Our latest Pioneer paper, in which we discuss the results from the Pioneer thermal model and its incorporation into the orbital analysis (the conclusion being that no significant anomalous acceleration remains once thermal radiation is properly accounted for) made it to the cover of Physical Review Letters. I am very grateful that I was given the opportunity to participate in this research, and I am very proud of this work and our results.
Yesterday, Venus transited the Sun. It won’t happen again for more than a century.
I had paper “welder’s glasses” courtesy of Sky News. Looking through them, I did indeed see a tiny black speck on the disk of the Sun. However, it was nowhere as impressive as the pictures taken through professional telescopes.
These live pictures were streamed to us courtesy of NASA. One planned broadcast from Alice Springs, Australia, was briefly interrupted. At first, it was thought that a road worker cutting an optical cable was the culprit, but later it turned out to be a case of misconfigured hardware. Or could it be that they were trying to fix a problem with an “intellectual property address”, a wording that appeared on several Australian news sites today? (Note to editors: if you don’t understand the text, don’t be over-eager replacing acronyms with what you think they stand for.)
I also tried to take pictures myself, holding my set of paper welder’s glasses in front of my (decidedly non-professional) cameras. Surprisingly, it was with my cell phone that I was able to take the best picture, but it did not even come close in resolution to what would have been required to see Venus.
The lesson? I think I’ll leave astrophotography to the professionals. Or, at least, to expert amateurs. Unfortunately, I am neither.
That said, I remain utterly fascinated by the experience of staring at a sphere of gas, close to a million and a half kilometers wide, containing 2 nonillion (2,000,000,000,000,000,000,000,000,000,000) kilograms of mostly hydrogen gas, burning roughly 580 billion kilograms of it every second in the form of nuclear fusion deep in its core, releasing photons amounting to about 4.3 billion kilograms of energy… and most of these photons remain trapped for a very long time, producing extreme pressures (so that the interior of the Sun is dominated by this ultrarelativistic photon gas) that prevent the Sun from collapsing upon itself, which will indeed be its fate when it can no longer sustain hydrogen fusion in its core a few billion years from now. And then, this huge orb is briefly occulted by a tiny black speck, the shadow of a world as big as our own… just a tiny black dot, too small for my handheld cameras to see.
I sometimes try to use a human-scale analogy when trying to explain to friends just how mind-bogglingly big the solar system is. Imagine a beach ball that is a meter wide. Now suppose you stand about a hundred meters away from it, like the length of a large sports field. Okay… now imagine that that beach ball is so bleeping hot, even at this distance its heat is burning your face. That’s how hot the Sun is.
Now hold up a large pea, about a centimeter in size. That’s the Earth. Another pea, roughly halfway between you and the beach ball would be Venus.
A peppercorn, some thirty centimeters or so from your Earth pea… that’s the Moon. Incidentally, if you hold that peppercorn up, at about thirty centimeters from your eye it is just large enough to obscure the beach ball in the distance, producing a solar eclipse.
Now let’s go a little further. Some half a kilometer from the beach ball you see a large-ish orange… Jupiter. Twice as far, you see a smaller orange with a ribbon around it; that’s Saturn. Pluto would be another peppercorn, more than three kilometers away.
But your beach ball’s influence does not end there. There will be specks of dust in orbit around it as far out as several hundred kilometers, maybe more. So where would the next beach ball be, representing the nearest star? Well, here’s the problem… the surface of the Earth is just not large enough, because the next beach ball would be more than 20,000 kilometers away.
To represent other stars, not to mention the whole of the Milky Way, we would once again need astronomical distance scales. If a star like our Sun was a one meter wide beach ball, the Milky Way of beech balls would be larger than the orbit of the Earth around the Sun. And the nearest full-size galaxy, Andromeda, would need to be located in distant parts of the solar system, far beyond the orbits of planets.
