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I am just wondering at what speed does Tesla Roadster is currently travelling toward the Asteroid Belt (Relative to Earth)?

Russell Borogove
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Sudhanshu Gaur
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    News reports before the revelation that the roadster was going to go even further out (to the asteroid belt) said the speed would be 25,300 mph. – JPhi1618 Feb 07 '18 at 15:36
  • @JPhi1618 Thanks for the update but where it is written 25,300 ?? – Sudhanshu Gaur Feb 07 '18 at 15:43
  • The article I saw just now was this one, but it's not a great/primary source. I know I saw the 25,000 number other places, but I don't think that figure is valid anymore since the "change of plans". – JPhi1618 Feb 07 '18 at 15:45
  • @JPhi1618 That's why i asked right now at what speed it is travelling. Thanks Anyways :) – Sudhanshu Gaur Feb 07 '18 at 15:54
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    That's why I didn't post as an answer. At least you know that it is faster than 25kmph. We may never have an answer because SpaceX doesn't seem to share every parameter of their missions. – JPhi1618 Feb 07 '18 at 16:09
  • @JPhi1618 Sorry, I didn't mean that Bdw thanks for at least telling me :) – Sudhanshu Gaur Feb 07 '18 at 16:18
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    It's not traveling to the asteroid belt. The tweet was wrong (though had the right C3). – Mark Adler Feb 08 '18 at 06:13
  • @MarkAdler please elaborate! Assuming the launch was prograde, $\sqrt{12.0 km^2/s^2}$ plus the Earth's velocity at a periapsis of 0.988 AU would make the apoapsis about 1.67 AU, or just past Mars' orbit as originally suggested. So you believe the $C_3$ in Musk's tweet is right and the apoapsis is wrong? – uhoh Feb 08 '18 at 09:20
  • @MarkAdler oh, it's in Horizons. It looks nice, and indeed the apoapsis is 1.67 as calculated. Math is great! – uhoh Feb 08 '18 at 12:01
  • Musk's tweet showed an aphelion (misspelled "apohelion") of 2.61 AU, which is way wrong. The C3 shown in that same plot of 12.0 is about right. – Mark Adler Feb 08 '18 at 15:09
  • @MarkAdler Apohelion is not misspelled, just an antiquated way of spelling it, from the original Greek "apo", meaning "from", and "helion", the Sun. Unless Musk was thinking aphelion and apoapsis in his head and jumbled the two. I don't know how he normally spells ap-helion, so I can't say for sure. – TylerH Feb 08 '18 at 19:17
  • I'm pretty antiquated myself, and I'd never seen it spelled that way. But ok. – Mark Adler Feb 08 '18 at 21:32

3 Answers3

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I've now created a website that will populate the speed in near real time, http://www.whereisroadster.com/. As of writing this, it states:

The current location is 332874 miles (535709 km) from Earth, moving away from Earth at a speed of 8243 miles/hour (13266 km/hour).

Also note that it isn't actually going to enter the Asteroid belt, that was an issue with Elon's initial tweet. No idea why he was wrong, but he was wrong.

PearsonArtPhoto
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    It would be good to include the information that this velocity is with respect to the Earth on the website. With respect to the Sun, or the Solar System barycenter, it's moving roughly 10x faster. This reminds me to fix my answer, since the OP does ask with respect to the Earth! (blush) – uhoh Feb 08 '18 at 15:01
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    @uhoh Good call, I've added a bit of clarification on that point. – PearsonArtPhoto Feb 08 '18 at 15:18
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EDIT (new):

Musk's tweet now appears to be wrong according to this comment! Now recalculating.

Using JPL's Horizons data for the planets and Roadster as I discuss more fully in this answer, I get the following:

           min     max
Roadster:  19.9    33.9  km/sec
Earth:     29.3    30.3  km/sec
Mars:      22.0    26.6  km/sec

You can see that the Roadster's velocity with respect to the Solar System barycenter has a wide swing, in accordance with it's more elliptical orbit.

With respect to the Earth, things are a lot different. Right now it's only about 3.5 km/s, and will drop down to about 3.2 km/s in a few weeks. There's a nice plot in this answer.

enter image description here

The following is obsolete!

Elon Musk's tweet shows the figure below, indicating a semi-major axis of about (2.61+0.98)/2 = 1.795 AU. Mars' semi-major axis is about 1.524 AU.

One AU is 149597870700. meters, and the standard gravitational parameter of the Sun $GM$ is about 1.327E+20 m^3/s^2. Using the values for perihelion and aphelion shown in the plot, and the vis-viva equation:

$$v^2 = GM \left( \frac{2}{r}-\frac{1}{a}\right),$$

I get a velocity at perihelion (which is at least darn close to where it is now) of 36,300 m/s, and at aphelion of about 13,600 m/s, which will happen in about 880 days, based on the two decimal place accuracy of the numbers shown in the tweet.

edit: Scott Manley gets a period of 910 days (2.404 years) at 05:40 in his new video (linked in this answer) rather than 880 days. I'm not sure of the reason for the discrepancy though. Perhaps he used the $C_3$ instead? In any even, he certainly knows a lot about asteroids!

enter image description here

Bothered by the periapsis being only 0.98 AU? I was. Then I remembered that the Earth is closest to the Sun during the Northern Hemisphere's winter.

uhoh
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    It's after midnight. Please someone check my arithmetic, correct if necessary. – uhoh Feb 07 '18 at 16:30
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    But as there is one more ∆v left. So don't you think it will increase the speed of the Roadster? Or you are also considering that in your solution. – Sudhanshu Gaur Feb 07 '18 at 16:38
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    @SudhanshuGaur The green line shows the projected orbit for the next two years, and it comes from SpaceX. I'm sure its after the 3rd and final burn, which happened many hours ago already. The tweet itself actually says "Third burn successful. Exceeded Mars orbit and kept going to the Asteroid Belt." – uhoh Feb 07 '18 at 16:47
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I plotted the HORIZONS data (SpaceX Roadster (spacecraft) (Tesla) [-143205]) over time since launch in order to answer the question for the near future:

Absolute velocity of the Tesla Roadster relative to Earth & SSB

The red line gives the answer to your question, by date (top axis labels) or days since launch (bottom axis labels). The velocity of Earth is taken from [399] (Earth [Geocenter]).

For reference, the blue line gives the velocity of the spacecraft relative to the solar system barycenter (SSB), and the green dashed line shows the velocity of Earth relative to the SSB.

JulianHzg
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