Question

Galileo's telescopes were not of high quality by modern standards. He was able to see the moons of Jupiter, but he never reported seeing features on Mars. Use the small-angle formula to find the angular diameter of Mars when it is closest to Earth. How does that compare with the maximum angular diameter of Jupiter? (Assume circular orbits with radii equal to the average distance from the sun.)

Answer 1

The angular diameter of Mars when it is closest to Earth can be calculated using the small-angle formula. It is approximately 3.56 arcseconds. The maximum angular diameter of Jupiter is about 35.06 arcseconds, which is larger than Mars by a factor of about 9.85.

Step-by-step explanation:

To find the angular diameter of Mars when it is closest to Earth, we can use the small-angle formula.
Mars' diameter is about half that of Earth, and the distance from Earth ranges from 0.4 to 2.7 times the Earth-Sun distance. Let's assume the average distance from Earth to Mars is 1.55 times the Earth-Sun distance.
Using the small-angle formula, we have:
Angular Diameter (in arcseconds) = Linear Diameter (in km) / Distance (in km) * 206,265 arcseconds
The linear diameter of Mars is half the diameter of Earth, so it is 0.5 times the diameter of Earth, which is about 12,742 km. The average distance from Earth to Mars is 1.55 times the Earth-Sun distance, which is about 1.55 * 149.6 million km = 231.38 million km.
Let's calculate the angular diameter of Mars when it is closest to Earth:
Angular Diameter = (0.5 * 12,742 km) / (231.38 million km) * 206,265 arcseconds = 3.56 arcseconds (rounded to two decimal places).

The maximum angular diameter of Jupiter can be calculated similarly. Jupiter's diameter is about 10 times larger than Earth's, so it is about 127,420 km. Jupiter is about 5 times farther from the Sun than Earth, so its average distance from Earth is about 5 times the Earth-Sun distance, which is 5 * 149.6 million km = 748 million km.
Using the small-angle formula:
Angular Diameter = (10 * 127,420 km) / (748 million km) * 206,265 arcseconds = 35.06 arcseconds (rounded to two decimal places).

Comparing the angular diameters, we can see that Mars has an angular diameter of 3.56 arcseconds when closest to Earth, while Jupiter has a maximum angular diameter of 35.06 arcseconds. Therefore, Jupiter appears larger in the sky than Mars by a factor of about 9.85.

Answer 2

Angular diameter of Mars = 15.80 * 10^5 arc seconds

The Angular diameter of Mars is 3 times the angular diameter of Jupiter

Step-by-step explanation:

Average distance of the earth from sun = 150.67 * 10^6 km

assuming the radius of Mars ( average distance from sun) = 209.33 * 10^6 km

assuming the radius of Jupiter(average distance from sun) = 768.71 * 10^6 km

The small-angle formula for mars

angular diameter = ( linear diameter / distance ) * (2.06 * 10^5 )

distance between earth and mars = 54.6 * 10^6 km

linear diameter = 2 * radius = 418.66 * 10^6 km

angular diameter = ( 418.66 / 54.6 ) * 2.06 * 10^5

= 15.80 * 10^5 arc seconds

small angel formula for Jupiter

Angular diameter = ( linear diameter / distance ) * (2.06 * 10^5)

distance between Jupiter and earth = 588 * 10^6 km

linear diameter = 2 * radius = 1537.42 * 10^6 km

Angular diameter = ( 1537.42 / 588) * 2.06*10^5

= 5.39 * 10^5 arc seconds

comparing the angular diameter of the Mars and that of Jupiter :

The angular diameter of mars / angular diameter of Jupiter

= 15.80 / 5.39 = 2.931 ≈ 3