• Fram2 Over the Poles

    John F. Lindner

    Historically, astronauts have launched roughly east to exploit Earth’s spin, as the terrestrial equator moves at nearly 1000 mph with respect to its center. But last week the Fram2 SpaceX Dragon crew launched south from the Kennedy Space Center to become the first humans to orbit Earth over its poles.

    The all-private, all-international, rookie crew consisted of entrepreneur and mission commander Chun Wang, cinematographer and Dragon commander Jannicke Mikkelsen, roboticist and Dragon pilot Rabea Rogge, and polar guide and medical officer Eric Philips. None of the crew were pilots, let alone professional astronauts, but they all had deep ties to polar exploration and each other, meeting at the Svalbard, Norway archipelago, one of Earth’s northernmost inhabited regions. Indeed, their mission was named after the Norwegian sailing ship Fram, which explored both poles circa 1900, and the crew carried a small wooden piece of Fram with them.

    The Fram2 crew performed 22 experiments from a list offered by SpaceX to help the company prepare for trips to Mars and beyond. One of these involved taking the first medical X-ray photographs in space, which on long spaceflights might help diagnose and repair injuries in people and faults in equipment. Another was high-resolution videography of Earth from their unique orbital perspective. Still another was the crew’s unassisted egress of the Dragon capsule shortly after splashdown.

    I especially enjoyed Chun’s social media updates, enabled via a laser connection to the Starlink satellite internet constellation. After adapting to microgravity, the crew seemed to enjoy themselves immensely and reported sleeping well. Addition of the cupola not only provided stunning views but notably expanded Dragon’s interior volume. The inspiring Fram2 mission expanded the human spaceflight envelope, advanced spaceflight privatization, and laid groundwork for extended spaceflight to Moon, Mars, and beyond.

    Cinematography from orbit
    Professional cinematographer and Dragon commander Jannicke Mikkelsen using a Red Digital Cinema Raptor 8K video camera in the Dragon’s cupola.

    First X-rays
    First X-ray photo on Earth by W. C. Röntgen in 1895 (left) and the first X-ray photo off Earth by the Fram2 crew in 2025 (right).

  • 2D Kepler Conjecture

    John F. Lindner

    Johannes Kepler asserted in 1611 that no packing of identical balls has density greater than the hexagonal close-packed “cannonball” packing of oranges at a grocery store’s fruit stand. But the gulf between intuition and proof was so large that the latter was not achieved until the work of Thomas Hales and Samuel Ferguson in 1998. And the resulting computer-assisted proof-by-exhaustion was so long and complicated that it was not finally accepted until 2017 after being formalized and certified by proof-checking computer software.

    Here I present an elementary proof due to Hales of the corresponding conjecture in two dimensions, that regular hexagonal packing is the densest disk packing in the plane, as asserted by Axel Thue in 1890 and first proved by László Fejes Tóth in 1940.

    Assume the (blue) disks have radius 1, so (light blue) disks circumscribing the circumscribing hexagon have radius 2 / \sqrt{3}>1. Overlapping circumscribing disks form (yellow) triangles connecting disk centers and circumcircle intersections. Partition the plane into 3 regions: (white) background, (light and dark blue) disks sans (yellow) triangles, and (yellow) triangles.

    2D Kepler Conjecture

    Density of a (dark blue) disk in the circumscribing hexagon is

    D_h = \frac{a_h}{A_h} = \frac{\pi 1^2}{6 \cdot (1 \cdot 2/\sqrt{3})/2} = \frac{\pi}{\sqrt{12}} \approx 0.9069.

    Density of a (dark blue) disk in the empty (white) background is

    D_b = 0.

    Density of a (dark blue) disk sans (yellow) triangle, with vertex angle \theta, in (light blue) a circumscribing disk sans (yellow) triangle is

    D_d = \frac{a_d}{A_d} = \frac{(2\pi-\theta)\, \pi 1^2}{(2\pi-\theta)\, \pi (2/\sqrt{3})^2} = \frac{3}{4} < D_h.

