HELP ME PLEASE WITH THIS Comet EXERCISE: Comet JT2023, known as the Olympic Comet, was discovered by astronomers Bruce
Brolin and Frankus Mascinni through the Ancient Lights search project on September 15, 2001. This
comet had its closest approach to Earth on February 29, 2020. Its orbit has an eccentricity of
e=0.99920 and an aphelion distance of ra=2,799 UA. Given that the mass of the Sun is MS-1.99x1030
kg.
a) Sketch the orbit and positions of the comet and the Sun, as well as the elements of the elliptical
orbit.
b) Work geometrically and algebraically to find a formula that relates the eccentricity of the orbit, the
aphelion, and the semi-major axis.
c) Determine the speed of the comet at its aphelion.
a) To sketch the orbit and positions of the comet and the Sun, as well as the elements of the elliptical orbit, you will need to understand the basic elements of an elliptical orbit. These elements include the eccentricity (e), the aphelion distance (ra), and the semi-major axis (a).
Start by drawing a coordinate system on a piece of paper with the Sun positioned at the origin (0,0). The comet will be positioned somewhere along the elliptical orbit around the Sun.
Next, draw the elliptical orbit itself. The shape of the ellipse will depend on the eccentricity (e). If the eccentricity is close to 0, the orbit will be nearly circular, while larger eccentricity values will result in more elongated orbits. Use the given eccentricity value of e=0.99920 to determine the shape of the ellipse. Remember that the aphelion distance (ra) is the distance between the Sun and the farthest point in the orbit.
Now, draw the Sun at the origin and mark the aphelion distance ra on the elliptical orbit. This point will represent the farthest distance between the comet and the Sun.
Finally, label the elements of the elliptical orbit, including the eccentricity (e), the aphelion distance (ra), and the semi-major axis (a). The semi-major axis can be calculated by dividing the aphelion distance by (1 + e), according to Kepler's laws.
b) To find a formula that relates the eccentricity of the orbit, the aphelion, and the semi-major axis, you can use Kepler's laws. The relationship between these elements is given by the formula:
a = ra / (1 + e)
where a is the semi-major axis, ra is the aphelion distance, and e is the eccentricity of the orbit. Rearranging the formula, we have:
ra = a * (1 + e)
This formula relates the eccentricity (e), the aphelion distance (ra), and the semi-major axis (a) of the elliptical orbit.
c) To determine the speed of the comet at its aphelion, you can use Kepler's second law, which states that equal areas are swept out in equal times. This means that the comet's speed will be fastest at its perihelion (closest point to the Sun) and slowest at its aphelion (farthest point from the Sun).
To find the speed at the aphelion, you need to know the semi-major axis (a) and the eccentricity (e). From the given information, you know the eccentricity is e=0.99920 and the aphelion distance is ra=2,799 UA. Use the formula from part b:
ra = a * (1 + e)
Rearranging the formula, we have:
a = ra / (1 + e)
Plug in the values:
a = 2,799 UA / (1 + 0.99920)
Solve for a to find the semi-major axis of the orbit. Note that 1 UA (astronomical unit) is the average distance between the Earth and the Sun, approximately 149.6 million km.
Once you have the semi-major axis, you can use Kepler's third law to find the period of the orbit (T). The formula for the period of an orbit is:
T = 2π * sqrt(a^3 / (G * M))
where π is approximately 3.14159, a is the semi-major axis, G is the gravitational constant (approximately 6.674 x 10^-11 m^3 kg^-1 s^-2), and M is the mass of the Sun (1.99 x 10^30 kg).
Now that you have the period (T) of the orbit, you can use it to find the speed at the aphelion. The speed at a given point in the orbit can be calculated using the formula:
v = (2π * a) / T
Substitute the values of a and T into the formula to find the speed of the comet at the aphelion. Note that the speed will be in units of distance per unit time (e.g., km/s).
Remember to convert units as necessary to ensure consistent measurements throughout the calculations.