What am I doing wrong? An easy gravity problem

Question

The following problem is giving me a headache:

Halley's comet follows an elliptical orbit around the sun. At perihelion, its distance from the sun ($r_P$) is $8.823 \cdot 10^{10}$ metres. At aphelion, its distance from the sun ($r_A$) is $6.152 \cdot 10^{12}$ metres. Its perihelion speed ($v_P$) is $5.46 \cdot 10^4$ metres per second. Compute its aphelion speed ($v_A$).

Using two methods:

1. Conservation of angular momentum:

$m v_A r_A = m v_P r_P$

$\displaystyle{v_A = \frac{r_P}{r_A} \, v_P \approx 783 \text{ m s}^{-1}}$

2. Conservation of energy:

$\displaystyle{\frac{1}{2} m {v_A}^2 - \frac{G m M_\odot}{r_A} = \frac{1}{2} m {v_P}^2 - \frac{G m M_\odot}{r_P}}$

$\displaystyle{v_A = \sqrt{ {v_P}^2 + 2 G M_\odot \!\left( \frac{1}{r_A} - \frac{1}{r_P} \right)} \approx 3939 \text{ m s}^{-1}}$

Question:

While conservation of angular momentum does give me the correct answer, conservation of mechanical energy doesn’t. Why is this so?

Answer 1

The formulas you derived from conservation of angular momentum and from conservation of energy both look correct. And of course both formulas should give the same result for $v_A$.

From your second formula for $v_A$, the given numbers, and the Standard gravitational parameter of the sun $$G M_\odot = 1.327\cdot 10^{20} \text{ m}^3/\text{s}^2$$ I get $$\begin{align}v_A &=\sqrt{ v_P^2 + 2GM_\odot \left(\frac{1}{r_A}-\frac{1}{r_P} \right)} \\ &=\sqrt{ (5.46\cdot 10^4\text{ m/s})^2 +2\cdot 1.327\cdot 10^{20} \text{ m}^3/\text{s}^2 \left( \frac{1}{6.152\cdot 10^{12}\text{ m}} -\frac{1}{8.823\cdot 10^{10}\text{ m}} \right)} \\ &= \sqrt{2.981\cdot 10^9\text{ m}^2\text{/s}^2 - 2.964\cdot 10^9\text{ m}^2\text{/s}^2} \\ &= 4 \cdot 10^3 \text{ m/s} \end{align}$$ Because of the small difference of two big numbers, the accuracy of this result is really bad.

Bear in mind, most comets have an eccentricity very close to $1$ (especially Halley's comet has $\epsilon=0.967$). Hence in the formula above the terms $v_P^2$ and $-2GM_\odot\frac{1}{r_P}$ nearly cancel. Therefore it would have been important to use very accurate values for $v_P$ and $r_P$, much more accurate than the given numbers we had.

Answer 2

As Thomas mentions, the data in the question aren't consistent with the standard gravitational parameter of the Sun, which is $\mu_\odot\approx 1.3271244×10^{11}\,\rm{km^3/s^2}$.

Using the given data and the vis-viva equation,
$$v^2=\mu\left(\frac2r-\frac1a\right)$$ (which can be derived from the orbital energy conservation equation), we get $\mu\approx 1.334×10^{11}\,\rm{km^3/s^2}$, so using that value for $\mu$ will give you $v_a=783\,\rm{m/s}$.

As nasu mentions, Halley's comet is probably not a great example for this exercise. No real orbit behaves like a nice Newtonian / Keplerian 2 body system, but comets are especially problematic because they have high eccentricity, so there's considerable variation between the radius & velocity at perihelion vs aphelion, and a comet is likely to be perturbed by planets as it traverses the inner Solar System. Also, a comet loses mass when it's near perihelion.

---

If you're curious, we can get better data for Halley's comet from NASA, in particular JPL and the Solar System Dynamics group. Here's the SSD database entry for 1P/Halley, which gives the osculating elements, a kind of Keplerian snapshot of the orbit at a particular epoch.