    Density of a (dark blue) disk in a (yellow) triangle is

    D_t = \frac{a_t}{A_t} \stackrel{T}{=} \frac{a_t^\prime}{A_t^\prime} \le \frac{a_t^{\prime\prime}}{A_t^\prime} = \frac{(1/6)\, \pi 1^2}{(1 \cdot 2/\sqrt{3})/2} = \frac{\pi}{\sqrt{12}} = D_h,

    where a linear transformation converts the acute (yellow) triangle to an equilateral triangle while preserving the area ratio, and the transformed (green) elliptical sector is smaller than the corresponding (green) circular sector.

    2D Kepler Conjecture

    Therefore, mean density of the (dark blue) disks is

    D=\frac{a}{A} = \frac{A_b D_b + A_d D_d + A_t D_t}{A_b + A_d + A_t}\le\frac{A_b D_h + A_d D_h + A_t D_h}{A_b + A_d + A_t} = D_h,

    the hexagonal density, which is the densest possible.

    2D Kepler Conjecture

  • Blue Ghost Eclipse

    John F. Lindner

    Last night’s lunar eclipse, as seen from Earth, looked like a solar eclipse, as seen from Luna.

    Firefly Aerospace‘s NASA-funded Blue Ghost lunar lander recently became the first commercial spacecraft to successfully land on Earth’s moon Luna. Blue Ghost’s mission is designed to last a single lunar day, about two terrestrial weeks. Last night it photographed the eclipse from its landing site at Mare Crisium.

    Due to is larger size, Earth appears almost four times larger from Luna than Luna appears from Earth, but in Blue Ghost’s eclipse photo, we can still see sunlight refracting around Earth’s atmosphere: a diamond ring of sunrises and sunsets.

    Eclipse from Earth and Luna
    Eclipse from Earth (left) and from Luna (right), 2025 March 14. NASA/Firefly Aerospace.

    Blue Ghost was not the first spacecraft to photograph an eclipse from Luna; that was Surveyor III on 1967 April 24.

  • Mach Cutoff

    John F. Lindner

    Two weeks ago, I watched live via Starlink as the Boom Supersonic XB-1 test aircraft broke the sound barrier in level flight, the first all-civilian aircraft to do so. This success promises the return of commercial supersonic flight, at least over ocean. This week, during the final test flight, I learned that no sonic boom had reached the ground in any of the XB-1’s supersonic passes. This suggests the possibility – for the first time – of commercial supersonic flight over land.

    Normally, a transonic aircraft near Mach 1 sheds pressure discontinuities called shock waves, which can disturb people on the ground. Such sonic booms have curtailed the development of supersonic commercial flight. However, atmospheric refraction, due to temperature and wind-speed gradients, can bend the shock wave rays, as in a mirage, so they never reach the ground.

    Temperature and wind gradients bend shock waves

    The XB-1 is a technology demonstrator for Boom’s planned Overture airliner, which Boom hopes will restore supersonic commercial flight for the first time since the retirement of the Concorde over 20 years ago. Boom has already built an Overture assembly plant in North Carolina, where I currently live.

    Despite widespread video conferencing, many people need to fly long distances. Hopefully, by the 2030s, Overture will shrink the globe by flying Mach 1.7 over ocean and Mach 1.0 to 1.3 over land using detailed altitude, temperature, and weather information to ensure that no boom reaches Earth. (Concurrently, NASA is developing the X-59 Quesst quiet-supersonic-technology low-boom demonstrator, which involves carefully shaping the aircraft to minimize the ground level boom.)

  • Mount Wilson Trek

    John F. Lindner

    Nobody walks in L.A., but as a Caltech grad student without a car in the mid 1980s, I once walked from my dorm room up Mount Wilson and touched the enclosure of the famous 100-inch (2.54-meter) Hooker telescope, for 32 years, the world’s largest, where Hubble discovered that the Andromeda “nebula” was actually a galaxy outside our Milky Way, where Hubble & Humason confirmed the expansion of the universe, and where Zwicky found early evidence for dark matter.