The only way we could reduce galaxies and groups of galaxies to a scale that humans can comprehend is by making stars and planets microscopic. So whereas the size of the solar system can perhaps be grasped by my beach ball and pea analogy, it is simply impossible to imagine simultaneously just how large the Milky Way is, not to mention the entire visible universe.
Or, as Douglas Adams wrote in The Hitchhiker’s Guide to the Galaxy: “Space is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.”
The Dragon spacecraft of SpaceX corporation successfully splashed down after a successful overall mission of hauling cargo to and from the International Space Station.
It’s an incredible success for a commercial venture.
I also found the appearance of the SpaceX mission control facility revealing. Ordinary office desks, ordinary office chairs, ordinary flat screen monitors. Nothing special, no horrendously expensive custom hardware.
This reinforces my impression that the SpaceX venture actually brought the 21st century to NASA and the ISS. An impression that was created by the ISS crew’s reaction to the Dragon capsule’s “new car smell” and its space-age interior complete with smooth surfaces, blue LEDs and whatnot.
Congrats to SpaceX. Well done. Here is to the future.
A few days ago, a bright 16-year old German student of Indian descent, Shouryya Ray of Dresden, won second prize in a national science competition with an essay entitled “Analytische Lösung von zwei ungelösten fundamentalen Partikeldynamikproblemen” (Analytic solution of two unsolved fundamental particle dynamics problems).
This story should have ended there. And perhaps it would have, were it not for the words in the abstract that said, among other things: “Das zugrundeliegende Kraftgesetz wurde bereits von Newton (17. Jhd.) entdeckt. […] Diese Arbeit setzt sich also die analytische Lösung dieser bisher nur näherungsweise oder numerisch gelösten Probleme zum Ziele.” (The underlying power law was discovered by Newton (17th century). The goal of this work is then the analytic solution of these until now only approximately or numerically solved problems.)
This was more than enough for sensation-seeking science journalists. The story was picked up first by Die Welt with the title “Mit 16 ein Genie: Shouryya Ray löste ein jahrhundertealtes mathematisches Problem” (Genius at 16: Shouryya Ray solves centuries-old mathametical problem) and then translated into English and other languages, even appearing in the Ottawa Citizen. In short order, even a biographical entry on Wikipedia was created; now nominated for deletion, many are voting to keep it because in their view, the press coverage is sufficient to establish encyclopedic notability.
Cooler heads should have prevailed. What science journalists neglected to ask is why, if this is such a breakthrough, the youth only received second prize. And in any case, what on Earth did he actually do? His essay or details about it were not published. The only clue to go by was a press photo in which the student holds up a large sheet of paper containing an equation:
As I discussed this very topic on a page I placed on my Web site a few years back (reacting to some bad math and flawed physics reasoning in an episode of the Mythbusters) I felt compelled to find out more. I guessed (correctly, as it turns out) that \(u\) and \(v\) must be the horizontal and vertical (or vertical and horizontal?) components of the projectile’s velocity, \(g\) is the gravitational constant, and \(\alpha\) is the coefficient of air resistance. However, I am embarrassed to admit that although I spent some time trying, I was not able to find a way to separate the variables and integrate the relevant differential equations to obtain Ray’s formula. I was ready to give up actually when I came across a derivation on reddit (and I realized that I was on the right track all along, I was just stubbornly trying to do a slightly different trick, which didn’t work). The formula is correct, and it is certainly an impressive result for a 16-year old, worthy of a second prize.
But no more. This is not a breakthrough. As it turns out, similar implicit solutions were well known in the 19th century. A formulation that differs from Ray’s only in notational details appeared in a paper by Parker (Am. J. Phys, 45, 7, 606, July 1977). Alas, such an implicit form is of limited utility; one still requires numerical methods to actually solve the equation.
Much of this was probably known to the judges of the competition, which is probably why they awarded the student second prize.
Hopefully none of this will deter young Mr. Ray from pursuing a successful career as a physicist or mathematician.
