Solutions To Stationary Points - Practical Focus Problems

The Problems Again and their solutions:

1) Mars reaches a stationary point 36 ½ days after opposition. Its elongation is then measured to be 136.2 deg. Given that the sidereal period is 687 days, find the distance of Mars from Earth in AU. Also, find the time to its next stationary point.

Solution: First obtain Mars synodic period, S:

Using : 1/ S = 1/P1 - 1/P2

With P1 = Earth's sidereal period (1 yr), and P2 = Mars sidereal  (687/365.25) = 1.88 yr

Then:

1/S = 1 – 1/1.88 = 1 – 0.5319 = 0.468 yr

Whence: S = 1/0.468 = 2.13 yrs. (or 779.9 days)

Mars’ distance from Earth is PE from the diagram in the  original blog post, so b may be obtained from:

b - a cos Θ = PE cos φ

and: PE = (b - a cos Θ)/ cos φ

Now (given a = 1 and b = 1.52):

cos Θ = [Ö a  Ö b (Ö a + Öb)]  / (a ^3/2 + b^3/2 )

= [Ö (1.52) ( Ö (1) + Ö (1.52))]/ ((1) ^3/2 + (1.52) ^3/2)

cos Θ = 0.958

so: Θ = 16.7 ^o

And the angle of elongation (136.2) is (in terms of the geometry):

136.2 = 180 – (Θ + φ)

So we can solve for φ:

φ = 180 – 136.2 – 16.7 = 27.1 ^o

Now:

PE = (b - a cos Θ)/ cos φ

= (1.520 – 0.958)/ cos (27.1) = (0.562)/0.890 = 0.631

So the distance of Mars from Earth denoted by PE (at stationary point) = 0.631 AU

The time to next stationary point:

T _r/2   = Θ/ 360 x S

where: (given a =1, b = 1.52):

So:

T _r/2   = Θ/ 360 x S = (16.7)/360 x (779.9 d) = 36.1 days

2) Find the length of time Jupiter has retrograde motion in each synodic period given its heliocentric distance is 5.2 AU and its sidereal period is 11.86 years.

Solution:

First, use: 1/ S = 1/P1 - 1/P2

And thereby obtain Jupiter’s synodic period, S:

Where P1 = Earth's sidereal period (1 yr), and P2 = Jupiter’s  sidereal period (11.86 yrs)

1/S = 1 – 1/(11.86) = 1 – 0.0843 = 0.9156

Then: S = 1/ 0.9156 = 1.09 yrs. = 398.1 d

The time for retrograde (r) motion in each synodic period is:

T _r    =2 Θ/ 360 x S

Where:

cos Θ = [Ö a  Ö b  (Ö a + Ö b)]  / (a ^3/2 + b ^3/2 )

and a =1, b = 5.2, so:

cos Θ= [Ö (5.2) (Ö  (1) + Ö  (5.2))]/ ((1) ^3/2  +  (5.2)  ^3/2)

And: cos Θ = 0.582

So: Θ = arc cos(0.582) = 54.4 ^o

and the time to move retrograde over each synodic period;

T _r  =2 Θ/ 360 x S = (2 x 54.4)/360 x (1.09 yr) = 0.329 yr. = 120.3 days

3) In the Epsilon Eridani star system a planet designated Epsilon Eridani III is determined to have the exact same orbital parameters as Earth (e.g. a, e, i etc.). In the same system, another exoplanet designated Epsilon Eridani IV is found to have V _p  = 4.5 km/s. a) Using your knowledge of the known parameters, plus the diagram shown, construct an appropriate parallelogram of velocities and hence obtain the angles: Θ and φ.

Solution:

We use  V _p = 4.5 km/s and V _E   = 47 km/s for the velocities.

Since Eridani III has the same orbital parameters as Earth we can employ Earth semi-major axis, etc. in the computations. Also, a check of tables (or previous problems from earlier sets) shows Eridani IV has the same orbital velocity as Neptune so will have approximately the same sidereal period of 163.73 yrs. Now to form the parallelogram we need to obtain the angles Θ and φ.

