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beamjockeyYou may be familiar with social-network "games," of which the best-known example seems to be Six Degrees of Kevin Bacon, though an older predecessor was the Erdős number of a mathematician.
Once upon a time, 3 May 1998 to be exact, Graham Nelson posted a wonderful article to Usenet's sci.space.history. He has given me permission to reproduce it here.
Mr. Nelson's excellent analysis engendered some comment threads, here (responding to an earlier version considering U.S. astronauts only) and here.
Once upon a time, 3 May 1998 to be exact, Graham Nelson posted a wonderful article to Usenet's sci.space.history. He has given me permission to reproduce it here.
From: Graham Nelson Subject: Six Degrees, U.S. & Russian Date: 1998/05/03
Message-ID: X-Deja-AN: 349892238 X-NNTP-Posting-Host: gnelson.demon.co.uk:194.222.103.187 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII X-Complaints-To: abuse@demon.net X-Trace: news.demon.co.uk 894222773 nnrp-07:26699 NO-IDENT gnelson.demon.co.uk:194.222.103.187 Organization: none MIME-Version: 1.0 Newsgroups: sci.space.history
Six Degrees of Neil Armstrong
-----------------------------
"I've danced with a man,
Who's danced with a girl,
Who's danced with the Prince of Wales..."
(song lyric, quoting Herbert Farjeon (1887-1945))
-----
Contents
1. The Rules
2. Can the game always be won?
3. Distances and centrality
4. The Armstrong Number
5. Six Degrees of Orville Wright
Appendix A. A One-Page History of Manned Spaceflight
B. Statistics to 3 May 1998
C. Proof of Main Theorem
-----
1. The Rules
In the game "Six degrees of Kevin Bacon", the player is given the
name of an actor -- Alec Guinness, say -- and challenged to think
of a series of films connecting him to Kevin Bacon, each connection
by a common cast member.
In the closely related game of "Six degrees of Neil Armstrong",
manned spaceflights replace films, those people aboard (henceforth
called "astronauts" regardless of nationality or employment status)
replace cast-members and Mr Neil Armstrong replaces Mr Kevin Bacon.
Calculating frivolous statistics replaces any more sensible
activity.
A "spacecraft" is any crewed vehicle launched with the intention
of reaching an altitude of 100 miles. (Early X-program rocket
planes do not count.) Crews from different craft are considered
merged into one crew only if they have crossed hatches into each
other's pressurised interior. So although Gemini 6 and Gemini 7
approached within less than twelve inches, and the four astronauts
remarked on each other's beards as seen through the windows, they
were never one crew. Similarly for the Shuttle's inspection-only
rendezvous with Mir. Astronauts on their first launch are called
"rookies".
-----
2. Can the game always be won?
"Neil Armstrong to Alexei Leonov" is an example where the player
can win, by nominating:
Alexei Leonov
~ Tom Stafford (Apollo/Soyuz)
~ John Young (Apollo 16)
~ Mike Collins (Gemini 10)
~ Neil Armstrong (Apollo 11)
which pleasingly joins the first man to walk in space, who then
drew a picture of it in the log-book with coloured pencils, with
the first to walk on the Moon, who was photographed only twice,
badly and by accident. The notation ~ abbreviates "was once in
the same crew as".
But the player is certain to lose if challenged to join Yuri
Gagarin to Neil Armstrong, because Gagarin flew only once, and on
his own. Gagarin is "related" to nobody: no astronaut can be
reached from him.
Main Theorem. Every astronaut is related to every other, except:
(i) The following are related to nobody at all:
John Glenn (Mercury 6) Yuri Gagarin (Vostok 1)
Scott Carpenter (Mercury 7) Gherman Titov (Vostok 2)
Valentina Tereshkova (Vostok 6)
Georgii Beregovoi (Soyuz 3)
(ii) Two pioneers are related only to subsequent rookie crews:
Al Shepard (Mercury 3) ~ Stu Roosa ~ Ed Mitchell (Apollo 14).
Pavel Popovich (Vostok 4) ~ Yuri Artyukhin (Soyuz 14).
(iii) Four rookie crews are related only to themselves:
Vladimir Komarov ~ Konstantin Feoktistov ~ Boris Yegorov
(Voskhod 1).
