# Orbital velocity of a planet - why is my calculation off by about 10%?

I am not sure if I am doing something wrong, or misunderstanding Reider and Kenworthy (2016).

I'm just trying to reproduce the orbital velocities listed in Table 1. The second paragraph of Section II list a mass of the primary and semi-major axis for the planet's orbit of 0.9 solar mass and 5.0 AU. From the table the mass of the planet ranges from 20 to 100 Jupiters, which is actually quite sizable, but I'll start without using the reduced mass.

The numerical values I'm using:

$$GM_{\odot}=\text{1.327E+20} \ \mathrm{m^3 kg^{-2}}$$ $$GM=0.9GM_{\odot}$$ $$\epsilon=0.65$$ $$1 \ \mathrm{AU} = \text{1.496E+11} \ \mathrm{m}$$ $$a=5.0 \ \mathrm{AU} \ = \text{7.480E+11} \ \mathrm{m}$$

The formulae I'm using:

$$r_{\text{peri}}=a(1-\epsilon)$$

$$v^2=GM(2/r-1/a)$$

$$v_{\text{peri}}=\sqrt{GM(2/r_{\text{peri}}-1/a)}$$

I get:

$$r_{\text{peri}}=\text{2.618E+11} \ \mathrm{m}$$

$$v_{\text{peri}}=\text{2.744E+4} \ \mathrm{m/s}$$

which is $27.44 \ \mathrm{km/s}$. But for $\epsilon=0.65$ the table below shows $29.5 \pm 0.4 \ \mathrm{km/s}$. Close but not really close enough, it's off by almost 10%.

If the mass of the planet (which is quite large) was considered, then the table would have to list a wider range of velocities, wouldn't it?

• Plugging in the other eccentricities calculates values 2-2.5 km/s less than the velocities given in the table. – HDE 226868 Oct 16 '16 at 17:59
• @HDE226868 right, thanks! I didn't see the need to add even more numbers to my question. I have a hunch whatever explain the disagreement in the middle number will apply to all of them. – uhoh Oct 16 '16 at 18:04
• @siddigan thanks for the edit suggestion, but it looks like the definition of the orbital-mechanics tag specifies spacecraft. This is such a simple two-body question I think the orbital-elements is enough. – uhoh Oct 18 '16 at 15:36

## 1 Answer

Well done you. I double checked the calculations and couldn't fault what you had done. So I contacted the lead author of the paper about it and here is the response:

"After checking the numbers in our paper, I found an error: we actually used a mass of 1.0 MSun for J1407 in our simulations, instead of the 0.9 MSun as stated. This accounts for the difference in pericentric velocities (as well as the different semi-major axes, which would be smaller in the 0.9MSun case). We will attempt to correct this in the published version, and send a correction to arXiv."

• Thanks for your help tracking this down! I think the results of this work are really exciting. I first read about them in this NPR news item Spin To Survive: How 'Saturn On Steroids' Keeps From Self-Destructing and the video of the simulation there in particular captured my interest. It is also shown here: vimeo.com/184968413 and the underlying context is shown beautifully here: vimeo.com/117757625 It's a great demo of retrograde orbit stability. – uhoh Oct 22 '16 at 0:43
• @uhoh You have genuinely made a contribution. Kudos to you. – ProfRob Oct 22 '16 at 0:43
• This is why my profile says "stackexchange rocks!" – uhoh Oct 22 '16 at 0:44