40 Eridani

40 Eridani is the designation of “a star” ( actually a 3 star system ) in the constellation Eridanus. One of these 3 stars is a White Dwarf, and famously the most easily accessible of this type to amateur observation.

All this became known to me in a recent conversation, and I noted, via my Starry Night software, that the current viewing window was drawing to a close this month of March, so I determined to get a photograph of it, as I have found that photography shows a lot that is undetectable at the eyepiece.

I use prime focus projection using my Nikon D5100 camera body mounted with an adapter on my Orion StarMax 127 mm, with a 1540 mm focal length. In this configuration it’s essentially a very long focal length camera lens, with an upright image through the view finder.

I have a “clocked” mount and I take a maximum 30″ exposure using a remote shutter release. This is pretty low end stuff compared to the deep sky shots you see with much more sophisticated amateur equipment and processing.

I can get good results, though, and in this case, the picture came out well.

40 Eridani

The bright yellowish star at lower right with the smaller white companion is 40 Eridani and the companion is the white dwarf. If you “click to enlarge”, you can easily see the small reddish close companion of the white dwarf, which I did not really expect to resolve, so of course I’m quite pleased with the picture.

Here’s an enlarged excerpt, just to celebrate.

40 Eridani close-up

There is a picture of 40 Eridani in Volume Two of the classic Burnham’s Celestial Handbook, called there Omicron 2, as “omicron” was the sequential designation, after alpha, beta, etc. of a pair of stars in the constellation Eridanus ( The River. ) Omicron 1 is not physically associated with the Omicron 2 system, aka 40 Eridani.

Just to complete the confusion, my Starry Night software shows Omicron 1 and Omicron 2 as “Beid” and “Keid”, which are “traditional star names”, although I read that the tradition in this case was officially adopted in 2016 ! Of course, Burnham’s does not use these names.

Anyway, the picture there is from an exposure at Lowell Observatory, of Pluto fame, taken around the same time as that discovery.

I took my photo and “pushed” the brightness and contrast to the maximum to bring out the background stars, then did a match-up with the photo illustration from Burnham’s. Here is a “blink comparison” using the Free Online Animated Gif Maker:

… click to run the animation.

It’s a little confusing at first glance, but hold your gaze on the stars above 40 Eridani ( in this orientation ) and you will see that they match almost perfectly. However, 40 Eridani “jumps up and down” .

I wasn’t looking for this and it threw me for a loop, but in reality, this is a record of the actual “proper motion” of this triple star system, relative to our solar system. In fact Burnham’s describes this motion and has a pair of illustrations showing the position circa 1934 and 1965.

I pasted these into one and added my observation ( in blue ) …

proper motion of Eridani 40

The blue imagery is a replication to scale taken from my high contrast image used in the blink comparison.

More on Proper Motion

The term “proper motion” in my experience, refers to the apparent angular movement of an object against an ideal inertial reference system centered on the sun. However, one might think that it ought to include the “third component” of the relative velocity, i.e. what is termed “radial velocity”.

Nevertheless, I think historical usage has treated these separately. This caused me some confusion in reference to a statement in Burnham’s that the “true space velocity is about 62 miles per second.”

I took this at face value at first, and assumed that the value must include the component of “radial velocity”, later mentioned. However, I don’t believe this is the case. The discussion was of “proper motion” as traditionally defined, so this “true velocity” was simply a conversion from angular to linear velocity along the apparent path in the sky.

The determination of radial velocity is a whole different question, since it involves physical analysis of the received light, and not any apparent motion of the object.

The question is mooted once we have the “proper motion” expressed in arcseconds per year, along with the “radial velocity” expressed in km/sec, or any such units for that matter.

In fact we may note that 186,000 = 62 X 3000, and I think this indicates that the value of 62 miles per second was being given as a “round number” for the transverse linear speed of 40 Eridani … i.e. c/3000 .

… but all this is a distraction! I just want to get an idea of the past and future apparent motion of 40 Eridani, and this can be based on a given specification of the proper ( angular ) motion, and the radial velocity.

With the understanding that the linear approximation won’t hold up forever, we see that we have a simple freshman physics problem of relative velocity between two objects, in this case the sun and 40 Eridani.

