Answer :
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].
### 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].