    Years later, I revisited Mount Wilson by car, but its observatories were secured by fencing, and I marveled at the opportunity I once had to get so close to the Hooker telescope. Today, as I write this, firefighters battle California wild fires that have killed at least eleven people, destroyed thousands of homes, and threatened the observatories. My 1980s route from Caltech up Mount Wilson is either evacuated or on fire.

    My route to Mount Wilson is currently under wild fire
    My memorable mid 1980s trek (blue curve) from my dorm room at Caltech to the Mount Wilson Hooker telescope is currently evacuated or on fire (yellow and red shading).

  • Outer Planet Cloud Colors

    John F. Lindner

    From my teens to my twenties, from junior high school to graduate school to young professor, I excitingly followed the first reconnaissance of the outer solar system by the Pioneer and Voyager spacecraft. But the exploration isn’t over.

    For the last decade, the Hubble Space Telescope has been systematically observing the colors and dynamics of the clouds of Jupiter, Saturn, Uranus, and Neptune. With resolution comparable to the Voyager flybys and sensitive to colors from ultraviolet to visible to near-infrared, Hubble is uniquely qualified for these observations.

    Apparently, Uranus and Neptune appear blue due to atmospheric methane, which preferentially absorbs red light, except for the polar haze above Uranus pole (pointing rightward in the photographs below due to the planet’s large obliquity or tilt).

    10 years of Hubble Space Telescope's Outer Planet Legacy images.
    10 years of Hubble Space Telescope’s Outer Planet Legacy (OPAL) images. Click for a larger version. NASA, ESA, Amy Simon (NASA-GSFC), Michael H. Wong (UC Berkeley); Image Processing: Joseph DePasquale (STScI).

  • Sum of Reciprocals

    John F. Lindner

    The sum of the reciprocals of the natural numbers diverges, but slowly, like the logarithm of the number of terms. The sum of the reciprocals of the prime numbers also diverges, but even more slowly, like the logarithm of the logarithm of the number of terms, as the primes are sparse in the naturals!

    Here I attempt elementary proofs that

    H_n = \sum_{m\le n} \frac{1}{m} = \mathcal{\Theta}(\log n),

    where m,n \in \mathbb{N}_1 = \mathbb{Z}^+, and

    S_n = \sum_{p\le n} \frac{1}{p} = \mathcal{\Theta}(\log \log n),

    where p \in \mathbb{P} is a prime number, as in Fig. 1. In both cases, I will try to bound the sums below and above to show they are of the same order, a g(x) \le f(x) \le b g(x) so f(x) = \Theta(g(x)). The results are not new, but the presentation is.

    Plots of inverses number and inverse prime sums
    Figure 1: Inverse numbers partial sums (red) increase like a log function, while inverse primes partial sums (blue) increase as the log of a log.

    Sum of Reciprocals of Naturals

    Lower Bound

    The harmonic inverse numbers sum overestimates the corresponding integral as

    H_n = \sum_{m=1}^n\frac{1}{m} >\int_1^{n+1}\hspace{-1.3em}\text{d}x\frac{1}{x} = \log(n+1),

    where the sum is the area bounded by blue rectangles to the hyperbola 1/x, as in Fig. 2.

    Hyperbola and bounding rectangles
    Figure 2: Tall blue rectangle sums overestimate the red integral area under the hyperbola, while short yellow rectangle sums underestimate it, for n=3.

    Upper Bound

    Similarly, the shifted sum underestimates the corresponding integral as

    -1+H_n = -1 + \sum_{m=1}^n\frac{1}{m}= \sum_{m=2}^n\frac{1} {m}≤\int^n_{1}\hspace{-0.6em}\text{d}x\frac{1}{x} = \log n.

    where the sum is the area bounded by short yellow rectangles to 1/x, and so

    H_n ≤ \log n+1.

