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Using Kepler's Third Law, what is the ratio of [tex]$\frac{p^2}{a^3}$[/tex] for Mars?

A. 4
B. 1
C. [tex]$1 \text{ AU}$[/tex]
D. 1 ly
E. [tex]$1.5 \text{ AU}$[/tex]


Sagot :

Sure, let's use Kepler's Third Law to find the ratio \( \frac{p^2}{a^3} \) for Mars.

### Step-by-Step Solution:

1. Kepler's Third Law: Kepler's Third Law of planetary motion states that the square of the orbital period \( p \) (in years) of a planet is directly proportional to the cube of the semi-major axis \( a \) (in astronomical units, AU) of its orbit. Mathematically, it is expressed as:
[tex]\[ \frac{p^2}{a^3} = \text{constant} \][/tex]

2. Constant Ratio: For all planets orbiting the Sun, this ratio is constant. For the Earth, we use the values \( p = 1 \) year and \( a = 1 \) AU. Therefore, the constant can be calculated as:
[tex]\[ \frac{p^2}{a^3} = \frac{1^2}{1^3} = 1 \][/tex]

3. Applying to Mars: For Mars, let’s denote its semi-major axis by \( a \) (which is 1.5 AU). Regardless of the value of \( a \), according to Kepler's Third Law, the ratio \( \frac{p^2}{a^3} \) remains the same constant for all planets in the solar system.

4. Conclusion: Therefore, the ratio \( \frac{p^2}{a^3} \) for Mars is:
[tex]\[ 1 \][/tex]

Hence, the ratio [tex]\( \frac{p^2}{a^3} \)[/tex] for Mars is [tex]\( \boxed{1} \)[/tex].