The trajectory of 40E wrt the sun is then a “great circle” in the sky, and wrt this path it is determined by the general equation for any such “fly by”, within a scale factor only for time.

The “zero point” of this trajectory is the point of closest approach, which provides the distance scale, and the time scale of the angular displacement is determined by this distance divided by the speed of the object.

That only leaves the determination of the point of closest approach of 40E to the sun, expressed as an angular displacement along its apparent path from its current position. I hope this diagram explains everything!

Here is my estimation of the apparent path in the sky of 40 Eridani over the next 19000 yrs, in accordance with the diagram above, and overlayed on a view from my Starry Night software:

Meet the Monster

That monster is the star, Deneb, familiar in the constellation Cygnus in the northern sky. It is among the brightest stars in the sky, but what cannot be obvious is that it far outstrips its visual rivals in intrinsic brightness, so that it is the bully in our neighborhood. It is a magnitude 1.25 star at a distance of 800 parsecs.

Now, the Sun has an absolute magnitude of 4.83, which is by definition its ideal visual magnitude at a distance of 10 parsecs. So to have a visual magnitude of 1.25, the Sun must appear 10^(.4*(4.83-1.25)) = 27 times brighter than it does at 10 parsecs. That means it must be 1/sqrt(27) times as distant , or about 3.7 parsecs away, or 216 times closer than Deneb, to show the same visual magnitude.

So, conversely, the earth would have to be orbiting Deneb at 216 AU if Deneb were to have the same apparent brightness as the Sun. That’s over five times the distance of Pluto from the Sun, which averages about 40 AU.

The period of an orbiting body is given by M/r^2 ~ r/T^2 or T ~ sqrt( r^3/M ), which embodies Kepler’s third law for a central body of a given mass, but allows us to account for a variable mass. Deneb has about 19 solar masses, so the “year” of this Deneb-earth would be sqrt( 216^3/19 ) = 728 earth years. Well, at least that’s comprehensible!

I was led into these contemplations after having taken the following photograph of Deneb, with its star background in the Milky Way, using my Nikon D5100 mounted on my telescope purely for its use as a “clocked” mount. The exposure is about 20 seconds, so the lack of guidance is not so bad, as long as I get reasonable alignment, for which purpose I have a well practiced system! … ( click to enlarge )


Slumming Andromeda

I’ve been doing prime focus photography with my Nikon D5100 and my Orion Starmax 127, millimeters, that is. The season has come around for the Andromeda Nebula, as I persist in calling it, and just tonight I got a good shot of it.

I had just been out the night before, but my shots were way out of focus, a chronic problem I have since the light is way too dim for autofocus, and the viewfinder is too small to visually discern a good focus in this light. Determined not to be thwarted, I went out again tonight with my aged Asus Eee, which I used to view my exposures and make iterative adjustments. Rather laborious, but effective.

So I got my in-focus shot of Andromeda. 30 seconds at ISO 6400 in a reasonably dark sky. The result looks pretty impressive on the viewfinder, as a nice bright blob. But let’s not get carried away! As might be expected, this is very down scale stuff.

Here’s an animated gif comparing my shot with a “pro” internet image ( Click to view ):output_GhKn2V

That’s M32 to the right, Andromeda itself being M31.

Well, there is actually detail visible in my exposure which is washed out in the bright core by the more sensitive exposure. You could say that, but really, it’s nice just to be out there and see it and do it yourself.

Where is Dawn Now?

The JPL DAWN mission page has a button for Where is Dawn now?, which is in a drop-down menu if you click on Mission in the main menu on the left of the home page. Dawn of course is the spacecraft that visited the asteroid Vesta, and is now fast approaching Ceres.

If you click on the graphic for Simulated View of Ceres from Dawn you get something like the following screen shot, from Feb 2, 2015
Well, this tells us that it is approaching Ceres, in the middle there, but what stars are those? I couldn’t recognize them, and it was driving me crazy. I thought if I tracked it for a while, I’d see something I recognized, but this got me nowhere.