Gerald Carr ~ Ed Gibson ~ Bill Pogue (Skylab 4).
Gennady Sarafanov ~ Lev Demin (Soyuz 15).
Vyacheslav Zudov ~ Valery Rozdestvensky (Soyuz 23).
Notes.
(i) In a few months' time, John Glenn will fly on STS-95
and cease to be an exception, leaving Scott Carpenter the
sole American astronaut never to have flown with any other.
(iii) Feoktistov and Yegorov were civilians sent up an an
experiment: one was an engineer, the other a doctor and
neither was very fully trained. Komarov would have had a
career in cosmonautics had he not been tragically killed in
the re-entry of Soyuz 1, of which he was the sole crewman.
Although Ed Gibson did not fly again, he did become head
of NASA's scientist-astronauts in the early 1980s.
It follows that of the 383 astronauts to date, all but 21 can
be joined to Neil Armstrong by an astute enough player.
-----
3. Distances and centrality
The "distance" between two astronauts is the length of the shortest
chain between them. The largest distance on record is 12 missions
long and occurs only between these two astronauts:
Vitaly Zholobov
~ Boris Volynov (Soyuz 21)
~ Alexei Yeliseyev (Soyuz 5)
~ Nikolai Rukavishnikov (Soyuz 10)
~ Vladimir Lyakhov (a Salyut 6 crew overlap in 1979)
~ Valery Polyakov (Soyuz TM 6)
~ Ulf Merbold (a Mir crew overlap in 1994)
~ John Young (STS-9)
~ Michael Collins (Gemini 10)
~ Neil Armstrong (Apollo 11)
~ David Scott (Gemini 8)
~ James McDivitt (Apollo 9)
~ Edward White (Gemini 4)
It is sad to relate that to almost all modern astronauts, Russian
or American, the most distant relation is usually either Ed White,
lost in the Apollo 1 ground fire, or Georgi Dobrovolsky, lost in the
re-entry of Soyuz 11, the first space station flight in history.
One measure of the "centrality" of an astronaut -- his degree of
involvement, if you like -- is the maximum distance to any other.
Thus Zholobov and White have centrality 12, which is highly non-
central. The most central are a whole cluster of career astronauts
involved in the Shuttle/Mir dockings, on both sides, sharing a
centrality rating of 6.
-----
4. The Armstrong Number
The "Armstrong number" of an astronaut is the distance to Neil
Armstrong, i.e., the shortest way the player can win the
Six Degrees game from that astronaut.
By definition only Neil Armstrong has A = 0.
Only Michael Collins, Dave Scott and Buzz Aldrin have A = 1.
Six different astronauts have A = 2, of whom John Young is the
only one to have flown on the Shuttle. The lowest Russian A-number
is 4, held by Alexei Leonov and a number of others.
The highest Armstrong number on record is A = 9, held jointly
by four Russians:
Andrian Nikolayev (Vostok 3, Soyuz 9)
Georgi Dobrovolsky (Soyuz 11)
Viktor Patsayev (Soyuz 11)
Vitaly Zholobov (Soyuz 21)
Within the U.S. ranks, the highest Armstrong number is now 7, shared
as one might expect by the five first-time crew members of the current
Shuttle flight, STS-90 or "Neurolab". Here's the shortest link between
one of them and Armstrong:
Scott Altman (the pilot of STS-90)
~ Richard Searfoss (STS-90)
~ John Blaha (STS-58)
~ James Buchli (STS-29)
~ Thomas Mattingly (STS-51C)
~ John Young (Apollo 16)
~ Michael Collins (Gemini 10)
~ Neil Armstrong (Apollo 11)
Mr Altman might yet see his degree drop as low as 5, if for instance
he should one day command a flight carrying Michael Foale, who shares
the record for the lowest degree of any currently active astronaut:
Michael Foale (first flight STS-45)
~ Byron Lichtenberg (STS-45)
~ John Young (STS-9)
~ Michael Collins (Gemini-10)
~ Neil Armstrong (Apollo-11)
Conceivably, if the ESA astronaut Ulf Merbold flies again, then
Altman could even drop to A = 4: Merbold's low A-number of 3, and
the fact that he stayed on Mir in 1994, means that current Mir crews
often have A-numbers about the same as those of the Shuttle crews
being sent up to dock with them. Some measure of the degree of
mixing caused by these dockings is that almost every astronaut of
the 1990s, of either nationality, has A between 4 and 6.