We must first obtain the synodic period  (S)  of Eridani IV:

1/ S = 1/P1 - 1/P2

Where P1 = 1 yr (for Eridani III), and P2 = 163.7 yrs. for IV

1/S = 1 – 1/163.7 = 1 – 0.0061 = 0.9939

Then: S = 1/0.9939 = 1.006 yr. = 367.7 d

To get Θ:

cos Θ = [Ö a  Ö b (Ö a + Ö b)]  / (a ^3/2 + b^3/2 )

where b is Eridani IV’s semi-major axis or:

b = {[P2] ²} ^1/3  = { [163.7] ²} ^1/3 = 29.9 AU

And we know a (semi-major axis for Earth) = 1 (AU) , so:

cos Θ= [Ö (29.9) (Ö (1) + Ö (29.9))]/ ((1) ^3/2 + (29.9)^3/2)

cos Θ = 0.215 and Θ = arc cos(0.215) = 77.6 ^o

Meanwhile, by using a scaled diagram (e.g. see the example shown ) we find: 

90 + φ = 105 deg so:

φ = 105 – 90 =  15 ^o


b) Hence, or otherwise, estimate the time planet Epsilon Eridani IV will be moving retrograde relative to Epsilon Eridani III, and also the time between its opposition and the next stationary point.

Solution:

The time for retrograde is:

T _r    =  2 Θ/ 360 x S = (2 x 77.6)/ 360 x (367.7 d) = 158. 5 d

Between opposition and next stationary pt.:

T _r/2  = (158.5 d)/2 = 79¼ d

c) Obtain the time during Epsilon Eridani IV's synodic period that it is moving direct.

Solution:

t(D) = (1 – Θ/ 180) x S = (1 - 77.6/ 180) (367.7 d) = 209 days

WSJ's Peggy Noonan Roasts Pete 'Howler' Hegseth For His 'Drama Queen' Speech To Generals Last Week

 

WSJ op-ed contributor Peggy Noonan, when she's in the zone about Trump and his lapdogs is incomparable. Such was the case with her recent column ('The Embarrassing Pete Hegseth', Oct. 4-5, p. A15). Basically skewering the pompous little dry drunk (Bill Maher's term) who delivered an absurd spiel to 800 top military brass last week, e.g.

Pete Hegseth Declares War on "Fat Generals" and "Woke Garbage"

As Noonan writes about this former FOX loser's performance:

"It was, as a former general said by phone, 'just flat out bizarre'. It was embarrassing to watch. He made everyone in the audience look smaller, which made their profession look smaller.  How does that help America?"

Of course, it solidifies the sentient world's opinion that we're a bunch of uneducated rubes and Philistines for putting the orange traitor and felon back in the White House. Because he, after all, is the moron who nominated Hegseth to be Secretary of Defense, and as Noonan notes:

"Hegseth has always had bad press, from the scandals that emerged after his nomination through fairly constant reports about chaos in his office.  I said early on he was a poor choice - a television host playing a culture warrior - who lacked the weight and gravitas the Pentagon needed."

True, but it wasn't what Captain Bonespurs needed. This 5-time draft dodger needed a compliant lapdog who'd follow his orders without question - whether militarizing Democratic cities, or ordering attacks on Venezuelan fishing boats and calling them "narco-terrorists".

As Edward Luce observes in his recent Financial Times column (Venezuela is Trump's Useful Ally):

"In July Hegseth halted U.S. weapons shipments to Ukraine only to discover Trump did not yet want that. He suffered a humiliating reversal. Since then Hegseth has been falling all over himself to please Trump, including ordering four televised air strikes on unidentified Venezuelan boats. Trump is so pleased with the optics of the apparently risk-free fireworks he is developing a taste for them."