Note the credit to MYSTIC simulator. This is a tool they hold pretty close to their vest. It is apparently very powerful and used in their actual trajectory planning. Searching for it, I found a link to the NASA/JPL SOLAR SYSTEM SIMULATOR, which is nice, but it provides a much different format, and I was not getting anywhere with it either. Since it was all I could use, I went back a year or so and looked at some wide fields of view, and I was able to track it back to the present, where the background was identifiable as being near the head of Draco, in our Northern sky, but I still had no luck correlating the stars to the MYSTIC image.

The reason for this, I finally realized, is that the SSS tool had not been kept up to date with the latest course corrections ( I presume ) as it gives a different distance from Ceres than MYSTIC, which is official, I guess.

It was finally just dumb luck and persistence that enabled me to spot the correlating stars in my Starry Night software, and I’m sure you’ll agree it’s a good match:


Note that the four stars of the almost-square “box” that Ceres is in ( and remained in for several days, ) is comprised of stars from three different constellations. Two in Serpens Cauda, and one each in Scutum and Ophiucus. A very obscure asterism !

Update: Feb 13

Here’s the latest WIDN image:Feb13

Ceres has “moved” ( in line of sight from DAWN ) “southward” ( in celestial coordinates ) into the constellation of Sagittarius. You might recognize the Sagittarian “teapot” asterism directly behind the bottom half of the representation of the DAWN space craft.

Now here’s the funny thing … DAWN is currently almost directly in line with Ceres from our terrestrial POV, as evidenced by this Starry Night view with Chicago set as the viewing location. Ceres is marked, along with it’s orbit. I adjusted the time to make the view align with the WIDN representation, and this view happens to occur at 9:46PM local time, on Feb 13, 2015. ( I did have to turn off the horizon mask, as this view is looking down through the ground at the given hour. Sagittarius rises just ahead of the sun at this season. )

I think this alignment has to be ascribed to coincidence, except that it does mean that DAWN is “rising” towards Ceres along an orbit which is “sunward” of Ceres’ orbit.


Update Feb 19

Here is an animated gif of a loosely spaced sequence of WIDN images, displaced to show a constant star background. This POV definitely gives the impression ( correct, I believe ) that DAWN is rising up through the ecliptic plane as it closes with Ceres. We can see Ceres sinking to the south below Sagittarius. It’s only going to get more interesting!output_4fYhUU

Feb 20 – A note on escape velocity

The Wikipedia article on Ceres gives a radius of 476 km and an escape velocity of .51 km/sec . This is the speed which gives an object a kinetic energy equal to the negative of the gravitational potential at this distance:

m v2/2 = m GM/R so v2/2 = GM/R

then the escape velocity at some r > R, say, is

vr = vR sqrt( R/r ) … very simple!

We note at Feb 20 03:16:52 we have r = 44580 km and v = .082 km/sec

but .51 km/sec sqrt(476/44580) = .053 km/sec

so DAWN is not gravitationally bound to Ceres at this point.

Note that the exact direction of the velocity is not important. Varying the direction just places the object with escape velocity at different places on various parabolic orbits.

Also note that the engine is not firing at this time. It seemed to be in braking mode the last few days, so has it shed any total energy?

On Feb 9 05:06:33 DAWN had r = 107010 km and v = .096 km/sec

The escape velocity at that distance is .51 km/sec sqrt(476/107010) = .034 km/sec

So the fact that it has shed considerable speed shows that it has gotten much closer to gravitational capture. If it were “drifting” it would have gained kinetic energy equal to the difference of the escape velocities at the two distances, requiring …

(v + delta_v )2 – v2 = .0532 – .0342

or v + delta_v = .104 km/sec

One more thing, braking with the same rocket is more effective at higher speeds, and one usually expects that a braking rocket will fire near close approach. The very low thrust of the ion rocket means that the approach has to be more “gentle”, which accounts for the braking having been in progress already. Let’s watch and see if DAWN does pick up a little more speed before the braking resumes, as it must.

… sure enough, DAWN is at 184 mph at Feb 20 13:22:16 vs. 182 mph at Feb 19 23:50:16
Kind of ironic that they give a more precise speed in mph vs. km/s, which is .08 for both times. 1 mph is ( exactly ) .00044704 km/s

Feb 21 – The Approach Begins

Here’s a diagram I drew “by hand” in MS Paint showing the approach of DAWN to “Ceres Space” I started with the MYSTIC graphics for the times shown. I used the given distances and measured the angle of 5 degrees from the background stars, using my STARRY NIGHT software. This gave me the blue line and the close approach distance of a straight line trajectory.