For NASA astronauts of the pre-Shuttle era the record is A = 6,
shared by Gordo Cooper, Joe Kerwin and Paul Weitz. When STS-95
flies, John Glenn will share this record with them.
-----
5. Six Degrees of Orville Wright
In a sense, the inter-relatedness of astronauts today is typical
of all acquired multi-person skills. If we had taken aeroplanes
instead of spacecraft, and pilots instead of astronauts, then
almost every pilot in history would be related to Orville Wright:
since the law of every nation requires trainee pilots to have
experienced ones sitting next to them, this will always be so.
Nearly all drivers are probably related to Henry Ford. Nearly
all concert pianists to J. S. Bach.
The interesting point is that being an astronaut seems to be
considered something that one learns by experience. This isn't
obvious. With a new vehicle, it may be better to train entirely
fresh recruits, nor is it clear that fighting-fit test pilots are
necessarily better stuff than plump, short-sighted spacecraft
designers: the Russian experience with space stations has been
that both are useful.
But I doubt very much if the cork will go back into the bottle.
Will there ever be more astronauts unrelated to Neil Armstrong?
Perhaps, if China builds a primitive Vostok-style one-man rocket;
perhaps also if rich enthusiasts succeed in building a working
orbital vehicle before their government succeeds in making them
undergo proper training. But I suggest that essentially every
astronaut who ever lives will be related to Armstrong, with
A-numbers which slowly -- but surprisingly slowly -- rise over
the decades.
Graham Nelson
3 May 1998
-----
Appendix A. A One-Page History of Manned Spaceflight
Only the U.S. and the USSR/Russia have launched people into space
or are likely to in the near future, though China, Japan and
Europe all announce this intention from time to time.
In Russia, one-man Vostok capsules 1961-3 were soon replaced with
craft capable of seating either two or three cosmonauts: briefly
Voskhod, 1964-5, but ever since 1967, variants of Soyuz. Russian
aims centred on long-duration flights aboard the 1973-85 Salyut
series of space stations, from 1986 on the modular, expanding Mir.
Permanent crews of two are visited by Soyuz capsules occupied by
two professional cosmonauts and one "guest", who would stay only
briefly. 1970s and 80s guests were InterCosmos air-force officers
from the Warsaw Pact, India, Syria and France. In the wake of the
Soviet collapse, paying guests ranged from a Japanese journalist
to an English chocolate-bar scientist; a practice dropped in 1994
when a third permanent berth was created instead, occupied by an
ESA or NASA scientist-astronaut.
The U.S. programme began with one-man Mercury capsules, 1961-3,
then two-man Gemini capsules 1965-6, then three-man Apollo 1968-75:
these visited the Moon, the Skylab space station and finally
docked with Soyuz 19 in the Apollo/Soyuz Test Project (ASTP).
America then lost access to space until 1981, when the Shuttle
became operational. Its crew is usually 5 to 7 and it has since
flown 90 times, carrying guests from friendly countries,
universities, aerospace companies and, twice, the U. S. Congress
committees funding NASA. In 1995, with STS-71, the Shuttle finally
found something to dock with: Mir.
-----
Appendix B. Statistics on Manned Spaceflight to 3 May 1998
383 different people have been astronauts, filling a total of
789 assignments to 208 manned launches; the "average" crew
thus has 3.80 members and the "average" astronaut has been
on 2.06 flights.
There have been 48 dockings of differently-launched crews:
the ASTP, eight of Shuttle with Mir, eleven of Soyuz with
Salyut 6, five with Salyut 7, twenty-three with Mir.
The current statistics for astronaut experience are:
Launches Number of astronauts
6 3 (*)
5 12
4 35
3 69
2 100
1 164
total 383
(*) John Young, Story Musgrave, Gennady Strekalov.
Almost all 6- and 5-launch astronauts are retired, as are
many of the 4-launch astronauts. Many 1-launch astronauts
are one-flight-only guests or from minor space programmes:
Europe, Japan, Canada, the Ukraine and so on.