But Peggy wasn't done with this arrogant little pissant, wannabe Sergeant York by any means, writing:

"Hegseth could have reiterated all his fulminations about 'DEI offices' by secure video conference, or just sent a video.  Instead, he dragged commanders from their stations to be his personal audience. So he could pose with a giant American flag behind him like George C.Scott in 'Patton'. Only Scott delivered a great speech. Hegseth gave a Ted Talk and paced the stage like a strutting, gelled bantam. Like an amped up actor with rehearsed gestures, expressions and voice shifts."

In other words, like a fraud, a pretender, like his master Trump.  Back to Peggy's take down of the imp:

"He used 'lethal' and 'lethality' a lot, like a young Hollywood script writer dreaming up some made right wing Army officer because he maybe watched 'Platoon' too much as a child. Then spouting 'We became the WOKE department. It's completely unacceptable to see fat generals."

Noonan adding:

"The retired general on the phone later sighed and said: "I would like you to know Norman Schwarzkopf, his hero, was fat. And George Patton wasn't exactly a gazelle.  Sound military leadership has little to do with physical fitness and everything to do with strategic judgment."

And finally, Peggy's most substantial point:

"What are we doing in this dangerous world having the head of the Defense Department prance around like this and embarrass the generals he used as a backdrop.  Why do his highly placed defenders in the administration think this is good for the White House?

When you are driven by a sense of urgency you must still try to act like a normal person.  Not like some pumped up drama queen who makes everything more jarring and fevered...It should have crossed his mind that he himself, when in service, never reached anywhere near the rank of those he was talking down to."

From my perspective Hegseth's posturing was all about creating the illusion of fearsome strength and threat. In line with Trump's threat of using blue cities as military "training grounds" or blasting small fishing boats out of Caribbean seas when they pose no threat at all to U.S. interests. As AOC noted on ALL In last week, this is the performance of frightened cowards who need the projection of strength to try to intimidate citizens into preemptively giving up their rights, e.g. ceding their free speech rights out of fear. In Hegseth's case, he wanted to instill in the generals this bogus "warrior ethos" and that if they opposed his (and Trump's) anti-Woke, anti-DEI agenda, they "should do the honorable thing and resign."  

No reference to this sorry ass goofball letting out the Pentagon's war plans (for an attack on Yemen) in a Signal chat some months ago. A genuine dishonorable deed if there ever was one, for which he never resigned. Or was held accountable.

As Noonan writes at the end:

"There are recent reports the Pentagon is putting forward new rules that journalists have their work approved before publication.  Where that stands is unclear but it's nuts. It makes America look like what our foes say we are, a place of make believe freedom in which even the press is controlled by the government."

As Timothy Snyder told Velshi on Saturday, it's time the American nation wakes up and begins massive resistance against these fiends - or we may soon find ourselves without any voice of resistance at all.

 See Also:

Constitution, Civil Rights, Civil Liberties

America’s Fourth Amendment Is on Life Support — What Do You Do When Your Government Becomes the Criminal?

by Thom Hartmann | October 7, 2025 - 4:48am | permalink

— from The Hartmann Report

Remember the old TV crime/drama shows? A cop would bang on a suspect’s door and the suspect would say, through the door, “Do you have a warrant?” The officer would then walk away, promising to come back later with the requisite paper signed by a judge.

No more. Now they’re kicking in doors, shooting pepper-gas balls into the open windows of cars driven by reporters, smashing windows and furniture, and concealing their faces and identities like the Klan did in days of old. In Chicago, they’ve shot two unarmed people, killing one. And there wasn’t a warrant signed by a judge to be seen anywhere.

People ask, “Are we there, yet? Has America gone fascist? Are we now in a militarized dictatorship?”

» article continues...

And:

Donald Trump

An Old Soldier Not Fading Away: Capt. Embree's Latest Letter to Trump

by Jaime O’Neill | October 6, 2025 - 5:11am | permalink

From The LA Progressive

5 October 2025
Dear Mr President,

I found your hour-long speech to the nation’s generals and admirals this week clarifying in the way that I suppose psychiatrists find that the script-free, order-free, grammar-free, image-filled rambles of their sicker patients illuminate the dark recesses of their troubled minds. Like those silently listening psycho-analysts that we imagine sitting beside their proverbial couches, the brass was content to listen and slowly assemble the clues that would tell them when to summon the aide with the butterfly net and the nurse with the hypodermic needle.

