Conservation of energy and momentum in planetary motion

[FranzDiCoccio]

Homework Statement

The period of a comet is 75.8 years. The perihelion distance is 0.596 AU (1 AU = 1.5 ⋅ 10¹¹ m).
The velocity at perihelion is v_p = 5.45 ⋅10⁴ m/s.

a) Find the length of the major semi-axis of the elliptical orbit.
b) Find the aphelion distance and the velocity at aphelion for the comet.

Homework Equations

T² = k a³ (Kepler's third law)
m v r = const. (conservation of angular momentum at aphelion and perihelion)

$$1/2 m v^2 - \frac{G m M}{r} = const$$ (conservation of energy)

The Attempt at a Solution

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I used the 3rd Kepler's law to find the semi-major axis a. For the constant I used
$$k=\frac{4 \pi^2}{G M}$$
where M = 1.99 ⋅10³⁰ kg as provided by the book.

I find
a = 17.9 AU = 2.68⋅10¹² m
r_a = 35.2 AU = 5.28 ⋅10¹² m

Then, from $$m v_a r_a = m v_p r_p$$ I find v_a=923 m/s.

So far, I'm in complete agreement with the solution provided by the book.

Now, I think that the same result could be derived from the conservation of energy.

$$v_a = \sqrt{v_p^2+2GM \left(\frac{1}{r_a}-\frac{1}{r_p}\right)}$$

However, this formula gives a completely different result: 7.15 ⋅10³ m/s.
Much larger than the previous one.

For the life of me, I cannot see the mistake. Here we are talking about a small comet orbiting the Sun, so the latter should be basically at rest. Center of mass should not be relevant.
The other thing I can think of is the constant in Kepler's law. But that should be ok in this case (again, the comet has a very small mass)

Can anyone help? Could it perhaps be a matter of roundoff? Difference between very small numbers?

EDIT: corrected typo in initial formula for energy

 

[mfb]

There is a typo (?) in your initial equation for the energy, it should be r, not r².

The approach later looks fine. The calculation via energy is sensitive to rounding errors (you subtract two large quantities to get a small one), so check that you keep enough significant figures. 7 km/s is way too fast.

 

[FranzDiCoccio]

Yes, sure, thanks, I've corrected the typo.

So you think it could be a rounding error... It occurred to me at the end of the post. With those big and small numbers multiplied by each other.
I tried collecting all of the powers of ten and managing them algebraically. Not much luck with that.

I'll give it another try. Thanks a lot for your reply. Rounding error is less scary than not understanding why it won't work.
Interesting problem...

 

[TSny]

The orbit is completely determined from the perihelion distance and speed. So, the period is determined from $v_p$ and $r_p$. It could be that the period specified in the problem is inconsistent with the data given for $v_p$ and $r_p$. I haven't checked this, though.

 

[mfb]

Going from 54.5 km/s to 54.4 km/s or 54.6 km/s changes the result significantly. Yes, rounding issue, not necessarily in the actual calculations.

 

[FranzDiCoccio]

The orbit is completely determined from the perihelion distance and speed. So, the period is determined from vpv_p and rpr_p. It could be that the period specified in the problem is inconsistent with the data given for vpv_p and rpr_p. I haven't checked this, though.

Going from 54.5 km/s to 54.4 km/s or 54.6 km/s changes the result significantly. Yes, rounding issue, not necessarily in the actual calculations.

Thanks a lot to both of you for your replies.
I'm not sure I know how to check whether the $v_p$ and $r_p$ are consistent with the period.
I can't simply equate the centripetal force and the gravitational force at perihelion, as if the orbit was circular, can I?

I'd like to go to the bottom of this. I'm not sure whether
1) using the conservation of energy is correct in principle but impractical because it would need data with a lot of significant digits.
2) the conservation of energy doesd not work in this specific case due to inconsistent data, but it would for consistent data (I mean, even with a reasonable number of significant digits).

Thanks again for your insight.
Francesco

 

[vela]

You can use the Vis-viva equation
$$v =\sqrt{GM\left(\frac 2r - \frac 1a\right)}$$ to solve for $v_p$, $r_p$, or $T$ after choosing the other two parameters.

I found the parameters you were given were inconsistent, but even with more accurate parameters, the energy calculation is very sensitive to the number of digits you use because the orbit is very eccentric.

 

[TSny]

Just to add to what vela and mfb have said regarding the sensitivity of the data in this problem,

If I take the given value for the period and perihelion distance, I find that the perihelion speed should be 54.1 km/s to 3 significant figures. Although this is fairly close to the given value of 54.5 km/s, this makes a significant difference in the aphelion distance and speed. I find the escape speed at the given perihelion distance to be about 54.6 km/s. So changing v_p from 54.1 to 54.6 km/s makes an "infinite" change in the aphelion distance!

 

[FranzDiCoccio]

Following vela's suggestion, I used the vis-viva equation by choosing $T=75.8\, \rm yrs$ and $r_p = 8.94\cdot 10^{10} \, \rm m$.
I found $a = 2.68 \cdot 10^{12} \,\rm m$ and $v_p = 54.0 \,{\rm km/s}$.

Repeating the original exercise with the original set of (corrected) data, I find $v_a = 917 \,{\rm m/s}$.
The last value is the same irrespective of the equation I use (conservation of a.m., vis-viva equation again or conservation of energy, which is basically the same as v.-v.).
Also, the energy at aphelion and perihelion are virtually the same.

Thus, as far as the exercise is concerned, if the data are compatible the result won't depend on the equation used to derive $v_a$.
I guess that if the data are compatible, the above discussed sensitivity to the velocity should not matter, right?
All in all it seems to me that it was a matter of sloppiness in the preparation of the exercise.

Thanks to all of you!
Francesco

 

[mfb]

I guess that if the data are compatible, the above discussed sensitivity to the velocity should not matter, right?

You saw how much the difference between 54.1 and 54.4 km/s made. Between 54.050 and 54.149, which both get rounded to 51.4, the difference is still notable.

 

[FranzDiCoccio]

Yes, I was too hurried.
The problem is that everything "worked" because I used all of the decimal digits given by the calculation with the vis-viva equation.

As soon as I truncate $v_p$ to fewer significant digits, the result from the conservation of energy starts to drift away from that of the conservation of angular momentum.
So I guess that in this situation it is preferable to use the conservation of angular momentum, which does not involve those products of huge and small numbers.
I mean, if $v_p$ is truncated, the conservation of a.m. still gives a reliable result for $v_a$, right?
It compares reasonably with the result of the vis-viva equation, which does not contain the truncated datum.

Thanks for your patience
Francesco

 

[mfb]

Right.