According to astronauts on board the ISS, the interior of the SpaceX capsule has a “new car smell“. Seeing the world’s first commercial spacecraft dock with the ISS successfully is, well, awesome I think is an appropriate word here.
The Dragon capsule of SpaceX Corporation is on its way after a successful launch towards the International Space Station. If all goes well, it will dock with the ISS in two days’ time, making it the first commercial spacecraft to do so, and paving the way to eventual human flight to the ISS on board commercial vehicles. This really is the beginning of a new era.
And the end of an old one. The ashes of James Doohan, better known as Scotty to Star Trek fans, are reportedly on board the Dragon capsule, to fulfil the actor’s final wishes.
I just read (link in Hungarian) that a far right member of the Hungarian parliament found it necessary to use a genetic test to prove that he is free of Jewish and Roma blood.
Even if it were possible to do so, I have no inclination to use a genetic testing service to find out the ethnicity of my ancestors. But, I do hope that I have Roma, Jewish, Hungarian, Slav, Russian, German, or for that matter Chinese or Indian ancestors. That is because there is only one group of people that I wish to belong to: the group of human beings. I have zero desire to join any subgroup whose sole purpose is to revel in the idea that they are somehow superior by birth to other subgroups. And, well, if all this makes me a mongrel or a tyke in the eyes of some with a better defined ethnicity… you know what, I don’t really like your purebred attitude either.
I am really disappointed to learn this morning that the world will not come to an end December this year. According to a new discovery, the Mayan calendar may have had at least 17 baktuns, not 13 as previously believed, so we are good for something like another two millennia.
Just as I was getting ready to sell my house and all my earthly possessions…
Here is a photograph of the cockpit of the Space Shuttle Endeavour, powered up for the very last time ever:
It is an emotional moment. But we must not let those emotions get in the way of reason. The Shuttle program swallowed up huge amounts of money and these orbiters, however wonderful, didn’t really take humanity anywhere.
Just consider: the Shuttle flew a few hundred kilometers from the surface of planet Earth. That is one one-thousandths (!) of the distance to the Moon, visited by Apollo astronauts over 40 years ago. But no human has been beyond Low Earth Orbit (LEO) since Apollo 17 flew in late 1972. Now if all goes well, in a few short years one of the very first missions of NASA’s new spacecraft, the Orion capsule, may take humans beyond the Moon, to the L2 Lagrange point. At last, this is real exploration again… not just a routine (albeit dangerous) taxiing between the surface and LEO.
And the taxiing is not going to stop for Americans. The Dragon capsule of SpaceX corporation is set to fly to the International Space Station next week. It is still an unmanned flight but if all goes well, perhaps the next time they’ll ferry not just cargo but also people.
In 2004, NASA landed two rovers on Mars, Spirit and Opportunity. Both far surpassed expectations, operating much longer than their planned design lifetime of 6 months.
Spirit was ultimately lost in 2010, but Opportunity, having spent the last five months in hibernation during the Martian winter, is now driving again. It is amazing that this machine is still functioning. Imagine leaving a solar powered remote control toy in the sandy desert somewhere. How long would it survive and remain drivable?
It’s one setback after another, sometimes with tragic consequences.
Last year it was Phobos-Grunt, Russia’s attempt to return to deep space beyond Earth orbit after a 15-year hiatus. Alas, Phobos-Grunt never managed to go too far… it failed to reach escape orbit and eventually fell back to the Earth.
And now, it’s the Sukhoi Superjet’s turn. After more than two decades, Russia is again trying to capture a small segment of the passenger jet market. Their demonstration model was on an Asian tour, trying to impress new customers. Well, they certainly created an impression… just not the impression they were hoping for. More tragically, 48 souls perished.
I suppose that from a Canadian (or Brazilian) perspective, this should be considered “good news”, since the Sukhoi Superjet 100 is intended to compete in a market that is dominated by Canada’s Bombardier and Brazil’s Embraer. But I don’t find this comforting. In fact, for the sake of the future of Russia’s commercial jet industry, I hope that this tragic accident will turn out to be a case of pilot error. Controlled flight into terrain.
