JPL has produced a further update for the orbit of the Tesla, although the rate of arrival of new observations has slowed. With the Tesla now below magnitude 20 and, with the Moon close to full, it is unlikely that there will be a new orbit update for at least another week or ten days. The latest orbit includes a first approximation at modelling nongravitational accelerations (the outward thrust of radiation pressure from sunlight hitting the car and of outgassing), improving the orbit accuracy and has allowed JPL to extend greatly the calculation of future encounters with the Earth, the Moon and Mars.
The table, below, summarises the last four orbit solutions:
JPL #6 
JPL #7 
JPL #8 
JPL #9 

Length of arc 
4.5 days 
5.2 days 
10.1 days 
14.2 days 
Number of observations 
269 
330 
353 

Perihelion distance 
0.986063 AU 
0.986063 AU 
0.986062 AU 
0.986061 AU 
Eccentricity 
0.261849 
0.261849 
0.260562 
0.2596427 
Semimajor Axis 
1.335855 AU 
1.335855 AU 
1.333529 AU 
1.3318719 AU 
Aphelion distance 
1.685648 AU 
1.685648 AU 
1.680996 AU 
1.6776826 AU 
Period 
1.544 years 
1.544 years 
1.5399 years 
1.5371 years 
JPL have calculated now future planetary encounters up to 2090, which is a good indication of the increased confidence in the orbit. As one would expect, the perihelion distance barely changes with each new orbit solution. What we see though is a trend to slowly decreasing eccentricity with the latest orbit solutions. This translates to a slight reduction in aphelion distance and hence in the orbital period.
The new solution also has a significant impact on future planetary encounters. The most important by far in terms of distance, is the Mars encounter on April 22^{nd} 2035. The latest orbit solution has reduced the miss distance and thus increased the importance of this encounter on the future orbit. Instead of a minimum possible encounter distance, within the uncertainties, of 2.9 Lunar Distances (2.9 LD), the minimum possible reduces to 1.6 LD. The most likely miss distance – the nominal value – reduces from 8.2 LD to 6.1 LD.
In contrast, the 2035 Mars encounter shifts the future orbit away from the Earth in 2047. The nominal miss distance increases from 9.2 LD to 12.2 LD, reducing quite significantly its effects.
As gravity has an inverse square relation, the perturbing effect is a function of the mass of the perturbing body and the inverse square of the encounter distance. So, an encounter with the Earth to 4 LD has the same perturbing effect as one with a planet one quarter of the Earth’s mass, at 2 LD. That is not exactly rocket science!
The three bodies that can encounter the Tesla – at present – are, in increasing order of size, the Moon, Mars and the Earth.
The Moon has such a small mass that distance of the future Earth encounters to 2090, none of which can get closer than 10 LD, its effects are completely negligible: for the Moon to have the same perturbing effect as the Earth, the Tesla would need to get to around 0.1 LD of the Moon, which is totally impossible in the next 75 years. Whereas, with about one tenth of an Earth mass, an encounter with Mars to 1 LD has approximately the same effect as one to 3 LD of the Earth. In other words, the Tesla has to get three times closer to Mars to feel the same perturbation as it does from an Earth encounter.
So, even though JPL calculates that there will be an impressivelooking five encounters of the Tesla with Mars, four of them will have almost no impact on the future orbit. If we say that an encounter with Earth to 1 LD gives a perturbation of 1000 units, none of these four more distant Mars encounters in 2020, 2052, 2067 and 2084 scores even 0.5 units!
For the nominal miss distances, the three most important encounters for determining the future orbit are, with the Earth in 2047 and 2088, with 6.5 and 2.8 units of perturbation respectively and with Mars in 2035, with 2.7 units of perturbation. No other encounter reaches 1 unit of perturbation.
However, the result is dramatically different if we consider the smallest possible encounter distances – these are calculated as 3 sigma, so there is only a 1in300 chance that the actual encounter distance will be closer – the 2035 Mars encounter could get as high as 40 units of perturbation, or as low as less than 1. In contrast, the perturbation from the 2047 Earth encounter is now fixed with considerable certainty, as it is certain that it will be relatively distant. Not so the 2088 Earth encounter, where the uncertainties mount so much that the encounter distance could be anywhere in the range from under 11 to more than 26 LD.
Would a hypothetical astronaut launched to Mars on a future Falcon rocket be able to see the Tesla fly by the planet in 2035?
Sadly, it seems unlikely. As seen from the surface of Mars, the Tesla will track through the constellation of Leo, on the outward leg of its orbit, as an evening object, only about 35% illuminated at closest approach. The nominal miss distance gives a magnitude of 21 at closest approach and 20 at brightest (when it will be in the constellation of Virgo), which would be about three days after closest approach – this assumes that the ultraviolet light has not spoilt the glossy paint job and reduced considerably the reflectivity of the Tesla. Even if the Tesla passes at the minimum possible miss distance, it will only get 3 magnitudes brighter so, a maximum of magnitude 17. So, no luck with observing the 2035 Mars encounter from the surface of the planet – always assuming that astronauts have made it there – but maybe the car will be retrieved for a future Museum of Mars… always assuming that the Air and Space Museum in Washington does not get it first!