Doubtless they were mildly amused to note how compulsively you repeated the reverential “sirs” by which your staff, the press, your generals, governors, and foreign heads of state address you; to be assured that you don’t want to win the Nobel Peace Prize, but that “it will be a big insult to our country” if you don’t; and to learn that you had $31 billion stashed on your “tariff shelf” – not, apparently, a metaphor, but a real shelf. By then they were probably thinking . . . idle chatter? . . . disturbed? . . . nuts? . . . bat shit?

» article continues...

And:

US Military Discovers It Is Led BY IDIOTS

And:

Pete Hegseth: Secretary Of MANSCAPING! | Armageddon Update

Development Of A Mathematical Ballistics Model Using Differential Equations (Part 1)

 

                                                              Fig. 1. Ballistic trajectories

                                                 Fig. 2.  Preliminary Mathematical Model

Ballistic trajectories have wide applications in our modern world, from lobbing missiles at enemy states to launching near Earth space craft. To proceed with the development of the mathematical model  (Fig. 1)we first  need to develop the basic equations of motion, stating with the much simpler Fig. 2.   Here M is a mass fixed at the origin in the x-y plane with a particle of mass m situated at the point P in the plane.  The rectangular coordinates of P as shown are (x,y).  The polar coordinates are (r, q).  According to Newtonian mechanics the mutual attraction between two masses, M and m, is given by the inverse square law:

1) F _Mm = GM m/ r²

Where F is the force vector and r is the distance vector.   If we assume the mass m is negligible compared to M, the force exerted on m is given by: 

2) F _m = GM A

Where A  is the acceleration imparted to m relative to the x-y coordinate system.  Since equations (1) and (2) are equivalent expressions of the same force, we can write:

3) F _Mm = F _m  or:   cm/ r²   =  m A

And the vector A  is anti-parallel to r.   Then the scalar of  A is given by:

4) A =  - c/ r²   

The components of acceleration from Fig. 2 are:

5)

A _x = x'' =  A cos q   =  - c/ r²   cos q

A _y = y'' =  A sin q   =  - c/ r²   sin q

Note equations (5) display a mix of rectangular and polar coordinates. Obtaining correct equations in rectangular and polar coordinates alone requires use of the following elementary transformations:

(6)

x =   r   cos q

y =  r sin q

r²  =  x²   +  y ²

To then obtain the equations in rectangular coordinates alone we solve for  sin q , cos q  and r then substitute into equations (5) to get; 

(7)

x''  =   - cx (  x²   +  y ²  ) ^3/2

y'' =  - cy (  x²   +  y ²  )  ^3/2

To then obtain the equations in polar coordinates the procedure entails differentiating the transformation set (6) twice with respect to t and then substituting the results into equations (5).

Problem to lead off next part:

Perform the procedures as described above to obtain the polar coordinates form of the equations.

Look For Runaway Greenhouse To Start A Decade Earlier With Trumpers Ready To Burn More Coal Via New Leases Issued Today

 

Coming 10 years sooner: A Hellhole planet thanks to Trump

Three years ago, the Supreme Court's rightist nuts and Trump collaborators ruled the EPA lacked the power to regulate CO2 in line with the Clean Air Act.  Now the 'chicken' are coming home to roost as we learned yesterday in an AP release, the Trump criminal administration plans - in the coming days - to hold the biggest coal sales in more than a decade. And, ironically, in that development, bringing the hell hole world of the runaway greenhouse a decade sooner to humanity.

According to the Associated Press report (appearing in The Denver Post):

"U.S. officials are offering for sale 600 million tons of coal next to strip mines in Montana and Wyoming.   The sales are a signature piece of Donald Trump's ambitions to dig more coal from federal lands and burn it for electricity. Yet most power plants served by those mines plan to quit burning coal within ten years, an Associated Press data analysis shows."