But what about the gravitational attraction of Ceres? It seems that DAWN is not yet bound to Ceres, but it should be on a discernible hyperbolic trajectory.

After some thrashing around with pencil and paper, I consulted my old copy of Marion’s CLASSICAL DYNAMICS and found the equation laid out for me, and highlighted to boot!

I used my “hand built” yacc based calculator which features a plug-in definition capability. ( It actually pushes the defining string into the input stream when the defined symbol is encountered. )

Here are the defines I used:

define k “g*R^2”
define l “v*rp”
define alpha “l^2/k”
define E “v^2/2-k/r”
define e “(1+2*E*l^2/k^2)^0.5”

“mu”, for the “reduced mass” can be ignored when one body is much less massive than the other, m << M . Then mu ~= m and it cancels out of the expressions for alpha and e.

k is then GM, but I used GM/R2 = g to get k = g R2

Then setting the variables:
g= 0.00028
v= 0.083
rp= 39700
r= 45000
R= 476

( all in kilometers and seconds ) I just typed “alpha” and “e” to get the orbit parameters used in Marion’s equation 10.32, shown above:


Then the distance of close approach and the angular displacement from the current position are given by:

180/pi * acos( (alpha/r – 1)/e )

Perhaps surprisingly, these make sense! It says DAWN would approach to 38400 km at an angular displacement of 36 degrees from the current position, assuming it continued to “coast”.

Here’s the result roughly drawn in purple on the diagram. Of course, this is all in the realm of rough estimation!



… Well, I say we are! According to a definite criterion that I will show you in this most recent animated gif: ( click to see the animation. )

The bright star at top right is Fomalhaut in Piscis Austrinus. It’s actually visible on the southern horizon from mid-northern latitudes, but the POV is shifting to the south. and that’s the constellation Phoenix at the bottom.

But to make my point, I draw your attention to the “Distance to Ceres” caption at the bottom. Note that it “bottoms out” right at 24000 miles, and starts to increase again. DAWN is passing through “periceris” or whatever it might properly be termed. That is, a closest approach to Ceres. Of course, as it continues to “shed energy” it will spiral closer, but right now it is retaining the semblance of an hyperbolic orbit.

As we learned in physics class, the energy of a body at rest infinitely far from a gravitating body is conventionally set to ZERO. If it falls toward a gravitating body it gains kinetic energy but loses potential energy as it sinks into the “gravitational well”. So, any body with net positive energy can escape to infinity with a finite velocity, and is not bound to the gravitating body.

I say all this because the net energy of DAWN ( per unit mass ) is easy to calculate from the MYSTIC captions, and we can gauge its approach to gravitational binding by the steady diminishment that we see in this value, which is equal to 1/2 V_inf2, where V_inf is magnitude of the aforementioned “velocity at infinity”.

The value of V_inf was 115 mph in the last frame of the animation at Feb 24 06:33:54 . As of Feb 25 05:34:47, V_inf was down to 107 mph

March 10 – APOAPSIS

The “arrival” or attainment of zero orbital energy, was well noted on March 6, but there was nothing in the motion or actions of the DAWN spacecraft that would have indicated this event, such as it was. It has been “firing” its thruster steadily before during and after this milestone, as it continues to work towards its Survey Orbit. DAWN is now approaching another event, which has a more obvious connection to  its motion. That is apoapsis, as it is generically termed, or maybe apocerium, if we follow Kepler’s coinage of apohelium, which was later “greekified” to the now conventional aphelion.

As an aside, this question of what to call the “apo” point of the orbit has been bubbling along for decades, if not centuries. I only knew of aphelion and apogee, to be honest, but now I’m seeing that apoapsis, that is apo-apsis, is supposed to be the standard term, as awkward as it is. Investigating, I find that apogee derives from apogaeum, a nineteenth century analogue to Kepler’s apohelium, based on gaia, for the earth, I presume.

Of course, I became familiar with apogee from the era of “satellites” in the 50’s and 60’s, when I was quite young. I don’t see why the term can’t be applied to any planetary body, just as one speaks of Martian geology.