Since the 1960s, JPL have been building a good mathematical model of the Solar System, which they publish as the Jet Propulsion Laboratory Development Ephemeris. It's produced using heavy-duty integration of the equations of motion (with relativistic corrections), fitted to ground and space-based observations. It's used for astronomy and spacecraft navigation. You can access it via the Horizons system.

Horizons has 30 files for Halley's comet, spanning over 2 millennia. We can get reasonable values for the perihelion & aphelion radii and velocities from the most recent file. The last perihelion was at 1986-Feb-9 11:00 TDB, and the next aphelion occurs this year at 2023-Dec-8 20:12 TDB. (TDB is Barycentric Dynamical Time).

Here are some graphs (created using Horizons) showing those events.

Here's the Sage / Python script I used to create those plots. The script can be used to create plots for other bodies.

Horizons doesn't give orbital speed, but it can give velocity vectors, so we need to use Pythagoras' theorem to calculate the speed. We can calculate the specific angular momentum from the cross product, $$\vec h = \vec r \times \vec v$$ In an ideal Kepler orbit, $\vec h$ is constant, but in the real world it varies a little bit.

Here's some data from those perihelion and aphelion events.

Perihelion

*******************************************************************************
Ephemeris / WWW_USER Tue May 23 08:15:18 2023 Pasadena, USA      / Horizons
*******************************************************************************
Target body name: 1P/Halley                       {source: JPL#73}
Center body name: Sun (10)                        {source: DE441}
Center-site name: BODY CENTER
*******************************************************************************
Start time      : A.D. 1986-Feb-09 10:55:00.0000 TDB
Stop  time      : A.D. 1986-Feb-09 11:05:00.0000 TDB
Step-size       : 1 minutes
*******************************************************************************
Reference Frame : 

  Ecliptic at the standard reference epoch

    Reference epoch: J2000.0
    X-Y plane: adopted Earth orbital plane at the reference epoch
               Note: IAU76 obliquity of 84381.448 arcseconds wrt ICRF X-Y plane
    X-axis   : ICRF
    Z-axis   : perpendicular to the X-Y plane in the directional (+ or -) sense
               of Earth's north pole at the reference epoch.

  
Output units: KM-S
Data for 1P/Halley relative to Sun (10)

 0  A.D. 1986-Feb-09 10:55:00.0000
Julian day: 2446470.954861111 Delta-T 55.184998
pos: (49541300.24923867, -68128722.2885063, 24862524.70850744) 
       87829427.67621097
range: 87829427.67621097
vel: (-42.71461856061069, -33.31270436334545, -6.191317447385252) 
 54.521622522742476 
 54.52162252274247
range rate: -0.005892579753330823
ang momentum: (1250044482.3137734, -755267342.7783911, -4560447074.553943) 
 4788602874.183476

 1  A.D. 1986-Feb-09 10:56:00.0000
Julian day: 2446470.955555555 Delta-T 55.184998
pos: (49538737.35465636, -68130721.02674797, 24862153.22069415) 
       87829427.35261033
range: 87829427.35261033
vel: (-42.71520078959276, -33.31190364611326, -6.19160965370952) 
 54.52162262008595 
 54.52162262008594
range rate: -0.004894107704062249
ang momentum: (1250044482.546071, -755267342.4459002, -4560447074.105693) 
 4788602873.764783

 2  A.D. 1986-Feb-09 10:57:00.0000
Julian day: 2446470.95625 Delta-T 55.184998
pos: (49536174.42514459, -68132719.7169433, 24861781.71534909) 
       87829427.08891803
range: 87829427.08891805
vel: (-42.71578298845905, -33.31110290538346, -6.191901855669553) 
 54.521622698525924 
 54.52162269852591
range rate: -0.003895635643391062
ang momentum: (1250044482.778329, -755267342.1134675, -4560447073.65747) 
 4788602873.346114

 3  A.D. 1986-Feb-09 10:58:00.0000
Julian day: 2446470.956944444 Delta-T 55.184998
pos: (49533611.46069843, -68134718.35909608, 24861410.19247157) 
       87829426.88513406
range: 87829426.88513406
vel: (-42.71636515720947, -33.31030214115619, -6.192194053265318) 
 54.5216227580624 
 54.52162275806241
range rate: -0.002897163571420877
ang momentum: (1250044483.01055, -755267341.7810934, -4560447073.209274) 
 4788602872.927471

