Methods to calculate speed and distance of objects in orbit

Question

I know that you can use Kepler's second law to calculate the speed an object travels at perihelion but arethere any other methods that I can use to calculate this? For example if I'm only given the objects orbital period (e.g. 15 days), its aphelion distance (e.g. 700000km), its mass (e.g. 3x earth mass), its radius (e.g. 2x earth radius), and its time interval at perihelion (e.g. 4 hours), how can I calculate calculate its speed at perihelion without Kepler's second law?

In addition to this, how can I calculate the distance the object travels with this information if it travels at constant speed during that time period? Do I need Kepler's law?

Answer 1

You can calculate the speed of an object in orbit using basic results from Newtonian orbital mechanics. A very useful relation is the vis-viva equation, which gives the orbital speed at any distance from the central body:

v² = GM (2/r − 1/a)

Here v is the orbital speed, G is the gravitational constant, M is the mass of the central body, r is the current distance from the central body, and a is the semi-major axis of the orbit.

For an elliptical orbit, the semi-major axis is simply the average of the perihelion and aphelion distances:

a = (rp + ra) / 2

where rp is the perihelion distance and ra is the aphelion distance.

To find the speed at perihelion, set r = rp in the vis-viva equation. This gives the perihelion speed:

vp = sqrt[ GM (2/rp − 1/a) ]

If you know the orbital period T but do not know the central mass, you can use Newton’s form of Kepler’s third law to find GM:

GM = (4π² a³) / T²

Substituting this into the vis-viva equation allows you to compute the perihelion speed using only the orbital period and the orbital distances.

For the distance traveled during a short time interval near perihelion, you can approximate it using simple kinematics if the speed does not change much over that short time. The distance is approximately

d ≈ vp × delta_t

where vp is the perihelion speed and Δt is the time spent near perihelion. In reality the speed changes continuously along the orbit, but for a short interval this constant-speed approximation is usually reasonable.

Answer 2

Hi! There are actually two other methods you could use to do this. I have listed them for you below, tell me if you need further help or tutoring.

method 1: How to Calculate Speed at Perihelion

You can calculate the speed at perihelion using the vis-viva equation:

v = √[GM * (2/r - 1/a)]

Where:

1. v = speed at distance r (perihelion distance)
2. G = gravitational constant
3. M = mass of the central body
4. r = distance from the central body at perihelion
5. a = semi-major axis of the orbit

Step 1: Estimate the Semi-Major Axis (a)

Use the orbital period and Kepler’s 3rd Law:

T² = (4π² * a³) / (G * M)

⇒ a³ = (G * M * T²) / (4π²)

Solve for a, then plug it into the vis-viva equation above.

method 2: Use Angular Momentum Conservation

m * v_p * r_p = m * v_a * r_a

⇒ v_p = v_a * (r_a / r_p)

however, this requires knowing both aphelion (r_a) and perihelion (r_p) distances.

Part 2: How to Calculate Distance Traveled at Constant Speed

Once you know the speed at perihelion v_p, and the time spent near perihelion t, just use basic kinematics:

distance = speed x time

Have a good one!