg human weigh on the surface of Kepler-10b if the weight in pounds is directly proportional to surface acceleration?

Answer: The acceleration is 21.8/9.8 = 2.2 times Earth's gravity, and since weight is proportional to gravitational acceleration we have the proportion:

21.8/9.8 = X/150lb and so the human would weigh 150 x 2.2 = 330 pounds!