 4  A.D. 1986-Feb-09 10:59:00.0000
Julian day: 2446470.957638889 Delta-T 55.184998
pos: (49531048.4613264, -68136716.95319971, 24861038.6520628) 
       87829426.74125841
range: 87829426.74125841
vel: (-42.7169472958409, -33.30950135343573, -6.192486246495237) 
 54.52162279869537 
 54.52162279869536
range rate: -0.001898691493514773
ang momentum: (1250044483.2427318, -755267341.4487774, -4560447072.761104) 
 4788602872.50885

 5  A.D. 1986-Feb-09 11:00:00.0000
Julian day: 2446470.958333333 Delta-T 55.184998
pos: (49528485.4270236, -68138715.49925798, 24860667.09412209) 
       87829426.65729108
range: 87829426.65729108
vel: (-42.71752940435326, -33.30870054222218, -6.192778435359278) 
 54.52162282042482 
 54.52162282042482
range rate: -0.0009002194098245485
ang momentum: (1250044483.4748755, -755267341.1165197, -4560447072.312958) 
 4788602872.090253

 6  A.D. 1986-Feb-09 11:01:00.0000
Julian day: 2446470.959027778 Delta-T 55.184998
pos: (49525922.35779855, -68140713.99726425, 24860295.51865065) 
       87829426.63323207
range: 87829426.63323209
vel: (-42.71811148274342, -33.30789970751986, -6.193070619855874) 
 54.52162282325076 
 54.52162282325075
range rate: 9.825267429316444e-05
ang momentum: (1250044483.7069805, -755267340.7843198, -4560447071.864839) 
 4788602871.671679

 7  A.D. 1986-Feb-09 11:02:00.0000
Julian day: 2446470.959722222 Delta-T 55.184998
pos: (49523359.25364635, -68142712.44722235, 24859923.9256478) 
       87829426.6690814
range: 87829426.6690814
vel: (-42.71869353101131, -33.30709884932887, -6.19336279998498) 
 54.52162280717321 
 54.52162280717322
range rate: 0.001096724758722537
ang momentum: (1250044483.9390473, -755267340.4521787, -4560447071.416746) 
 4788602871.253131

 8  A.D. 1986-Feb-09 11:03:00.0000
Julian day: 2446470.960416667 Delta-T 55.184998
pos: (49520796.11457215, -68144710.84912825, 24859552.31511426) 
       87829426.76483904
range: 87829426.76483904
vel: (-42.71927554915457, -33.30629796765245, -6.193654975745417) 
 54.52162277219214 
 54.52162277219214
range rate: 0.002095196839384477
ang momentum: (1250044484.1710756, -755267340.1200953, -4560447070.9686775) 
 4788602870.834603

 9  A.D. 1986-Feb-09 11:04:00.0000
Julian day: 2446470.961111111 Delta-T 55.184998
pos: (49518232.94058114, -68146709.20297794, 24859180.68705081) 
       87829426.92050503
range: 87829426.92050503
vel: (-42.71985753717082, -33.30549706249387, -6.19394714713599) 
 54.521622718307576 
 54.521622718307576
range rate: 0.003093668912241966
ang momentum: (1250044484.4030662, -755267339.7880707, -4560447070.5206375) 
 4788602870.416103

 10  A.D. 1986-Feb-09 11:05:00.0000
Julian day: 2446470.961805556 Delta-T 55.184998
pos: (49515669.73166837, -68148707.50877522, 24858809.04145671) 
       87829427.13607931
range: 87829427.13607931
vel: (-42.72043949505998, -33.30469613385323, -6.194239314156668) 
 54.52162264551948 
 54.52162264551948
range rate: 0.004092140977163232
ang momentum: (1250044484.635018, -755267339.4561038, -4560447070.07262) 
 4788602869.997623

