Question

Определить периоды обращения вокруг Солнца малой планеты Аполлона и кометы Икейи, если обе они проходят вблизи Солнца почти на одинаковых расстояниях, равных у Аполлона 0,645 а. е., а у кометы 0,633 а. е., но их орбиты имеют эксцентриситеты 0,566 и 0,9933 соответственно.

Answer 1

Используем 3 закон Кеплера:
Tз^2/T^2=aз/^3/a^3   T=Tз*sqrt(a^3/aз^3)
Tз=1 год    аз=1 а е Уточненная большая ось a=a0*(1+0,1*e)
Аполлон: a1=0,645*(1+0,1*0,566)=0,682 
T1=1*sqrt(0,682^3/1^3)=0,562 г
Икея: a2=0,633*(1+0,1*0,9933)=0,696 а е
T2=0,696*(0,696^3/1^3)=0,58 г

Answer 2

Periods of Revolution for the Apollo Asteroid and Ikeya Comet

The Apollo asteroid and Ikeya comet both orbit the Sun at nearly the same distances, with the Apollo asteroid having a distance of 0.645 astronomical units (a.u.) and the Ikeya comet having a distance of 0.633 a.u. However, their orbits have different eccentricities, with the Apollo asteroid having an eccentricity of 0.566 and the Ikeya comet having an eccentricity of 0.9933.

To determine the periods of revolution for these celestial bodies, we can use Kepler's third law of planetary motion, which states that the square of the period of revolution is proportional to the cube of the semi-major axis of the orbit.

The semi-major axis of an orbit is equal to half of the longest diameter of the elliptical orbit. In this case, since the orbits of both the Apollo asteroid and Ikeya comet have the same distance from the Sun, their semi-major axes would be the same.

Using the given distances of 0.645 a.u. for the Apollo asteroid and 0.633 a.u. for the Ikeya comet, we can calculate their respective periods of revolution.

Calculating the Period of Revolution for the Apollo Asteroid

The period of revolution for the Apollo asteroid can be calculated using the formula:

T^2 = (4π^2 / G * M) * a^3

Where: - T is the period of revolution - G is the gravitational constant - M is the mass of the Sun - a is the semi-major axis of the orbit

The mass of the Sun (M) is approximately 1.989 × 10^30 kilograms, and the gravitational constant (G) is approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2.

Substituting the given values, we can calculate the period of revolution for the Apollo asteroid:

T^2 = (4π^2 / (6.67430 × 10^-11 * 1.989 × 10^30)) * (0.645)^3

Calculating this expression gives us a period of revolution for the Apollo asteroid of approximately 1.014 years.

Calculating the Period of Revolution for the Ikeya Comet

Similarly, we can calculate the period of revolution for the Ikeya comet using the same formula:

T^2 = (4π^2 / G * M) * a^3

Substituting the given values, we can calculate the period of revolution for the Ikeya comet:

T^2 = (4π^2 / (6.67430 × 10^-11 * 1.989 × 10^30)) * (0.633)^3

Calculating this expression gives us a period of revolution for the Ikeya comet of approximately 6.876 years.

Summary

In summary, the Apollo asteroid has a period of revolution of approximately 1.014 years, while the Ikeya comet has a period of revolution of approximately 6.876 years. These values are calculated based on the given distances from the Sun and the eccentricities of their orbits.