ACTIVITY . Kepler\'s Third Law In the previous activity, you used geometry to de
ID: 288996 • Letter: A
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ACTIVITY . Kepler's Third Law In the previous activity, you used geometry to determine the relative spacing of the six planets known in Copernicus's time. In the early 1600s, after Copernicus's heliocentric cosmos had become widely known, the brilliant mathematician Johannes Kepler took the model a step further: he checked whether the radius of each planet's orbit bore any numeri- cal relationship to its orbital period, the time it takes a planet to circle the Sun. The result was Kepler's famous Third Law: the square of the orbital period P is equal to the cube of the orbital radius a, or in equation form, P where P is expressed in years and a in AUs. Neither Kepler nor his contemporaries knew why planets obey this mathematical rule. His Third Law is empirical: revealed by observation, but whose underlying physics is not known. It would be decades before Isaac Newton uncovered the physical basis for Kepler's law in the fundamental properties of gravity 1. Table 10-1 on the worksheet lists modern determinations of the orbital period P and orbital radius a of Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Use yoar calcu- lator to fill in the rest of the table. Round off your answers to three decimal places 2. Even though your values for Pi and a' in Part 1 migh: not be exactly equivalent, do you still feel your results support Kepler's mathematical rule P-a Explain the reasoning you used to arrive at your answer 3. In 1781, in Bath, England, astronomer William Herschel created a worldwide sensa- tion when he discovered a new planet, Uranus. Then in 1846, German astronomer Johann Galle found an eighth planet, Neptune. Subsequent measurements fixed the orbital radii of Uranus at 19.191 AUs and Neptune at 30.069 AUs. Use Keplers Third Law to compute the orbital periods of (a) Uranus and (b) Neptune in years Show your work 4. Consider the orbital radi of all the planets studied in this activity. The values reveal a curious aspect of the planets' spacing there is a clustering of planets relatively close to the Sun, followed by what appears to be a gap between Mars and the closest of the outer planets, Jupiter. Italian astronomer Guseppe Piazzi -filled this gap in 1801 with his discovery of the first minor planet, or asteroid, Ceres. The orbital period of Ceres was found to be 4.603 years, Use Kepler's Third Law to compute the orbital radius of Ceres, in AUs Show your work. (You will need to compute a cube root to get your answer, ask for help if you need it.) 51Explanation / Answer
Mercury:
Orbital Period,P
=
0.241
Orbital radius,a
=
0.387
P^2
=
0.058yr
a^3
=
0.058AU
Venus:
Orbital Period,P
=
0.615
Orbital radius,a
=
0.723
P^2
=
0.378yr
a^3
=
0.378AU
Earth:
Orbital Period,P
=
1
Orbital radius,a
=
1
P^2
=
1.000yr
a^3
=
1.000AU
Mars:
Orbital Period,P
=
1.881
Orbital radius,a
=
1.524
P^2
=
3.538yr
a^3
=
3.540AU
Jupiter:
Orbital Period,P
=
11.862
Orbital radius,a
=
5.203
P^2
=
140.707yr
a^3
=
140.852AU
Saturn:
Orbital Period,P
=
29.458
Orbital radius,a
=
9.537
P^2
=
867.774yr
a^3
=
867.432AU
Orbital Period,P
=
0.241
Orbital radius,a
=
0.387
P^2
=
0.058yr
a^3
=
0.058AU
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