So offering the biggest coal sale in ten years when the power plants are planning to stop burning it in ten years.  This is what I mean by "Trumptards". 

The Denver Post account goes on to note:

"Three other plants earmarked for expansions or new leases under Trump also face declining demand as power plants use less of their coal and in some cases shut down. This according to data from the U.S. Energy Information Administration and the nonprofit Global Energy Monitor."

And just who, pray tell, is going to buy all the 600 million tons of coal these Trumptards plan to sell? According to the AP:

"This question looms over this administration's enthusiastic embrace of coal, a leading contributor to climate change. It also shows the uncertainty inherent in inserting those policies into markets where energy producing customers make long term decisions with massive implications, not just for their own viability but for the future of the planet."

Again, Trumptards - or should we say, planetary arsonists? 

The AP report in the Post goes on to note the upcoming lease sales in Montana and Wyoming are in the Powder Rive region, home to the most productive coal fields. Also noting Trump officials plan to go forward with it today, despite the government shutdown. In other words, they want to use the shutdown not only to fire tens of thousands more federal workers, and halt funding of green energy projects in blue states, but also get vastly more coal burning underway. To finish burning the planet to a crisp and the humans on it.

According to a Department of Energy formula which the Trump renegades have since erased from websites, burning the coal from the two leases being sold would generate more than one billion tons of planet warming carbon dioxide. This, added to what is already fueling climate change at the rate of   2.7 W/ m²   per year, making it likely the runaway greenhouse will be underway by 2090, not 2100.

Recall that the past ten years, from 2015 to 2024, have been the hottest on record, with 2024 being the warmest year overall, according to scientific and weather organizations. This marks a significant shift, as all the warmest years in recorded history have occurred within this recent decade. The World Meteorological Organization (WMO) has confirmed that 2024 surpassed previous records, making it the first year to exceed the 1.5 degrees Celsius warming threshold above pre-industrial levels. 

Recall also the demented excuse for a U.S. president called climate change a "con job" when he addressed the U.N. General Assembly in a Sept. 23 speech. He also praised coal as "beautiful" and boasted about the abundance of U.S. supplies while deriding solar and wind power.

The outburst left Barbados' Prime Minister Mia Mottley speechless, though when she recovered her equilibrium, she told the Bajan press it was the "most unhinged, ignorant speech by a leader" she'd ever seen or heard.

I am in total agreement, and remain embarrassed that so many idiot, ignorant fool voters could put this orange slimeball traitor and felon back into power.

See Also:

White House offers ‘concierge’ service to fossil fuel firms - The Washington Post

Excerpt:

The White House is offering “concierge, white glove service” to oil, coal and other fossil fuel companies that are seeking to gain fast approval for their projects, according to an energy official, while simultaneously slowing down or blocking solar and wind projects.

And:

IPCC report calls for rapid, deep, and immediate emissions reductions - Bulletin of the Atomic Scientists (thebulletin.org)

And:

After Portland and Lytton Are Humans Even Ready For The Cusp Of The 'Runaway Greenhouse'?

And:

Record heat to bake Calif. as April snowpack nears 70-year low

And:

Environmental Policy

Whiplashed: From Drought to Deluge on a New Planet

by Jane Braxton Little | March 22, 2022 - 6:43am | permalink

— from TomDispatch

And:

 The Intelligent Irrationalists: Why So Many High IQ Folks Deny Sound Science 

And:

Five Major Climate Tipping Point Manifestations & What They Mean 

 And:

Bjorn Lomberg - Climate Change Clown Has No Clue Concerning Adaptation To A Rapidly Warming World 

And:

And:

Is There A Climate Emergency? All The Indicators Point To A Serious Crisis Nearing Emergency 

And:

As CO2 Concentration Hits A New Record The Threat Of Tipping To A More Hostile Global Climate Has Increased 

And:

Energy Policy | Environmental Policy

The Later We Stop, The Worse the Result

by Thomas Neuburger | August 7, 2023 - 6:02am | permalink

— from God's Spies

Theologian William Lane Craig Bumbles Again In His "Fine Tuning" Arguments To Prove A Deity

 

                                Wesleyan Theologian William Lane Craig - at it again.

