Suppose there's a collison way out in the Oort cloud, say 30,000 AU away from the Sun. This collison causes one of the comets to alter its orbit and head almost directly at the Sun, such that the comet just skims the surface of the Sun and heads back out to its starting point. Assuming it can survive the intense heat that close to the Sun, how long will it take the comet to complete one orbit?

This calls for an easy application of Kepler's third law. With the semimajor axis (a) in a.u. and the period (P) in years,

P^2 = a^3
In this case, a = 15,000 au is the average distance from the sun, so
P = 1.84*10^6 years

http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

To calculate the time it takes for the comet to complete one orbit, we need to know its orbital period. The orbital period of an object depends on its semi-major axis, which is the average distance between the object and its primary body, in this case, the Sun.

In this scenario, the comet's initial position is in the Oort cloud, approximately 30,000 astronomical units (AU) away from the Sun. Let's assume the comet's orbit changes but retains the same semi-major axis for simplicity.

The semi-major axis of the comet's orbit would still be 30,000 AU, but now with a highly eccentric orbit due to the close encounter with the Sun. Since the comet's orbit is highly elliptical, we cannot directly determine its orbital period using standard formulas.

However, we can approximate the time it takes for one orbit using specific assumptions. Let's assume the comet's speed remains roughly constant throughout its orbit, so it spends equal amounts of time in each portion of its elliptical orbit.

1. Perihelion: The point of closest approach to the Sun.
2. Aphelion: The point of greatest distance from the Sun.

If we assume the comet skims the Sun just slightly above its surface (perihelion), we can calculate the approximate time it takes for the comet to travel from perihelion to perihelion.

First, let's approximate the distance from the Sun's surface to the perihelion of the comet's orbit using our assumption. The Sun's radius is approximately 696,340 kilometers.

Let's consider the comet's orbit to be a perfect circle (though it is highly elliptical). So, the distance from the Sun's center to its surface, plus the distance from the surface to the perihelion, would be equal to 696,340 km (the Sun's radius).

Therefore, the distance from the Sun's surface to the perihelion would be approximately 2 × 696,340 km = 1,392,680 km.

Now, we need to calculate the time it takes for the comet to travel from the perihelion to the perihelion again. Since we don't know the comet's exact speed, we'll assume it remains constant.

Next, we can use Kepler's third law of planetary motion to approximate the orbital period. Kepler's third law states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (a) of the orbit.
Mathematically, it can be expressed as:

T^2 = k * a^3,

where k is a constant.

Since we assumed the semi-major axis to be 30,000 AU, we can rewrite the equation as:

T^2 = k * (30,000 AU)^3.

Now, let's convert the units to kilometers, as it would be more practical in this case. 1 AU is approximately 150 million kilometers.

T^2 = k * (30,000 AU * 150 million km/AU)^3.

Simplifying:

T^2 ≈ k * 33.75 * 10^18 km^3.

Now, let's consider the distance from perihelion to perihelion, which we approximated as 1,392,680 km (as mentioned above). Since the comet's speed is constant, it takes the same amount of time to travel between these two points. Therefore, we can use this distance as the approximate circumference of the orbit.

C = 2πr ≈ 2π * 1,392,680 km ≈ 8,747,549 km.

Now we can set up another equation using the distance and the time it takes to complete the orbit.

T = (C / v),

where T is the time, C is the circumference of the orbit, and v is the comet's constant velocity.

Substituting the values:

T ≈ (8,747,549 km / v).

Now we can equate the two expressions for T:

k * (33.75 * 10^18 km^3) ≈ (8,747,549 km / v)^2.

Simplifying:

kv^2 ≈ 33.75 * 10^18 km^3.

Since we don't have exact values for k or v, we cannot accurately determine the exact time it takes for the comet to complete one orbit. However, the provided calculations and assumptions allow us to approximate the time based on the given scenario.