Exploring Earth's Orbit with a Changing Sun

In summary, If the mass of the sun were 1/2 it's current mass suddenly show the orbit of the Earth would be a parabola. However, because the approximations used in this calculation are not exact, the Earth's average speed does exceed this value.
  • #1
AdkinsJr
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Homework Statement



If the mass of the sun were 1/2 it's current mass suddenly show the orbit of the Earth would be a parabola...

The Attempt at a Solution



I'm not sure what kind of approach to apply here, if the total energy equals zero then the orbit is a parabolic trajectory, if it is less than zero the the orbit is circular or elliptical, if I plug in numbers for the mass of the earth, sun, the radius and the average speed I can find escape velocity,

[tex]E=\frac{1}{2}mv^2 -\frac{GMm}{r}[/tex]

and solve for the velocity [tex]v=\sqrt\frac{GM}{r}[/tex]

and show that the Earth's average speed exceeds this value, but since these numbers are going to be approximations it's difficult to tell whether it should be parabola or bound orbit...E will not be exactly zero...
 
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  • #2
AdkinsJr said:
and show that the Earth's average speed exceeds this value, but since these numbers are going to be approximations it's difficult to tell whether it should be parabola or bound orbit...E will not be exactly zero...
You can do it analytically, assuming Earth has a perfectly circular orbit* you can express the radius as function of the velocity (or vice versa) and calculate the total energy afterwards.

*otherwise the whole statement is not true anyway
 
  • #3
That makes sense, by radius I presume you mean the orbital equations solns. Yeah I think that will simplify things down a bit. [tex]r(\theta)=\frac{\alpha}{1+e cos(\theta)}[/tex]...for a circle e=o so [tex]r_c=\frac{ml^2}{k}[/tex] where [tex]l=L/m[/tex] and [tex]k=GMm[/tex]

[tex]r_c=\frac{m(v_cr_c)^2}{GMm}[/tex] if I solve for [tex]V_c[/tex] I get [tex]v_c^2=\frac{GM}{r}[/tex]

If I plug this into energy equation it does actually work when I plug in 1/2 M for the mass of the sun in the potential energy term.
 

FAQ: Exploring Earth's Orbit with a Changing Sun

How does the changing sun affect Earth's orbit?

The sun's changes in energy output, known as solar variability, can affect Earth's orbit in a few different ways. The sun's radiation can cause Earth's atmosphere to expand, leading to increased drag on satellites and a decrease in their orbits. Additionally, changes in solar radiation can impact Earth's climate, altering the planet's rotation and potentially affecting its orbit. These changes are relatively small, but can have a cumulative effect over time.

How does Earth's orbit change over time?

Earth's orbit around the sun is not a perfect circle, but rather an ellipse. This means that the distance between Earth and the sun varies throughout the year. The eccentricity of Earth's orbit, or how elongated it is, also changes over time due to gravitational interactions with other planets. These changes can affect the length of our seasons and the amount of sunlight reaching different parts of Earth.

Can Earth's orbit be affected by other celestial bodies?

Yes, Earth's orbit can be influenced by other planets, as well as the moon and other objects in our solar system. The gravitational pull of these bodies can cause slight changes in the shape and orientation of Earth's orbit, leading to variations in our planet's climate and seasons over long periods of time.

How does Earth's orbit impact life on our planet?

Earth's orbit plays a crucial role in determining the conditions necessary for life on our planet. Its distance from the sun and the tilt of its axis affect the amount of solar radiation and seasonal changes, which impact the growth of plants and the survival of various species. Changes in Earth's orbit can also contribute to long-term climate shifts and potentially lead to mass extinction events.

Are there any upcoming changes in Earth's orbit that we should be aware of?

Currently, there are no major changes expected in Earth's orbit in the near future. However, scientists continue to monitor the sun's activity and its potential impact on our planet's orbit. Additionally, the gradual effects of gravitational interactions with other planets may lead to small variations in Earth's orbit over longer timescales, but these changes are unlikely to have a significant impact on life on our planet.

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