Well, at any rate, we are approaching the turning point of DAWN from its continuing motion away from Ceres ( despite it’s inevitable capture ) into a motion falling towards it. Following the Where Is Dawn Now? graphic on the DAWN mission page, we see that its motion is at a low point, suggesting a thrown object at its apex.. In the approximately six hours between Mar. 10,2015 00:03:54 and Mar. 10,2015 06:01:34, DAWN has decelerated from 78 mph to 76 mph, and moved 290 miles further from Ceres; from 43060 miles away to 43350 miles away. These are numbers appropriate to highway driving, not space travel!

So what happens when it “makes the turn” and starts falling towards Ceres ? I anticipate that they will turn off the thruster for a while and let it fall, but we’ll see.

March 17 – Making the Turn

DAWN is still approaching apoapsis, but it is very near, as its speed relative to Ceres of a mere 40 mph would indicate. Here is an animated gif comprised of  1 frame per day for the 14 days March 4 thru March 17. I think you can see a slight bow in the arc of the apparent motion of Ceres against the constellation Orion as DAWN adjusts the plane of its orbit. I did the best I could to keep the star background constant, but its still not perfect, as you can see. I hope it’s good enough to allow you to picture the apparent motion of Ceres against the fixed stars from DAWN’s POV. The sun is almost coming into view just above Orion, as you can see by the almost completely dark disk of Ceres.

Note that the distance from Ceres increases from 55020 km to 77740 km during this interval as DAWN climbs to its apex. You can see the disk of Ceres diminish in size as DAWN recedes from it.

output_MbifyqMarch 25, 2015

On March 19 DAWN achieved apoapsis at 48740 miles from Ceres moving at 34 mph, and is now starting to move closer to Ceres. As of today at 10:33 UTC it has “fallen” to 46460 miles away, and is moving at 40 mph! Here we go!

BTW, DAWN is moving towards a point almost directly opposite the sun from Ceres. WHERE IS DAWN NOW? provides views of the sun ( and other objects ) from DAWN’s vantage point, and here is an overlay showing that the sun is just about to come into view in the Ceres frame of reference … along with the earth and mars too!


Note that I had to expand the “sun view” and slightly rotate it to get a matching overlay.



Here’s an excerpt of a sol 351 Curiosity Mastcam image catching Phobos ( at bottom ) passing by Deimos high overhead in the wee hours, local time, 2013-08-01 08:44:11 UTC as labelled in the Raw Image Gallery. You can clearly see the features of Phobos, which make it very reminiscent of the Death Star.

I can recreate this event with my Starry Night software by setting my viewing coordinates to the Bradbury landing site on Mars. However, it doesn’t seem to quite match up in detail, and I can’t even come close to identifying the background stars. So that’s a bit of a mystery to me.

I haven’t seen any publicity for this yet on the Mars Science Laboratory web site, but I guess they’ll get around to it. There was publicity for a series of Navcam photos from Sol 317 made into a movie, but that was at low resolution.

Kinematics of a close encounter

At the time of the August 2003 Mars opposition much was made of the “moment of closest approach” and I began thinking about how long this “moment” would last, within various limits, such as the diameter of Mars, or even a mile, or a foot.  The distance between the planets is not a dynamically significant measure in this case, since the force between them is small and doesn’t substantially affect their relative motion, so this is a problem in “kinematics”, meaning mere description of the motion. Furthermore, we only require an approximate formula for this distance around the time of closest approach.

With the May 31 2013 close approach to earth of “(1998) QE2”, this subject again presented itself. This time I watched a webcast of a telescopic camera image which had live commentary excitedly announcing the moment, or at least the minute of close approach, so I got to thinking about it again.

This time, I decided that a good model was a straight line pass at constant speed, and derived a formula for the distance as a function of time based on that. In 2003, I had used the approximation of circular motion obeying Kepler’s Third Law to derive the relative acceleration of Earth and Mars. Revisiting the idea, I realized that I had neglected the effect of overtaking, which my new straight line model emphasized. So I thought I should try a derivation, and putting pencil to paper, I immediately wrote down the following, and I was very pleased to see that it settled the whole thing in the simplest possible way:
… the two terms represent the kinematical “acceleration” of the distance value, and the true dynamic acceleration projected along the unit vector of the displacement between the bodies, or planets. ( Note d = |x12t0 ).