    Tight Bound

    Combine the lower and upper bounds to find

    \log(n+1)<H_n≤\log n + 1,

    and so

    H_n = \sum_{m\le n} \frac{1}{m} = \mathcal{\Theta}(\log n).

    Sum of Reciprocals of Primes

    Lower Bound

    The inverse numbers sum is dominated by

    H_n = \sum_{m\le n} \frac{1}{m} < \sum_{n\ge p|m} \frac{1}{m} = \prod_{p\le n}\left(\sum_{k=0}^\infty \frac{1}{p^k}\right),

    where the second sum is over all natural numbers m = p_1^{c_1} p_2^{c_2} \cdots p_r^{c_r} divisible by primes p \le n , which includes arbitrarily large m generated by the finite product of the infinite sum. The latter is a geometric series with start 1 and ratio 1/p, so

    H_n < \prod_{p\le n} \left( 1-\frac{1}{p} \right)^{-1} < \prod_{p\le n} e^{2/p} = \exp\left({2\sum_{p\le n} 1/p}\right),

    as e^{2/p} >1+2/p > (1-1/p)^{-1} for p\ge 2, and the product of exponentials is the exponential of the sum. Take logs of both sides to find

    \log H_n < 2 \sum_{p\le n} \frac{1}{p} = 2 S_n

    and since \log n < \log(n+1) < H_n from earlier,

    S_n > \frac{1}{2} \log H_n > \frac{1}{2}\log\log n.

    Upper Bound

    Raise the inverse primes sum to the kth power and write it as a multidimensional sum over the inverse prime factors product

    S_n^k = \left(\sum_{p\le n} \frac{1}{p}\right)^k = \hspace{-0.7em} \sum_{p_1,p_2,\cdots,p_k < n} \frac{1}{p_1 p_2\cdots p_k } < \sum_{m\le n^k} \frac{k!}{m},

    as any reciprocal 1/m appears at most k! times with one prime from each of the k factors in any order in the product p_1p_2 \cdots p_k \le n^k. Since H_x \le \log x + 1 from earlier, write

    S_n^k < k! H_{n^k} \le k! (\log n^k + 1) \le k!\, k\, 2\log n,

    as 1 \le \log x for x \ge e > 2. Take the kth root of both sides to find

    S_n < (k!\, k\, 2\log n)^{1/k} < k\, e\, 2(\log n)^{1/k},

    as

    \begin{aligned}(k!)^{1/k} &\le (k^k)^{1/k} = k, \\ k^{1/k} &= \left(e^{\log k}\right)^{1/k} = e^{(\log k) / k} < e^1 = e,\\ 2^{1/k} &\le 2,\end{aligned}

    for k \ge 1. Now take k = \lfloor 1 + \log\log n \rfloor to get

    \begin{aligned}S_n &< (1 + \log\log n) e 2 (\log n)^{1/(1+\log\log n)}\\ &= (1 + \log\log n) e 2 e^{\log\log n/(1+\log\log n)}\\ & \le (1 + \log\log n) e 2 e.\end{aligned}

    Finally, for n \ge 3,

    \begin{aligned}S_n &< 2e^2 \left(\frac{1}{\log\log n} + 1\right) \log\log n \\& \le 2e^2 \left(\frac{1}{\log\log 3} + 1\right) \log\log n \\& = C \log\log n,\end{aligned}

    where the constant C = 2e^2(1/\log\log 3 + 1) \approx 172.

    Tight Bound

    Combine the lower and upper bounds to find

    \frac{1}{2}\log\log n < S_n \lesssim 172 \log \log n,

    and so

    S_n = \sum_{p\le n} \frac{1}{p} = \mathcal{\Theta}(\log \log n).

  • There and Back Again

    John F. Lindner

    I awoke yesterday at dawn in a log cabin in Vermont. Fortunately, the wifi was good.

