Conceptual Idea: “According to Newton, Kepler's 3rd Law (P2 = D3) should be modified to read M+m = D3/P2 where M+m is mass in units of Sun+Earth mass, D is the orbital semi-major axis in astronomical units (AU), and P the time for one orbital revolution in Earth years." If one were trying to measure the mass of the Sun, MS, by observing a planet of mass mP orbiting the Sun, why could this equation be used to get a good estimate of MS? If one were trying to measure the mass of Jupiter, MJ, by observing the moon Ganymede of mass mG orbiting Jupiter, why could this equation be used to get a good estimate of MJ?

Conceptual Idea: “According to Newton, Kepler's 3rd Law (P2 = D3) should be modified to read M+m = D3/P2 where M+m is mass in units of Sun+Earth mass, D is the orbital semi-major axis in astronomical units (AU), and P the time for one orbital revolution in Earth years." If one were trying to measure the mass of the Sun, MS, by observing a planet of mass mP orbiting the Sun, why could this equation be used to get a good estimate of MS? If one were trying to measure the mass of Jupiter, MJ, by observing the moon Ganymede of mass mG orbiting Jupiter, why could this equation be used to get a good estimate of MJ?

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter11: Gravity, Planetary Orbits, And The Hydrogen Atom

Section: Chapter Questions

Problem 21P: Comet Halley (Fig. P11.21) approaches the Sun to within 0.570 AU, and its orbital period is 75.6 yr....

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Comet Halley (Fig. P11.21) approaches the Sun to within 0.570 AU, and its orbital period is 75.6 yr. (AU is the symbol for astronomical unit, where 1 AU = 1.50 1011 m is the mean EarthSun distance.) How far from the Sun will Halleys comet travel before it starts its return journey?
Mars' period (its "year") was noted by Kepler to be about 687 days (Earth days), which is (687d / 365 d) = 1.88 yr. Determine the distance of Mars from the Sun using the Earth as reference. (The distance of Earth from the Sun is 1.50 x 10 m) !3! Thu TMS IMS TE TES TES TE 2 28 x 10 m TES yr After reading and understanding the concept Gravity, please do the following problems: 1. What keeps a satellite up in its orbit around the Earth?
Use Kepler's Law, which states that the square of the time, T, required for a planet to orbit the Sun varies directly with the cube of the mean distance, a, that the planet is from the Sun.Using Earth's time of 1 year and a mean distance of 93 million miles, the equation relating T (in years) and a (in million miles) is 804375T2=a3.Use that relation equation to determine the time required for a planet with mean distance of 206 million miles to orbit the Sun. Round to 2 decimal places. years
A new planet is discovered orbiting a distant star. Observations have confirmed that the planet has a circular orbit with a radius of 12 AU and takes 117 days to orbit the star. Determine the mass of the star. State your answer with appropriate mks units. [NOTE: AU ..stands.for...astronomical unit". It is the average distance between Earth & the Sun. 1 AU≈ 1.496 x 1011 m.] Enter a number with units. I be quite large and your calculator will display the answer as a power of 10. If, as an example, your answer was 8.54 x 1056, you would type "8.54e56" into the answer box (remember to state your units with your answer).]
If a planet has a period of 8 years, what is its average distance from the Sun, in astronomical units? Assume thatthe mass of the planet is small compared to the Sun and thatthe average distance from the Sun is the same as the semimajoraxis. Make sure you express your answer with at least one digitafter the decimal point.
Consider the Earth's orbit around the Sun to be circular with radius R = 9.30 x 107 mi and it takes 365 days to complete one revolution. What is the distance Earth traveled for one revolution (circumference of a circle is 2??2πR )?
(a) Jupiter's third-largest natural satellite, Io, follows an orbit with a semimajor axis of 422,000 km (4.22 ✕ 105 km) and a period of 1.77 Earth days (PIo = 1.77 d). To use Kepler's Third Law, we first must convert Io's orbital semimajor axis to astronomical units. One AU equals 150 million km (1 AU = 1.50 ✕ 108 km). Convert Io's a value to AU and record the result. aIo = AU (b) One Earth year is about 365 days. Convert Io's orbital period to Earth years and record the result. PIo = yr (c) Use the Kepler's Third Law Calculator to calculate Jupiter's mass in solar units. Record the result. MJup(Io) = MSun (d) Based on this result, Jupiter's mass is about that of the Sun. Jupiter has a similar fraction of the Sun's volume. The two objects therefore have rather similar density! In fact, Jupiter has a fairly similar composition as well: most of its mass is in the form of hydrogen and helium.
The value of Gme/re2, where me is the mass of the earth, re is the radius of the earth, and G is the universal gravitational constant is (enter your answer with three significant figures)
Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm3 of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?
Calculate the angle of (a) 10 (degree) (b) 1′ (minute of arc or arcmin) and (c) 1″(second of arc or arc second) in radians. Use 3600=2π rad, 10 =60′ and1′ = 60 ″
Newton’s law of gravitation and the formula for centripetal acceleration can be used to show that: T^2=(4π^2/Gms)R^3 where G is the universal constant of gravitation and MS is the mass of the Sun. Take logarithms to base 10 of both sides of the equation to complete the expression for 2 lg T.2 lg T = ……………… × lg R + ……………………
The Earth is travelling around the sun at a speed of about 67,000 miles per hour [mph]. If it takes one year [yr] for the Earth to circle the Sun and we assume the Earth?s orbit is circular, what is the radius of the orbit in units of meters [m]? Use an appropriate SI prefix to report your answer so there are one, two, or three digits to the left of the decimal point.