The most experienced crew ever launched was on STS-80, whose
members were, on average, on their 3.6th flight; which did
not help them to get the EVA hatch door open, though the
mission was otherwise a success. The next most experienced
was Soyuz T-13, averaging 3.5, two experts sent up to the
desperately stricken Salyut 7 in order to revive it.
The average Shuttle crew's experience level is 1.95.
There have been 31 all-rookie crews, but that is to count 14
early one-man missions; once Apollo became operational,
the only two U.S. rookie crews were Skylab 4 and STS-2;
once Soyuz did, there were ten on the Russian side.
(Russia has often placed technical expertise, or military
rank, over experience.) The last two were Soyuz 25, 1977
and Soyuz TM-19, 1994. Rookie missions have tended to be
unlucky: Soyuz 15, Soyuz 23, Soyuz 25 and STS-2 all came
down in a hurry, for instance, while Skylab 4 was an unhappy
experience almost leading to a mutiny.
-----
Appendix C. Proof of Main Theorem
The Russian part is best checked by computer, but isn't too
surprising, given (nearly) permanently manned space stations
since 1980 or so and many opportunities for crew overlap. The
American half can just about be demonstrated by hand, and
without needing to use any Russian missions.
Write A ~~ B if a chain of missions can be constructed between
A and B.
Every Shuttle mission from STS-7 to the present day has been
commanded by someone who had already flown aboard the Shuttle.
Thus every Shuttle astronaut ~~ a crew member of one of
STS-1, STS-2, STS-3, STS-4, STS-5 or STS-6.
STS-1 was commanded by John Young.
STS-2 was commanded by Engle ~ Covey (STS-51 I) ~ Hauck (STS-26)
~ Crippen (STS-7) ~ Young (STS-1).
STS-3 was piloted by Fullerton ~ Musgrave (STS-51 F)
~ Covey (STS-51 I) hence ~~ Young as above.
STS-4 was commanded by Mattingly ~ Young (Apollo 16).
STS-5 was commanded by Brand ~ Stafford (ASTP)
~ Young (Apollo 10).
STS-6 was commanded by Weitz ~ Musgrave (STS-6) hence ~~ Young
as above.
Thus, every Shuttle astronaut ~~ John Young. Which of the
earlier ones do?
Young
~ Collins (Gemini 10)
~ Armstrong (Apollo 11)
~ Scott (Gemini 8)
~ McDivitt (Apollo 9)
~ White (Gemini 4)
~ Schweickart (Apollo 9)
~ Worden (Apollo 15)
~ Irwin (Apollo 15)
~ Aldrin (Apollo 11)
~ Lovell (Gemini 12)
~ Borman (Gemini 7)
~ Anders (Apollo 8)
~ Swigert (Apollo 13)
~ Haise (Apollo 13)
~ Grissom (Gemini 3)
~ Stafford (Apollo 10)
~ Schirra (Gemini 6)
~ Eisele (Apollo 7)
~ Cunningham (Apollo 7)
~ Brand (Apollo/Soyuz Test Project)
~ Slayton (Apollo/Soyuz Test Project)
~ Cernan (Apollo 10)
~ Evans (Apollo 17)
~ Schmitt (Apollo 17)
~ Mattingly (Apollo 16)
~ Duke (Apollo 16)
More deviously,
Young
~ Garriott (STS-9)
~ Lousma (Skylab 3)
~ Bean (Skylab 3)
~ Gordon (Apollo 12)
~ Conrad (Apollo 12)
~ Cooper (Gemini 5).
This leaves only Glenn, Carpenter, Shepard, Roosa, Mitchell,
Carr, Gibson and Pogue, who divide up as claimed above.
That demonstration exaggerates the importance of John Young:
if he had retired in the late 70s and never flown STS-1 and STS-9,
the theorem would still be true (though trickier to prove). Even
if he, Mattingly and the other Apollo survivors had all retired,
the theorem would be true thanks to links between Apollo and the
Russian Soyuz programme, and subsequently between Shuttle and the
Soyuz/Mir programme.
--------------------------------------------------------------------
--
Graham Nelson | gra...@gnelson.demon.co.uk | Oxford, United KingdomMr. Nelson's excellent analysis engendered some comment threads, here (responding to an earlier version considering U.S. astronauts only) and here.