    Each successive test of the SpaceX Superheavy Starship has been a significant improvement over the previous one, and test five was no exception, with both the booster and the ship demonstrating soft pinpoint landings — except this time, the booster falling from the edge of space was caught in mid air by the mechazilla launch tower itself!

    Software controls and coordinates the booster’s position, velocity, and orientation, as well as the chopstick‘s height and opening angle, moving a single point in an abstract parameter space, to smoothly join booster to tower. This thrilling and unprecedented engineering achievement was another big step toward perfecting history’s most powerful and reusable launch system, a potentially transformative technology.

    Mechazilla captures Superheavy booster
    SpaceX Texas Starbase launch tower catches a Superheavy booster returning from the edge of space after propelling a Starship to a pinpoint soft landing in the Indian Ocean, 2024 October 13. (SpaceX multiple exposure photograph.)

  • Rey’s Theme

    John F. Lindner

    Yesterday, as part of the Polaris Dawn mission, SpaceX engineer Sarah Gillis became the youngest person to walk in space. Today, on a space-qualified violin, she performed Rey’s Theme, composed by John Williams as the musical leitmotif for Rey, the central character in the Star Wars movie The Force Awakens.

    The performance audio and video were transmitted to Earth via SpaceX’s Starlink satellite network and were accompanied by performances of Earth-based musical ensembles. As space travel expands, musicians, artists, and poets will increasingly travel and live beyond Earth.

    Sarah Gillis playing violin in Dragon spacecraft
    Sarah Gillis plays a Star Wars theme aboard the Polaris Dawn SpaceX Dragon spacecraft during orbital night.

    Sarah Gillis playing violin in Dragon spacecraft
    Sarah completes the performance during a daytime pass. (Polaris Dawn experiences a new dawn about every 106 minutes.)

  • Skywalker

    John F. Lindner

    Up before dawn this morning for the Polaris Dawn space walk, the first commercial space walk and the furthest from Earth since the Apollo program over half a century ago. After stalling for so long, human space flight is again advancing.

    Polaris Dawn’s Commander Jared Isaacman, Pilot Scott “Kidd” Poteet, Mission Specialist Sarah Gillis, and Mission Specialist and Medical Officer Anna Menon had already broken the 1966 Gemini XI altitude record for Earth-orbital missions (excluding Apollo cis-lunar missions). This morning, the crew vented the SpaceX Dragon Resilience‘s cabin, exposing all four of them to the vacuum of space, realizing the first four-person spacewalk, and eclipsing the 1992 STS-49 mission, which included history’s only three-person spacewalk.

    At orbital sunset and near orbital apogee, Isaacman then floated outside to the Dragon’s skywalker mobility aid. For about 12 minutes, he tested the new SpaceX extravehicular activity (EVA) space suit in a carefully practiced choreography. Gillis followed next during orbital night and repeated the same tests, becoming the youngest person to perform a spacewalk, younger even than the very first spacewalker, Alexei Leonov. Later today, in yet another milestone, the crew transmitted images via high-bandwidth laser light beams to SpaceX’s Starlink satellite network.

    Excitedly and inspirationally, Gillis and Menon are the first SpaceX engineers to fly in space. They have been intimately involved in human spaceflight development and training and will soon return to Earth to leverage their experience and help accelerate SpaceX’s progress toward making life and consciousness interplanetary.

    Commander Jared Isaacman tests the new SpaceX EVA suit at orbital sunset near apogee.
    Commander Jared Isaacman tests the new SpaceX EVA suit at orbital sunset near apogee. All four astronauts participated in the spacewalk by being exposed to the vacuum of space.

    Polaris Dawn Crew
    From left to right, the Polaris Dawn crew is Sarah Gillis, Kidd Poteet, Jared Isaacman, and Anna Menon. Gillis and Menon are the first SpaceX engineers to fly in space

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