QE2 Close Approach

With this in hand we may examine the NASA ephemeris data for the close approach of (1998) QE2, which conveniently features “delta” and “deldot”, which are the distance in AUs and the time derivative of same. “dot” is here the Newtonian convention of indicating the time derivative of a variable.

This table is excerpted from results obtained with the JPL HORIZONS web interface. You’ll have to search for “1998 QE2” and then change the start and stop times, and the step size to 1 minute, then click Generate Ephemeris. There is a lot of other data in the table, as you’ll see.

In the excerpted data below, note that the deldot values are almost evenly spaced with a constant difference of about 0.0011649 km/sec, representing a pseudo-acceleration along the line of sight of 0.0011649 km/sec/minute. So the appearance along the line of sight is similar to that of a thrown ball, say, rising towards a viewer positioned at a height, then falling away. Note that the acceleration is 1.11649/60 m/sec2 or about 0.0019 g , so this is a rather slow motion affair. It takes about 1 minute for the speed of approach/recession to decrease/increase by 1 meter/second .

Date               Time ( UT )           delta ( AU )                     deldot ( km/sec)

2013-May-31 20:44:58.068        0.03917532423283        -0.0163077
2013-May-31 20:45:58.068        0.03917531792555        -0.0151429
2013-May-31 20:46:58.068        0.03917531208546        -0.0139781
2013-May-31 20:47:58.068        0.03917530671257        -0.0128133
2013-May-31 20:48:58.068        0.03917530180688        -0.0116484
2013-May-31 20:49:58.068        0.03917529736839        -0.0104836
2013-May-31 20:50:58.068        0.03917529339710        -0.0093188
2013-May-31 20:51:58.068        0.03917528989301        -0.0081540
2013-May-31 20:52:58.068        0.03917528685613        -0.0069891
2013-May-31 20:53:58.068        0.03917528428646        -0.0058243
2013-May-31 20:54:58.068        0.03917528218400        -0.0046594
2013-May-31 20:55:58.068        0.03917528054875        -0.0034946
2013-May-31 20:56:58.068        0.03917527938071        -0.0023297
2013-May-31 20:57:58.068        0.03917527867989        -0.0011649
2013-May-31 20:58:58.068        0.03917527844628         0.0000000
2013-May-31 20:59:58.068        0.03917527867989         0.0011649
2013-May-31 21:00:58.068        0.03917527938073         0.0023297
2013-May-31 21:01:58.068        0.03917528054878         0.0034946
2013-May-31 21:02:58.068        0.03917528218406         0.0046595
2013-May-31 21:03:58.068        0.03917528428656         0.0058244
2013-May-31 21:04:58.068        0.03917528685629         0.0069893
2013-May-31 21:05:58.068        0.03917528989324         0.0081542
2013-May-31 21:06:58.068        0.03917529339743         0.0093191
2013-May-31 21:07:58.068        0.03917529736885         0.0104840
2013-May-31 21:08:58.068        0.03917530180750         0.0116489
2013-May-31 21:09:58.068        0.03917530671338         0.0128138
2013-May-31 21:10:58.068        0.03917531208650         0.0139787
2013-May-31 21:11:58.068        0.03917531792686         0.0151436
2013-May-31 21:12:58.068        0.03917532423446         0.0163085

If we change the step size to 1 hour, we see that the spacing of deldot still remains close to a constant, but a systematic drift is evident, represented by a linear increase in the spacing with time, meaning the (pseudo) acceleration increases with time in the direction away from the earth POV, so the approach slows down, but the retreat speeds up.