Aphelion

*******************************************************************************
Ephemeris / WWW_USER Tue May 23 08:20:47 2023 Pasadena, USA      / Horizons
*******************************************************************************
Target body name: 1P/Halley                       {source: JPL#73}
Center body name: Sun (10)                        {source: DE441}
Center-site name: BODY CENTER
*******************************************************************************
Start time      : A.D. 2023-Dec-08 20:07:00.0000 TDB
Stop  time      : A.D. 2023-Dec-08 20:17:00.0000 TDB
Step-size       : 1 minutes
*******************************************************************************
Reference Frame : 

  Ecliptic at the standard reference epoch

    Reference epoch: J2000.0
    X-Y plane: adopted Earth orbital plane at the reference epoch
               Note: IAU76 obliquity of 84381.448 arcseconds wrt ICRF X-Y plane
    X-axis   : ICRF
    Z-axis   : perpendicular to the X-Y plane in the directional (+ or -) sense
               of Earth's north pole at the reference epoch.

  
Output units: KM-S
Data for 1P/Halley relative to Sun (10)

 0  A.D. 2023-Dec-08 20:07:00.0000
Julian day: 2460287.338194444 Delta-T 69.183245
pos: (-2969083297.868456, 4075236407.241996, -1488997611.068494) 
       5257387306.482162
range: 5257387306.482162
vel: (0.7117114059901035, 0.5576172685416729, 0.1069742795213711) 
 0.9104469452260261 
 0.9104469452260258
range rate: 1.388329725401449e-06
ang momentum: (1266236259.2930634, -742121036.6610487, -4556004351.77037) 
 4786572213.051204

 1  A.D. 2023-Dec-08 20:08:00.0000
Julian day: 2460287.338888889 Delta-T 69.183245
pos: (-2969083255.165767, 4075236440.699026, -1488997604.650035) 
       5257387306.482237
range: 5257387306.482238
vel: (0.7117115586505691, 0.5576170339915946, 0.1069743614476542) 
 0.910446930535514 
 0.910446930535514
range rate: 1.106562034845343e-06
ang momentum: (1266236243.9175308, -742121020.7261598, -4556004277.4991455) 
 4786572135.8195915

 2  A.D. 2023-Dec-08 20:09:00.0000
Julian day: 2460287.339583333 Delta-T 69.183245
pos: (-2969083212.463069, 4075236474.156041, -1488997598.23157) 
       5257387306.482295
range: 5257387306.482295
vel: (0.7117117113110666, 0.5576167994414974, 0.1069744433739345) 
 0.9104469158451081 
 0.9104469158451081
range rate: 8.247943120062491e-07
ang momentum: (1266236228.5419626, -742121004.7913287, -4556004203.228009) 
 4786572058.588061

 3  A.D. 2023-Dec-08 20:10:00.0000
Julian day: 2460287.340277778 Delta-T 69.183245
pos: (-2969083169.760362, 4075236507.613041, -1488997591.813101) 
       5257387306.482336
range: 5257387306.482336
vel: (0.7117118639715968, 0.5576165648913801, 0.106974525300212) 
 0.9104469011548082 
 0.9104469011548084
range rate: 5.430265553196157e-07
ang momentum: (1266236213.1663575, -742120988.8565575, -4556004128.956962) 
 4786571981.356616

 4  A.D. 2023-Dec-08 20:11:00.0000
Julian day: 2460287.340972222 Delta-T 69.183245
pos: (-2969083127.057646, 4075236541.070029, -1488997585.394628) 
       5257387306.482361
range: 5257387306.482361
vel: (0.7117120166321583, 0.5576163303412452, 0.1069746072264861) 
 0.9104468864646149 
 0.9104468864646149
range rate: 2.612587680165742e-07
ang momentum: (1266236197.7907176, -742120972.9218463, -4556004054.686006) 
 4786571904.125257

 5  A.D. 2023-Dec-08 20:12:00.0000
Julian day: 2460287.341666667 Delta-T 69.183245
pos: (-2969083084.35492, 4075236574.527001, -1488997578.976148) 
       5257387306.482367
range: 5257387306.482368
vel: (0.7117121692927525, 0.5576160957910903, 0.1069746891527576) 
 0.9104468717745275 
 0.9104468717745273
range rate: -2.050905311700638e-08
ang momentum: (1266236182.4150398, -742120956.9871924, -4556003980.415136) 
 4786571826.893979