We had first met Wesleyan theologian William Lane Craig back in June, 2011, when he tried to use Bayesian statistics to prove the Resurrection, i.e.

• William Lane Craig’s Probability Gibberish

Blissfully unaware that statistical or probabilistic arguments are incapable of "proving" a claimed objective reality. According to Craig in his latest iteration, both the strength of the gravitational force and the weak nuclear force had to be "fine - tuned" to one part in ten to the power 100 - else life could not have been possible.

This is insipid nonsense since no physical measurements can possibly have such precision. Also, long before the cosmological constant (L) was determined, cosmologists had already estimated the upper bound at   10 ^-120 in the same units  (m^-2). An alternate computation using quantum mechanics, however, obtained a value   » 1, a huge discrepancy.

Why the difference? Well, because quantum physics must factor in quantum particles. The problem is that not all the contributing quantum particles are known in respect of their contributions to the vacuum of space.  We know there is about 5 times more dark matter than ordinary matter, but that comes in different forms with some particles producing positive contributions, other negative. For example, there is baryonic and non-baryonic dark matter.

The former includes protons and neutrons while the latter includes electrons and neutrinos.  The non-baryonic dark matter further breaks down into cold dark matter and hot dark matter.  The terms not so much indicative of current temperatures as the phase of the early universe at which each 'decoupled' from the hot radiation background following the Big Bang. Cold dark matter particles tend to have larger mass and among the candidates considered are: gravitinos, magnetic monopoles and primordial black holes.

The negative and positive contributions each affect  L  differently but we know the negative and positive contributions sum to infinity, and infinity minus infinity is something our current math can't handle without making extra assumptions. Physicists solve the impasse by cutting off associated energy calculations when quantum mechanics interferes with general relativity.  Hence, all such calculations are highly speculative.  Transl.  We do not take them literally.

However, Craig appears to take them very literally else he'd not claim fine tuning to one part in ten to the power 100 for both the force of gravitational attraction and weak nuclear force. Where did Craig get this idea? Likely, from Paul Davies book: 'The Accidental Universe', p. 107,  which itself bumbles via use of a nonsensical 'quantum'  cosmological constant (L_q). Davies, incidentally, did not use or reference his 'creation' in any future works.

So Craig is basing his fine-tuning effort on a 'macguffin' constant from which emerges an  incomprehensible 'infinity'.  This would be bad enough but then is compounded in his 2008 book 'Reasonable Faith' by confusing cosmic inflation with the cosmological constant.

Craig's desperate efforts remind me of Sir Arthur Eddington's own venture into numerical babble. Sir Arthur once arrived at a value for the fine structure constant of a   = 1/136 by taking the ratio of two "naturally occurring units of action". ('Great Ideas and Theories of Modern Cosmology', 1961, p. 178).  He chose one unit of action as the quantum for radiation, or  ħ  =   h/ 2p  and the second as the action for elementary particles, or e² / c.   Then, taking:

{e² / c}/  ħ  =  1/ 136

And holy moly, we're almost at a! (Which has an actual measured value of a  = 0.007297352569, or around 1/137.036)

Curiously, Eddington wasn't bothered by the divergence (from even the crude value a   = 1/137 )  , and just introduced a "fudge factor". This "was for obscure reasons that are difficult to understand". Perhaps, in the end, he was simply mesmerized by a kind of 'numerology' as Craig is mesmerized by his.

Eddington also came up with a quadratic equation: 

10x²  +    136x +  1 = 0, 

 linking his  fine structure result with the mass ratio of the proton to electron, i.e. in terms of the ratio of its two roots. From there, Eddington parlayed his fine structure and other pure number results into a kind of "universal theory" linking every aspect of the cosmos in a kind of romantic quest. Much like Kepler before him, with his "harmonic geometry"  in which the five Pythagorean regular polyhedra dictate the structure of the universe and reflect God's plan through geometry.

We shouldn't be too hard on Sir Arthur  (or Johannes Kepler) as he wasn't the first scientist to be taken in by numerical relationships, "harmonic" ratios, and "precision" theoretics. Nor will he likely be the last.  Even today we behold "scientists" seriously working on the so-called "anthropic principle". This  nonsense is based on the fallacy (due to a misunderstanding of physics units, dimensions) that there is an implicit "fine tuning". This in turn depends on a putative "fine precision" - but that is almost always based on the choice of units.

The bottom line takeaway from all this?  Theologians ought not depend on fine tuning to prove a supernatural creator.  Better yet, stay out of cosmology and physics, period. 

See Also:

The Vatican's Big Bang Conference: Does George Lemaitre's "Theory" Prove A Supernatural God?

And:

Response To Tom McLeish's Counter Arguments In Physics Today

And:

Brane Space: "At Its Heart Science Is Based On Faith?" -- Errr....No!

And:

Bishop Richard Barron Misfires On God & Metaphysics

Practical Astronomy Focus: Stationary Points

 

One of the key tasks for planetary astronomers - amateur or professional - is to ascertain the point at which a superior planet's motion changes from being retrograde to direct. I are referring here to the planet's geocentric (Earth-centered or referenced) angular velocity, and identifying when it becomes direct - between opposition and quadrature (see previous blog instalments for these configuration definitions).

Clearly, if such a change occurs, there will be some point the motion is neither retrograde or direct and this is called a "null" point or better, "stationary point". The problem then, is to obtain an expression for the planet's elongation E at a stationary point in terms of the distances of the planet and Earth from the Sun. Using such an expression and the measured elongation, the planet's heliocentric distance can be computed. (In addition, we can also obtain an analytical expression giving the angle Θ between the heliocentric radius vectors of planet and Earth at the stationary point) Further, such an analytic expression together with the synodic period of the planet, allows us to predict its next stationary point.

To fix ideas here, we use the accompanying diagram and let the positions of Earth and planet at the stationary point be denoted by E and P. The velocity of P relative to E must lie along the geocentric radius vector (EP) if the planet appears stationary. As we learned in the "Basic Physics" blogs, we can form a "parallelogram" of velocities - just as we can for forces. In this case, we can form the velocity parallelogram PABC, with the resultant given by PB.(E.g. the resultant of the vectors (- V _E) and the planet's tangential velocity V_p. Note that: PA = V_p and PC = -V _E

Since we assume the orbits are circular (a close enough approximation) the angle Θ between the heliocentric radius vectors must be angle PCB between the velocity vectors. In addition, note that angle APB = 90 + φ. Now, if we extend CP to meet the line SE (produced at D) then angle DPA = Θ and angle EPD = 90 - (Θ + φ). Focusing now on the triangle PCB, we may use the sine formula of trigonometry to write:

sin[90 - (Θ + φ)] / V_p = sin (90 + φ) / V _E

or (using trigonometric identities):

cos (Θ + φ) = (V_p  / V _E) cos φ

Making use of another trig identity and triangle SEP, we may write:

SP = PE cos φ + SE cos Θ

or:

(I) b = PE cos φ + a cos Θ

Applying it once more to triangle SEP we get:

SE = SP cos Θ + PE cos (SEP)

or:

(II) a = b cos Θ + PE cos (Θ + φ)

From the geometry of the configuration, since angle SEP = 180 - (Θ + φ), we can write equations (I) and (II) as:

(Ia) b - a cos Θ = PE cos φ

and:

(IIa) a - b cos Θ = PE cos (Θ + φ)

Dividing through eqn. (Ia) by eqn. (IIa):

[b - a cos Θ]/ [a - b cos Θ] = cos φ/ [cos (Θ + φ)]