Astronomy is supposed to be a relatively safe profession. I suppose observational astronomers may occasionally injure themselves when working on a telescope, but it’s probably rare. For them to become murder victims is even more unlikely.
So why would a Japanese astronomer, working in Chile on the Atacama Large Millimeter Array, be murdered on the street just outside his apartment?
Everyone who saw the 1986 disaster of the space shuttle Challenger remembers the words from mission control: “Flight control is here looking very carefully at the situation, obviously a major malfunction.”
I was watching a newly surfaced home video of the explosion courtesy of The Huffington Post. (Well worth watching. In particular, notice just how cold it must have been that morning, as evidenced by the clothing people wore.) This led me to a link about Steve Nesbitt, the NASA communications officer who uttered those sad but memorable words.
By the time NASA was ready to fly shuttles again, Nesbitt was already promoted to a new position. But he asked his bosses to be the announcer for the next flight, because “the last one ended rather badly.” Thus he became the voice of NASA in September 1988, when Discovery flew.
Nesbitt retired last year and the shuttles are now heading to museums. But, I admit, the emotional impact of the failed launch of Challenger remains just as strong today as it was 26 years ago.
I was having a discussion with a lawyer friend of mine. I was trying to illustrate the difference between the advocating done by lawyers and the scientist’s unbiased (or at least, not intentionally biased) search for the truth. One is about cherry-picking facts and arguments to prove a preconceived notion; the other about trying to understand the world around us.
I told him that anything and the opposite of anything can be proven by cherry-picking facts. Then it occurred to me that it is true even in math. For instance, by cherry-picking facts, I can easily prove that \(2\times 2=5\). Let’s start with three variables, \(a\), \(b\) and \(c\), for which it is true that \(a=b+c\). Then, multiplying by 5 gives
$$5a=5b+5c.$$
Multiplying by 4 and switching the two sides gives
$$4b+4c=4a.$$
Adding these two equations together, we get
$$5a+4b+4c=4a+5b+5c.$$
Subtracting \(9a\) from both sides, we obtain
$$4b+4c-4a=5b+5c-5a,$$
or
$$4(b+c-a)=5(b+c-a).$$
Dividing both sides by \(b+c-a\) gives the final result:
$$4=5.$$
And no, I did not make some simple mistake in my derivation. In fact, I can use computer algebra to obtain the same result, and computers surely don’t lie. Here it is, with Maxima:
(%i1) eq1:5*a=5*b+5*c$ (%i2) eq2:4*b+4*c=4*a$ (%i3) eq3:eq1+eq2$ (%i4) eq4:eq3-9*a$ (%i5) eq5:factor(eq4)$ (%i6) eq6:eq5/(b+c-a); (%o6) 4 = 5
All I had to do to make this happen was to ignore an inconvenient little fact, which is precisely what lawyers (not to mention politicians) do all the time. Surely, if I can prove that \(2\times 2=5\), I can prove anything. So can lawyers and they know it.
April 23, 2012, and this is what I see through my window:
A month ago, I had to run my air conditioner. Grumble.
The Space Shuttle Discovery is on its way to its final resting place.
Many lament the end of the Shuttle program. They shouldn’t. Beautiful as these machines were, they really stifled the American space program. For decades, countless billions of dollars were spent… on going around, and around, and around, in low-Earth orbit, ultimately getting nowhere.
When Barack Obama opted for a variant of the Augustine Commission‘s “flexible path” approach, some pundits called it the end of America’s space dominance. I think the contrary is true. Instead of opting for an overly ambitious but ultimately unrealistic space program that would eventually die on the floors of Congress due to lack of funding, Obama chose a space program that places the emphasis on sustainable development: a long term vision of expanding human presence beyond Earth orbit in the solar system, not necessarily with spectacular landings on Mars (however desirable such a landing may be, it’s also insanely risky and expensive) but with building the infrastructure for a permanent human presence beyond the protective shield of the Earth’s radiation belts.