2013-May-31 08:58:58.067       0.03929608858667       -0.8350130
2013-May-31 09:58:58.067       0.03927682497794       -0.7658902
2013-May-31 10:58:58.067       0.03925922624191       -0.6966541
2013-May-31 11:58:58.067       0.03924329499832       -0.6273135
2013-May-31 12:58:58.067       0.03922903365661       -0.5578770
2013-May-31 13:58:58.067       0.03921644441785       -0.4883535
2013-May-31 14:58:58.067       0.03920552926676       -0.4187518
2013-May-31 15:58:58.067       0.03919628997722       -0.3490808
2013-May-31 16:58:58.067       0.03918872810782       -0.2793494
2013-May-31 17:58:58.067       0.03918284500170       -0.2095666
2013-May-31 18:58:58.067       0.03917864178520       -0.1397414
2013-May-31 19:58:58.067       0.03917611936989       -0.0698829
2013-May-31 20:58:58.067       0.03917527844677        0.0000000
2013-May-31 21:58:58.067       0.03917611949120        0.0698981
2013-May-31 22:58:58.067       0.03917864276007        0.1398025
2013-May-31 23:58:58.067       0.03918284829222        0.2097040
2013-Jun-01 00:58:58.067        0.03918873590808        0.2795937
2013-Jun-01 01:58:58.067        0.03919630521201        0.3494624
2013-Jun-01 02:58:58.067        0.03920555558842        0.4193011
2013-Jun-01 03:58:58.067        0.03921648620645        0.4891009
2013-Jun-01 04:58:58.067        0.03922909601848        0.5588527
2013-Jun-01 05:58:58.067        0.03924338376138        0.6285477
2013-Jun-01 06:58:58.067        0.03925934795697        0.6981768
2013-Jun-01 07:58:58.067        0.03927698691478        0.7677314
2013-Jun-01 08:58:58.067        0.03929629872991        0.8372025

Some fairly simple considerations of the nature of our approximate calculations will show why it works as well as it does, and give a value for the “next order” of deviation from it.


Just as the Space Shuttle bay provided a “free fall” environment as it orbited the earth, the vicinity of earth is a “free fall” environment for objects near it. That is, we do not need to know what orbit they are following, but can consider them to be in free fall with the earth. Two forces limit this assumption. The first of course is the gravity of the earth itself, and the second is the “tidal” or differential field of the sun’s gravity near the earth, which in first approximation grows in proportion with distance from the earth.

Which of these is greater for our case of a close approach at about 0.04 AU ?

The force of earth’s gravity, measured by g at the earth’s surface, diminishes in inverse proportion to the square of the distance from the earth, measured in earth radii, re . Since 0.04 AU is about 940 re, We expect an acceleration due to earth’s gravity of about 1/88400 g or 0.000011 m/sec2 at that distance.

The acceleration due to the sun’s gravity at 1 AU is measured by the centripetal acceleration of the earth in its orbit, that is ( 2pi/1 year )2 1 AU, which comes to 0.006 m/sec2 . The tidal force is measured by the distance in AU times this value, or 0.000240 m/sec2, about 22 times as great as the earth’s gravitational pull. The exact direction and magnitude of the solar tide depends on the relative position of the object, just as the tides of the earth ocean vary with the position of the moon and the sun in the sky, but we may consider the value, once calculated, to be constant in the next order of approximation.

Here is a derivation extending our approximation to the “third order” in time:

The expression at the bottom has a “kinetic” term linear in the (dynamic) acceleration and a term representing a variable dynamic, or true, acceleration. If we assume a constant dynamic acceleration, the second term vanishes of course. The first term is intuitively due to the acceleration along the path perpendicular to the line of close encounter, since time, in this case, signifies only the position along that line, so we may regard acceleration as a displacement in the time scale. Got it? OK then! Actually, this is how I conceived the idea in the first place, and I did the derivation just to provide a formal justification.

Terminat hora diem, terminat auctor opus

Comet of the Century

It cleared today and I figured this was my one and only chance to observe the comet PanStarrs, so I sojourned to the site of Lew’s Garage, proper, this night of March 13, 2013. There I did observe the comet, and managed to obtain this image of it, although I must say it’s a little rough around the edges, and certainly doesn’t do justice to the entire experience. It was getting colder and the ice in the fields was freezing hard and letting me know about it with a background of crackling sounds. I might have thought I was in the arctic. I certainly felt cold enough.

It looked a little better through my Celestron 9X63 binoculars than the image would indicate, but it was fleetingly visible, as it was on the verge of being obscured on the horizon just as the sky darkened sufficiently to see it. So, all in all, a major triumph.