Question

Problem A.1: Parabolic Trajectory (4 Points)

As presented in the qualification round, the comet P/2023 IAAC circles the Sun in an elliptical or-
bit. There are other comets with a parabolic trajectory, for example the comet C/2023 IAAC.
(a) Explain the meaning of the letters P and C in the names of the two comets.
The vis-viva-equation can be extended for different types of trajectories as follows:
for ellipses
for parabolas
for hyperbolas
v(x) =
1
√ « (² – ²-)
a
with
a> 0
a = ∞
a < 0
and μ = G(m₁ + m₂). Here, m₁ is the comet's mass, m₂ the Sun's mass (1.9 x 10³⁰ kg), x the dis-
tance between the comet and the Sun, and G is the gravitational constant (6.67 x 10-¹¹ m³kg-¹s²).
(b) Determine the velocity (in km/s) of C/2023 IAAC for a distance of 0.8 AU to the Sun.

Answer 1

Answer: The letters P and C in the names of the two comets indicate the type of orbit that they have around the Sun. P stands for periodic, which means that the comet has an elliptical orbit and returns to the inner solar system at predictable intervals. C stands for non-periodic, which means that the comet has a parabolic or hyperbolic orbit and is not expected to return to the inner solar system.

(b) To determine the velocity of comet c/2023 IAAC for a distance of 0.8 AU to the sun, we can use the Vis-Viva equation:

v(x) = õ(2/x - 1/a)

where µ = G(m1 + m2) and G is the gravitational constant.

For comet c/2023 IAAC, we do not know the mass of the comet, so we cannot calculate µ directly. However, we can assume that the comet has a negligible mass compared to the mass of the Sun, which is 1.9 x 10^30 kg. Therefore, we can simplify the equation as follows:

v(x) = √(Gm2)(2/x)

where m2 = 1.9 x 10^30 kg and x = 0.8 AU = 1.2 x 10^11 m.

Plugging in these values, we get:

v(1.2 x 10^11 m) = √[(6.67 x 10^-11 m3kg^-1s^-2)(1.9 x 10^30 kg)][2/(1.2 x 10^11 m)]

v(1.2 x 10^11 m) = 57.7 km/s (rounded to one decimal place).

Therefore, the velocity of comet c/2023 IAAC for a distance of 0.8 AU to the sun is approximately 57.7 km/s.

Step-by-step explanation: