• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Stability of a circular orbit

15
0
1. Homework Statement
In a classical model of a multi-electron atom, electrons are assumed to move in a modified electrostatic potential $V(r)$, given by;

$$V(r)=\dfrac{-k}{r}e^{-r/a}$$

Show that the effective potential is ;

$$V_e(r)=\dfrac{J^2}{2mr^2}+\dfrac{-k}{r}e^{-r/a}$$

Then show that the circular orbit is unstable unless;

$$ 0.5* (1+\sqrt{5}) \textgreater \dfrac{r}{a} $$

2. Homework Equations

Take the derivative of the effective potential and using the fact that it is zero at the radius of the circular orbit, express the constant k.

Then take the second derivative and because the orbit is stable the stationary point, at the radius of the circular orbit, must be a minima, hence the second derivative evaluated at the point is greater than 0 for orbit to be stable.

You insert the k from line 1 in order to simplify the equation and some terms cancel.

3. The Attempt at a Solution
I have tried to solve the problem multiple times and obtained;

$$k=e^{r/a}*\dfrac{J^2}{mr^3}(\dfrac{1}{r^2}+\dfrac{1}{a})^{-1}$$

this lead me to the inequality;

$$ \sqrt{1+\sqrt{2}} \textgreater \dfrac{r}{a}$$


Could anyone tell me whether my approach is correct ??

Thank you
 

DEvens

Education Advisor
Gold Member
1,008
302
What did you do to get that inequality?

What you were supposed to do, as the statement of the problem said, was use the expression for k to remove k from the potential. Then take the second derivative of that expression, now with no k in it. Then determine what makes that second derivative greater than 0.

What did you do?
 
15
0
That is exactly what I have done; I am attaching a figure with my working;
ieic0l.jpg
.
 

DEvens

Education Advisor
Gold Member
1,008
302
Check your algebra when you take the first derivative. In particular, check what the derivative w.r.t. r of exp(-r/a) is.
 
15
0
Check your algebra when you take the first derivative. In particular, check what the derivative w.r.t. r of exp(-r/a) is.
Thank you for pointing this out, I cannot believe I made such a basic mistake.
 

Want to reply to this thread?

"Stability of a circular orbit" You must log in or register to reply here.

Related Threads for: Stability of a circular orbit

  • Posted
2
Replies
30
Views
2K
  • Posted
Replies
1
Views
4K
Replies
1
Views
288
  • Posted
Replies
0
Views
967
  • Posted
Replies
1
Views
4K
  • Posted
Replies
4
Views
3K
  • Posted
Replies
1
Views
985
  • Posted
Replies
1
Views
864

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top