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At time to a comet is observed at radius ro traveling with speed vo at an acute angle α to the line from the comet to the sun. Put the sun at the origin O, with the comet on the x axis (at to ) and its orbit in the xy plane, and then show how you could calculate the parameters of the orbital equation in the form r=c/[1+ϵcos(ϕδ)]. Do so for the case that r0=1.0×1011m,vo=45km/s, and α=50 degrees. [The sun's mass is about 2.0×1030kg, and G=6.7×1011Nm2/s2.]

Short Answer

Expert verified
Use given data to determine angular momentum, specific energy, eccentricity, semi-major axis, and fit these into the orbital equation form.

Step by step solution

01

Calculate Angular Momentum

To find the comet's angular momentum per unit mass, use the formula: h=r0v0sin(α)First, convert the speed to meters per second if it's given in other units: v0=45×1000=45000 m/sSubstitute the given values into the equation:h=(1.0×1011 m)×45000 m/s×sin(50)3.450×1015 m2/s
02

Calculate Specific Orbital Energy

The specific orbital energy ϵ is calculated using:ϵ=v022GMr0Substitute the known values into the equation:ϵ=(45000)22(6.7×1011)(2.0×1030)1.0×1011This simplifies to:ϵ=1.0125×1091.34×109=3.275×108 J/kg
03

Calculate Eccentricity

Use the relationship:ϵ=GMϵ2aSince specific energy ϵ<0, the orbit is an ellipse. Solve for ϵ:ϵ=1+2ϵh2GM2Substitute the known values:ϵ=1+2×3.275×108×(3.450×1015)26.7×1011×(2×1030)2Calculate ϵ, typically a value between 0 and 1 for an ellipse.
04

Calculate Semi-Major Axis

We can now find the semi-major axis a using the relationship:a=GM2ϵSubstitute the known values:a=6.7×1011×(2×1030)2×3.275×108Solve for a which will give the semi-major axis of the comet's orbit.
05

Calculate the Orbital Equation Parameters

Combining the previous results, write the orbital equation in:r=c1+ϵcos(ϕδ)Where c=a(1ϵ2). Using the calculated values of ϵ and a, find the parameter c. The parameter δ involves more detailed initial conditions analysis, often using angular positions. For simplicity, assume initial positioning angle leads to δ=0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Momentum
Angular momentum is a key concept when calculating a comet's orbit. It measures the rotational momentum of an object that is traveling in a circular or elliptical path around another object, such as the sun.
The formula we use is h=r0v0sin(α), where:
  • r0 is the distance from the comet to the sun at the observation time t0.
  • v0 is the speed of the comet.
  • α is the angle between the comet's direction and the line from the sun to the comet.
We represent angular momentum per unit mass, which helps in simplifying calculations related to gravitational forces.
When you multiply these values together, you get a measurement of the comet's momentum as it travels along its path.
This tells us how much the comet is changing its motion as it orbits the sun.
Specific Orbital Energy
Specific orbital energy is a measure of how much energy a comet has as it travels through space. This energy makes sure a comet stays in orbit rather than drifting off into space. It's calculated using the formula:ϵ=v022GMr0
  • v0 is the initial speed of the comet.
  • G is the universal gravitational constant, 6.7×1011 Nm2/kg2.
  • M is the mass of the sun.
  • r0 is the radius or the distance from the comet to the sun.
When you calculate specific orbital energy and get a negative value, like we have here, it means the comet is in a bound orbit, typically an ellipse.
This means the comet is always gravitationally tied to the sun and will continue to orbit it.
Eccentricity
Eccentricity is a measure of how much a comet's orbit deviates from a perfect circle. The formula for eccentricity is:ϵ=1+2ϵh2GM2
  • When eccentricity is exactly 0, the orbit is a perfect circle.
  • When it's between 0 and 1, the orbit is an ellipse.
  • A value of 1 or more suggests a parabolic or hyperbolic trajectory, not usually seen in bound orbits.
A comet with a high eccentricity means a long, stretched-out orbit. Conversely, low eccentricity implies a rounder shape.
To find the eccentricity of an orbit, we substitute the values from angular momentum and specific energy calculations.
Semi-Major Axis
The semi-major axis is an important measurement in determining the size of an elliptical orbit. It represents the average distance from the center of the ellipse to its perimeter over one complete orbit.
In the universe of orbits, the semi-major axis is crucial because:
  • It provides a measurement of the orbit's size.
  • It is directly linked to the orbital period through Kepler's laws.
To calculate the semi-major axis a, the formula is:a=GM2ϵA positive value indicates that the comet will continue to orbit in an elliptical path around the sun, maintaining a consistent orbital period.
Orbital Equation Parameters
To fully understand and describe a comet's orbit, we use an equation in the form:r=c1+ϵcos(ϕδ)
  • r is the radius or the distance from the comet to the focus of the orbit (e.g., the sun).
  • ϵ is the eccentricity, indicating how stretched out the orbit is.
  • c is a constant that integrates the semi-major axis and eccentricity, given by c=a(1ϵ2).
  • ϕ is the true anomaly, the angle from the closest point in orbit.
  • δ is a phase angle, which might often be set to zero if it simplifies the problem.
This equation helps to map the path a comet takes through space, showcasing the essential parameters that paint the full picture of its journey around the sun.

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Most popular questions from this chapter

What would become of the earth's orbit (which you may consider to be a circle) if half of the sun's mass were suddenly to disappear? Would the earth remain bound to the sun? [Hints: Consider what happens to the earth's KE and PE at the moment of the great disappearance. The virial theorem for the circular orbit (Problem 4.41) helps with this one.] Treat the sun (or what remains of it) as fixed.

Prove that for circular orbits around a given gravitational force center (such as the sun) the speed of the orbiting body is inversely proportional to the square root of the orbital radius.

Consider two particles of equal masses, m1=m2, attached to each other by a light straight spring (force constant k, natural length L ) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r1 and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r,ϕ) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X,Y. (c) Write down the Lagrange equations for r and A. Solve these for the two special cases that r remains constant and that ϕ remains constant. Describe (i) the contant the corresponding motions. In particular, show that the frequency of oscillations in the second case is ω=2k/m1.

Verify that the positions of two particles can be written in terms of the CM and relative positions as r1=R+m2r/M and r2=Rm1r/M. Hence confirm that the total KE of the two particles can be expressed as T=12MR˙2+12μr˙2, where μ denotes the reduced mass μ=m1m2/M.

Consider two particles of equal masses, m1=m2, attached to each other by a light straight spring (force constant k, natural length L ) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r1 and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r,ϕ) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X,Y. (c) Write down the Lagrange equations for r and A. Solve these for the two special cases that r remains constant and that ϕ remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is ω=2k/m1.

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