At the very end of tonight’s episode of The Simpsons, just before the end credits, we caught a brief glimpse of Pioneer 10 (or was it 11?), along with an extraterrestrial intently studying Carl Sagan’s famous golden plaque.
But wait a minute… stupid alien is holding the plaque upside-down. No wonder he can’t make sense of it.
And they didn’t get the shape of the RTG fins right. Can’t really blame them; way too many artistic depictions of Pioneer show the generators with the small, rectangular fins that, I believe, were on (non-nuclear) engineering mockups used during testing.
Now here is a way to use physics more cleverly than Sheldon Cooper to avoid a costly ticket for a moving violation: http://arxiv.org/abs/1204.0162.
The brief, two-sentence abstract reads: A way to fight your traffic tickets. The paper was awarded a special prize of $400 that the author did not have to pay to the state of California.
Perhaps unsurprisingly, the paper is dated April 1. But the story, it appears, is real nonetheless.
I just came across this delightful imaginary conversation between a physicist and an economist about the unsustainability of perpetual economic growth.
The physicist uses energy production in his argument: growth at present rates means that in a few hundred years, we’ll produce enough energy to start boiling the oceans. And this is not something that can be addressed easily by the magic of technology. When waste heat is produced, the only way to get rid of it is to radiate it away into space. After about 1400 years of continuous growth, the Earth will be radiating more energy (all man-made) than the Sun, which means it would have to be a lot hotter than the Sun, on account of its smaller size. And in about 2500 years, we would exceed the thermal output of the whole Milky Way.
This, of course, is nonsense, which means terrestrial energy production will be capped eventually by basic physics. If GDP would continue to grow nonetheless, it would mean that the price of energy relative to other stuff would decrease to zero. This is also nonsense, since a limited resource cannot become arbitrarily cheap. But that means GDP growth must also be capped.
What I liked about this argument is that it is not emotional or ideological; it’s not about hugging trees or hating capitalism. It is about basic physics and elementary logic that is difficult to escape. In fact, it can be put in the form of equations. Our present energy production \(P_0\) is approximately 15 TW, which is about 0.002% of the Sun’s output that reaches the Earth:
\begin{align}
P_0&\simeq 1.5 \times 10^{13}~\rm{W},\\
P_\odot&\simeq 7 \times 10^{17}~\rm{W},\\
\eta_0&=P_0/P_\odot \sim 0.002\%.
\end{align}
For any other value of \(\eta\), there is a corresponding value of \(P\):
\begin{align}
P=\eta P_\odot.
\end{align}
Now all we need is to establish a maximum value of \(\eta\) that we can live with; say, \(\eta_{\rm max}=1\%\). This tells us the maximum amount of energy that we can produce here in the Earth without cooking ourselves:
\begin{align}
P_{\rm max}=\eta_{\rm max}P_\odot.
\end{align}
On the economic side of this argument, there is the percentage of GDP that is spent on energy. In the US, this is about 8%. For lack of a better value, let me stick to this one:
\begin{align}
\kappa_0\sim 8\%.
\end{align}
How low can \(\kappa\) get? That may be debatable, but it cannot become arbitrarily low. So there is a value \(\kappa_{\rm min}\).
The rest is just basic arithmetic. GDP is proportional to the total energy produced, divided by \(\kappa\):
\begin{align}
{\rm GDP}&\propto \frac{\eta}{\kappa}P_\odot,\\
{\rm GDP}_{\rm max}&\propto \frac{\eta_{\rm \max}}{\kappa_{\rm min}}P_\odot,
\end{align}
And in particular:
\begin{align}
{\rm GDP}_{\rm max}&=\frac{\eta_{\rm max}\kappa_0}{\eta_0\kappa_{\rm min}}{\rm GDP}_0,
\end{align}
where \({\rm GDP}_0\) is the present GDP.
