Period and velocity at perihelion/aphelion

In summary, the conversation discusses finding the period of Halley's comet using Kepler's laws. The attempt at a solution involves solving for the semi-major axis and using t^2 = a^3, but it is incorrect to say they are equal. The correct approach involves finding the constant of proportionality using the Earth's period and distance from the Sun, and then relating the period of Halley's comet to its semi-major axis. To find the velocity at a certain point, one needs to understand the concept of mechanical energy and relate it to the orbit size and gravitational parameter of the primary. Another question is also discussed, which involves relating total energy to find the velocity at a certain point by using the conservation of angular momentum.
  • #1
Cocoleia
295
4

Homework Statement


I need to find first of all the period of Halley's comet given that:
eccentricity: 0.967
perihelion: 8.8x10^7 km
and AU = 1.496x10^8km

Homework Equations


Kepler's laws

The Attempt at a Solution


I think that the
Perihelion = a(1-e) where a is the semi major axis. So I solved for a and then used t^2 = a^3 for period
 
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  • #2
Cocoleia said:

Homework Statement


I need to find first of all the period of Halley's comet given that:
eccentricity: 0.967
perihelion: 8.8x10^7 km
and AU = 1.496x10^8km

Homework Equations


Kepler's laws

The Attempt at a Solution


I think that the
Perihelion = a(1-e) where a is the semi major axis.
Good.
So I solved for a and then used t^2 = a^3 for period
Not good. Time squared is not the same as a distance cubed unit-wise. What is true is that t^2 varies as (is proportional to) a^3. To make them equal you would need the constant of proportionality.

You'll either need to use the full Newton's version of Kepler's 3rd law which includes the gravitational parameter for the solar system (essentially GM for the Sun), or do something tricky using another well known orbit and period :wink:
 
  • #3
gneill said:
Good.

Not good. Time squared is not the same as a distance cubed unit-wise. What is true is that t^2 varies as (is proportional to) a^3. To make them equal you would need the constant of proportionality.

You'll either need to use the full Newton's version of Kepler's 3rd law which includes the gravitational parameter for the solar system (essentially GM for the Sun), or do something tricky using another well known orbit and period :wink:
I meant for it to be a capital T for period, sorry
 
  • #4
T squared is proportional to a cubed, but it incorrect to say they are equal. The two sides of the "equation" do not even have the same unit. You may be able to get the proportionality knowing the distance of the Earth to the Sun (1.a.u = 1.5 x 10 ^ 8 km), and the period of the orbit (1 year).
 
  • #5
mpresic said:
T squared is proportional to a cubed, but it incorrect to say they are equal. The two sides of the "equation" do not even have the same unit. You may be able to get the proportionality knowing the distance of the Earth to the Sun (1.a.u = 1.5 x 10 ^ 8 km), and the period of the orbit (1 year).
What do you mean by finding the proportionality?
 
  • #6
Cocoleia said:
What do you mean by finding the proportionality?
mpresic said:
T squared is proportional to a cubed, but it incorrect to say they are equal. The two sides of the "equation" do not even have the same unit. You may be able to get the proportionality knowing the distance of the Earth to the Sun (1.a.u = 1.5 x 10 ^ 8 km), and the period of the orbit (1 year).
Now, I figured it out. How would I find the velocity at this point?
 
  • #7
If you have the constant of proportionality (k) from the means already discussed, knowing the Earth's period (1 year) and the distance from the Sun, you can relate the period to the semimajor axis (a). T squared = k times semimajor axis cubed. You can relate the period of Halley's comet to the semimajor axis of Halley's comet, using the same equation. To get the semimajor axis, you know eccentricity and the perigee, you should be able to get apogee, and calculate the semimajor axis. You never need the velocity.
 
  • #8
If you want to find the velocity then you will need to know something about the mechanical energy of the orbit. That depends upon the size of the orbit and the gravitational parameter μ = GM of the primary (the Sun in this case).