Now, cos (Θ + φ) = (V_p  / V _E)  cos φ

And we know:

(V2/V1) = Ö (a1/a2 ) 

Combining these in (Ia) and (IIa) we have:

[b - a cos Θ]/ [a - b cos Θ] =   (V_p  / V _E)  =  Ö (b /a)

Algebraically re-arranging we find:

cos Θ = [Ö a  Ö b (Ö a + Öb)]  / (a ^3/2 + b^3/2 )

The preceding can be vastly simplified if all units are normalized to those in terms of Earth's to the Sun. Thus, a = 1 and b is in terms of a. So let:

Ö Q =  Ö a  Ö b 

(1 + Ö Q) = (a ^3/2 + b^3/2 )

Then re-write as:

cos Θ = [Ö Q  (1 + Ö Q) ]/ (1 + Q ^3/2)

As may be discerned from the diagram, when Earth is at point E' where angle ESP = 90 degrees, we have another stationary point. It is of interest here to compute the total time during which a planet will be seen to move retrograde and this is just the time it takes Earth's radius vector to advance through an angle equal to 2 Θ, with respect to the planet's radius vector. This will be given by t( Θ), where:

t( Θ) = 2 Θ/ 360 x S X (ΘS)/ 180

where S is the synodic period.

The time that elapses between opposition and the next stationary point is just t( Θ)/2.

Meanwhile, the time interval during a synodic period that a planet's motion is direct is t(D) where:

t(D) = (360 - 2 Θ)/ 360 X S X (1 - Θ/ 180)S


Example Problem:

Mars has a synodic period S = 779.9 days. At a particular stationary point Mars' radius vector is determined to be 1.52 AU. Find the angle Θ that Earth would have advanced to reach this point and time elapsed between opposition and the next stationary point .

Solution:

We have:

cos Θ = [Ö Q  (1 + Ö Q) ]/ (1 + Q ^3/2)

where:

Ö Q   =     Ö a  Ö b  = Ö (1) (1.52) = 1.23

 (1 + Q ^3/2)   =  Ö a +  b ^3/2  = 1 + (1.52) ^3/2  = 1 + 1.87 = 2.87

Then:

cos Θ = [ Ö (1.23)  (2.23) ] / (2.87)

cos Θ = [(1.10)(2.23)]/ (2.87) = 1.64/2.87 = 0.571

Θ = arc cos [0.571] = 55.1 deg

The time that elapses between opposition and the next stationary point is just:

t( Θ)/2= ½ [2 Θ/ 360 x S X (ΘS)/ 180 ]

Where: S = 779.9d/ 365.25 d = 2.13 yrs

So:


t( Θ)/2= ½ [(110.2)/ 360 x (2.13) X (55.1 x 2.13)/ 180 ]

t( Θ)/2= ½ [(0.306) (2.13) (0.652)] = 0.21 yrs.

Application Problems:

1) Mars reaches a stationary point 36 ½ days after opposition. Its elongation is then measured to be 136.2 deg. Given that the sidereal period is 687 days, find the distance of Mars from Earth in AU. Also, find the time to its next stationary point.

2) Find the length of time Jupiter has retrograde motion in each synodic period given its heliocentric distance is 5.2 AU and its sidereal period is 11.86 years.

3) In the Epsilon Eridani star system a planet designated Epsilon Eridani III is determined to have the exact same orbital parameters as Earth (e.g. a, e, i etc.).

In the same system, another exoplanet designated Epsilon Eridani IV is found to have V_p = 4.5 km/s.

a) Using your knowledge of the known parameters, plus the diagram shown, construct an appropriate parallelogram of velocities and hence obtain the angles: Θ and φ.

b) Hence, or otherwise, estimate the time planet Epsilon Eridani IV will be moving retrograde relative to Epsilon Eridani III, and also the time between its opposition and the next stationary point.

c) Obtain the time during Epsilon Eridani IV's synodic period that it is moving direct.