We know \(\eta_0\sim 0.002\%\). We know \(\kappa_0=8\%\). We can guess that \(\eta_{\rm max}\lesssim 1\%\) and \(\kappa_{\rm min}\gtrsim 1\%\). This means that
\begin{align}
{\rm GDP}_{\rm max}\lesssim 4,000\times {\rm GDP}_0.
\end{align}
This is it. A hard limit imposed by thermodynamics. But hey… four thousand is a big number, isn’t it? Well… sort of. At a constant 3% rate of annual growth, the economy will increase to four thousand times its present size in a mere 280 years or so. One may tweak the numbers a little here and there, but the fact that physics imposes such a hard limit remains. The logic is inescapable.
Or is it? The word “escape” may be appropriate here for more than one reason, as there is one obvious way to evade this argument: escape into space. In a few hundred years, humanity may have spread throughout the solar system, and energy amounts enough to boil the Earth’s oceans may be powering human colonies in the hostile (and cold!) environments near the outer planets.
That is, if humans are still around a few hundred years from now. One can only hope.
Our second short paper has been accepted for publication in Physical Review Letters.
I have been involved with Pioneer 10 and 11 in some fashion since about 2002, when I first began corresponding with Larry Kellogg about the possibility of resurrecting the telemetry data set. It is thanks the Larry’s stamina and conscientiousness that the data set survived.
I have been involved actively in the research of the Pioneer anomaly since 2005. Seven years! Hard to believe.
This widely reported anomaly concerns the fact that when the orbits of Pioneer 10 and 11 are accurately modeled, a discrepancy exists between the modeled and measured frequency of the radio signal. This discrepancy can be resolved by assuming an unknown force that pushes Pioneer 10 an 11 towards the Earth or the Sun (from that far away, these two directions nearly coincide and cannot really be told apart.)
One purpose of our investigation was to find out the magnitude of the force that arises as the spacecraft radiates different amounts of heat in different directions. This is the concept of a photon rocket. A ray of light carries momentum. Hard as it may appear to believe at first, when you hold a flashlight in your hands and turn it on, the flashlight will push your hand backwards by a tiny force. (How tiny? If it is a 1 W bulb that is perfectly efficient and perfectly focused, the force will be equivalent to about one third of one millionth of a gram of weight.)
On Pioneer 10 and 11, we have two main heat sources. First, there is electrical heat: all the instruments on board use about 100 W of electricity, most of which is converted into heat. Second, electricity is produced, very inefficiently, by a set of four radioisotope thermoelectric generators (RTGs); these produce more than 2 kW of waste heat. All this heat has to go somewhere, and most of this heat will be dissipated preferably in one direction, behind the spacecraft’s large dish antenna, which is always pointed towards the Earth.
The controversial question was, how much? How efficiently is this heat converted into force?
I first constructed a viable thermal model for Pioneer 10 back in 2006. I presented results from custom ray-tracing code at the Pioneer Explorer Collaboration meeting at the International Space Science Institute in Bern, Switzerland in February 2007:
With this, I confirmed what has already been suspected by others—notably, Katz (Phys. Rev. Letters 83:9, 1892, 1999); Murphy (Phys. Rev. Letters 83:9, 1890, 1999); and Scheffer (Phys. Rev. D, 67:8, 084021, 2003)—that the magnitude of the thermal recoil force is indeed comparable to the anomalous acceleration. Moreover, I established that the thermal recoil force is very accurately described as a simple linear combination of heat from two heat sources: electrical heat and heat from the RTGs. The thermal acceleration \(a\) is, in fact
$$a=\frac{1}{mc}(\eta_{\rm rtg}P_{\rm rtg} + \eta_{\rm elec}P_{\rm elec}),$$
where \(c\simeq 300,000~{\rm km/s}\) is the speed of light, \(m\simeq 250~{\rm kg}\) is the mass of the spacecraft, \(P_{\rm rtg}\sim 2~{\rm kW}\) and \(P_{\rm elec}\sim 100~\rm {W}\) are the RTG heat and electrical heat, respectively, and \(\eta_{\rm rtg}\) and \(\eta_{\rm elec}\) are “efficiency factors”.