You could determine μ from what you know about Earth's orbit (equate gravitational acceleration to centripetal acceleration for the Earth). Then you'd need to know how to relate the orbit size (semi-major axis) to the total mechanical energy of the orbit.

If this is a question that is part of the same assignment then you'll have to provide the relevant equations from your studies and show what you've already tried.
 
  • #9
gneill said:
If you want to find the velocity then you will need to know something about the mechanical energy of the orbit. That depends upon the size of the orbit and the gravitational parameter μ = GM of the primary (the Sun in this case).

You could determine μ from what you know about Earth's orbit (equate gravitational acceleration to centripetal acceleration for the Earth). Then you'd need to know how to relate the orbit size (semi-major axis) to the total mechanical energy of the orbit.

If this is a question that is part of the same assignment then you'll have to provide the relevant equations from your studies and show what you've already tried.
Ok, I think I understand. I am also working on a another question where I am given:
upload_2016-11-15_21-42-50.png

And asked to find the velocity at point P, which is the perigee. I assume A is the apogee. My question is, How can I relate total energy to find the velocity at this point?
 
  • #10
What is v? I'm guessing here, but is it the velocity of a circular orbit with radius R?
 
  • #11
gneill said:
What is v? I'm guessing here, but is it the velocity of a circular orbit with radius R?
Yes. At the point A, it begins to take the ellipse shaped path and decelerates to a velocity of v/2
 
  • #12
You should start a new thread for this question. It requires a different approach and concepts (much more Newton than Kepler :smile:).
 
  • #13
You can use the conservation of angular momentum. at apogee and perigee the angle between the momentum and the force is 90 degrees, so
M Va Ra = M (v/2) R = M Vp Rp. You also need to relate R = Ra to semimajor axis and the eccentricity.
 
  • #14
mpresic said:
You can use the conservation of angular momentum. at apogee and perigee the angle between the momentum and the force is 90 degrees, so
M Va Ra = M (v/2) R = M Vp Rp. You also need to relate R = Ra to semimajor axis and the eccentricity.
Neither eccentricity nor semi-major axis are given in this case (assuming that you are addressing the problem of post #9).

I've suggested to the OP that this problem should be relocated to a separate thread since it is a new, different question from the original problem that started this thread. We should await its appearance in a new thread and not discuss it further here.
 

1. What is perihelion and aphelion?

Perihelion and aphelion are two terms used to describe the closest and farthest points in the orbit of a celestial body around the sun. Perihelion is the point where the object is closest to the sun, while aphelion is the point where it is farthest from the sun.

2. How does the distance at perihelion and aphelion affect the velocity of a celestial body?

The distance at perihelion and aphelion has a direct impact on the velocity of a celestial body. According to Kepler's Second Law of Planetary Motion, a planet travels faster when it is closer to the sun (perihelion) and slower when it is farther away (aphelion).

3. What is the significance of calculating the period and velocity at perihelion and aphelion?

Calculating the period and velocity at perihelion and aphelion helps us understand the orbital dynamics of a celestial body. It allows us to determine the shape and size of the orbit, as well as the time it takes for the object to complete one orbit around the sun. This information is crucial in studying the behavior and movement of planets, comets, and other objects in our solar system.

4. How do scientists measure the period and velocity at perihelion and aphelion?

Scientists use various methods to measure the period and velocity at perihelion and aphelion depending on the object being studied. For planets, the orbital period can be calculated using Kepler's Third Law of Planetary Motion, which relates the orbital period to the distance from the sun. The velocity can then be determined by dividing the distance traveled by the period. For comets, scientists use telescopes and radar to track their movement and calculate their orbital period and velocity.

5. Can the period and velocity at perihelion and aphelion change over time?

Yes, the period and velocity at perihelion and aphelion can change over time due to various factors such as gravitational interactions with other celestial bodies or changes in the shape of the orbit. For example, comets can have highly elliptical orbits that can change over time as they get closer to the sun, affecting their period and velocity at perihelion and aphelion.

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