 6  A.D. 2023-Dec-08 20:13:00.0000
Julian day: 2460287.342361111 Delta-T 69.183246
pos: (-2969083041.652185, 4075236607.983961, -1488997572.557665) 
       5257387306.482359
range: 5257387306.482358
vel: (0.7117123219533779, 0.5576158612409178, 0.1069747710790256) 
 0.9104468570845468 
 0.9104468570845469
range rate: -3.022769049520304e-07
ang momentum: (1266236167.0393274, -742120941.0525994, -4556003906.14436) 
 4786571749.66279

 7  A.D. 2023-Dec-08 20:14:00.0000
Julian day: 2460287.343055556 Delta-T 69.183246
pos: (-2969082998.949441, 4075236641.440905, -1488997566.139176) 
       5257387306.482331
range: 5257387306.482331
vel: (0.7117124746140352, 0.5576156266907264, 0.1069748530052906) 
 0.910446842394672 
 0.9104468423946722
range rate: -5.840447890643871e-07
ang momentum: (1266236151.6635778, -742120925.1180646, -4556003831.873671) 
 4786571672.4316845

 8  A.D. 2023-Dec-08 20:15:00.0000
Julian day: 2460287.34375 Delta-T 69.183246
pos: (-2969082956.246688, 4075236674.897836, -1488997559.720683) 
       5257387306.482288
range: 5257387306.482288
vel: (0.7117126272747253, 0.5576153921405148, 0.106974934931553) 
 0.9104468277049035 
 0.9104468277049036
range rate: -8.65812707041303e-07
ang momentum: (1266236136.2877927, -742120909.1835896, -4556003757.60307) 
 4786571595.200663

 9  A.D. 2023-Dec-08 20:16:00.0000
Julian day: 2460287.344444444 Delta-T 69.183246
pos: (-2969082913.543926, 4075236708.354752, -1488997553.302184) 
       5257387306.482227
range: 5257387306.482228
vel: (0.7117127799354466, 0.5576151575902857, 0.1069750168578118) 
 0.9104468130152412 
 0.910446813015241
range rate: -1.14758065571967e-06
ang momentum: (1266236120.9119701, -742120893.2491733, -4556003683.33256) 
 4786571517.969725

 10  A.D. 2023-Dec-08 20:17:00.0000
Julian day: 2460287.345138889 Delta-T 69.183246
pos: (-2969082870.841155, 4075236741.811655, -1488997546.883681) 
       5257387306.482151
range: 5257387306.48215
vel: (0.7117129325961998, 0.5576149230400378, 0.1069750987840677) 
 0.9104467983256854 
 0.9104467983256855
range rate: -1.429348636601686e-06
ang momentum: (1266236105.5361128, -742120877.3148162, -4556003609.06214) 
 4786571440.738874

And here's the script that generated that data.

As you can see, the (specific) angular momentum at perihelion and aphelion are fairly similar, but they do vary at the 4th digit.

Answer 3

The problem is not with the input data but with the choice of the system to apply the conservation laws. I suppose that the problem was supposed to be an example of using conservation of angular momentum or/and energy. However, these laws apply to a a system isolated from external foces which the Sun-comet is not. NASA provides the data for the aphelion in 1948 and the following perihelion in 1986. So the closest two extreme points. https://solarsystem.nasa.gov/asteroids-comets-and-meteors/comets/1p-halley/in-depth/#:~:text=Halley%20was%20last%20seen%20in,year%20journey%20around%20the%20Sun. With these values, which are actually pretty close to the ones in the OP problem, I get an energy decreases by about 6.5%. The angular momenta at the two positions are much closer. I get a difference of about 0.1%, which may be just error in the measured values.

In conclusion, I think the author of the problem should have picked a better system for the student to practice conservation laws. Or maybe he did this on purpose, to make the student realize that the energy is not always conserved. There are some conditions for it to hold.