This simple force model is very useful because it can be incorporated directly into the orbital model of the spacecraft.
In the years since, the group led by Gary Kinsella constructed a very thorough and comprehensive model of the Pioneer spacecraft, using the same software tools (not to mention considerable expertise) that they use for “live” spacecraft. With this model, they were able to predict the thermal recoil force with the greatest accuracy possible, at different points along the trajectory of the spacecraft. The result can be compared directly to the acceleration that is “measured”; i.e., the acceleration that is needed to model the radio signal accurately:
In this plot, the step-function like curve (thick line) is the acceleration deduced from the radio signal frequency. The data points with vertical error bars represent the recoil force calculated from the thermal model. They are rather close. The relatively large error bars are due primarily to the fact that we simply don’t know what happened to the white paint that coated the RTGs. These were hot (the RTGs were sizzling hot even in deep space) and subjected to solar radiation (ultraviolet light and charged particles) so the properties of the paint may have changed significantly over time… we just don’t know how. The lower part of the plot shows just how well the radio signal is modeled; the average residual is less than 5 mHz. The actual frequency of the radio signal is 2 GHz, so this represents a modeling accuracy of less than one part in 100 billion, over the course of nearly 20 years.
In terms of the above-mentioned efficiency factors, the model of Gary’s group yielded \(\eta_{\rm rtg}=0.0104\) and \(\eta_{\rm elec}=0.406\).
But then, as I said, we also incorporated the thermal recoil force directly into the Doppler analysis that was carried out by Jordan Ellis. Jordan found best-fit residuals at \(\eta_{\rm rtg}=0.0144\) and \(\eta_{\rm elec}=0.480\). These are somewhat larger than the values from the thermal model. But how much larger?
We found that the best way to answer this question was to plot the two results in the parameter space defined by these two efficiency factors:
The dashed ellipse here represents the estimates from the thermal model and their associated uncertainty. The ellipse is elongated horizontally, because the largest source of uncertainty, the degradation of RTG paint, affects only the \(\eta_{\rm rtg}\) factor.
The dotted ellipse represents the estimates from radio signal measurements. The formal error of these estimates is very small (the error ellipse would be invisibly tiny). These formal errors, however, are calculated by assuming that the error in every one of the tens of thousands of Doppler measurements arises independently. In reality, this is not the case: the Doppler measurements are insanely accurate, any errors that occur are a result of systematic mismodeling, e.g., caused by our inadequate knowledge of the solar system. This inflates the error ellipse and that is what was shown in this plot.
Looking at this plot was what allowed us to close our analysis with the words, “We therefore conclude that at the present level of our knowledge of the Pioneer 10 spacecraft and its trajectory, no statistically significant acceleration anomaly exists.”
Are there any caveats? Not really, I don’t think, but there are still some unexplored questions. Applying this research to Pioneer 11 (I expect no surprises there, but we have not done this in a systematic fashion). Modeling the spin rate change of the two spacecraft. Making use of radio signal strength measurements, which can give us clues about the precise orientation of the spacecraft. Testing the paint that was used on the RTGs in a thermal vacuum chamber. Accounting for outgassing. These are all interesting issues but it is quite unlikely that they will alter our main conclusion.
On several occasions when I gave talks about Pioneer, I used a slide that said, in big friendly letters,
PIONEER 10/11 ARE THE MOST PRECISELY NAVIGATED DEEP SPACE CRAFT TO DATE.
And they confirmed the predictions of Newton and Einstein, with spectacular accuracy, by measuring the gravitational field of the Sun in situ, all the way up to about about 70 astronomical units (the distance of the Earth from the Sun).