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Computer Lab – Planets & Conjunctions
(Virtual Lab Remote Edition)
Discovery of Neptune
[Parts of this discussion were adapted from Scientific American
, Dec. 2004 and DIO
,
June 1999]
"That star is not on the map!" Those were the famous words of astronomy
student Heinrich d'Arrest to staff astronomer Johann Galle at the Berlin Observatory
announcing the discovery of Neptune. Galle was testing an extraordinary prediction
made by French mathematician Urbain Jean Joseph Leverrier.
Leverrier was investigating the motion of Uranus, the outermost known planet
of the time. To account for the observed path of Uranus, he had hypothesized an eighth
planet and calculated where such a planet must be to explain the path of Uranus. He
had determined the necessary location for such a planet and sent that information to
Galle. In less than half an hour of observing on September 23, 1846, Galle spotted a
small blue disk very close to Leverrier's prediction. Leverrier later named the planet
Neptune.
Start the Voyager 4 program and click the ABC button at the right edge of the
screen to turn off labels (turn the button from black to white).
Select the
Display/Magnitude Limits… menu and move the left slider vertically so that the limit
for Narrow Field reads 11.0. Click the OK buttons to dismiss the window. Now set your observing location to Berlin in Europe (I believe you can do this
without my giving you detailed instructions). Set the date to Sept. 23, 1846 at, say, 11
PM. Zoom to 1°, this gives a view on screen about six times as wide as the telescopic
view that Galle had but we don't want your search to take half an hour.
Select
the
Center/Coordinates… menu and
select the Ecliptic button. Enter
325,0,0 for Longitude, 0,0,0 for Latitude. Click OK.
This is where Leverrier said to look but Neptune does not appear in this view. So
you need to move your 'telescope' around a bit. Hold down the spacebar on the
keyboard and click and drag within the Voyager window (hold the mouse button down
while moving the mouse). The cursor on screen turns into a hand and you can slide the
field of view around. Search until you find Neptune, a blue spot larger than any stars.
P & C – 1
John Couch Adams, a young British mathematician, was working on the
Neptune calculation just like Leverrier. About a week after Galle had found Leverrier's
planet, British Astronomer Royal George Airy announced the following;
(a) Adams had also successfully calculated Neptune's position;
(b) Adams was shy and had delayed communicating or publishing his results;
(c) British astronomer James Challis had also now found Neptune using Adam's results;
and
(d) Challis had delayed his search because he lacked the detailed star charts available in
Berlin.
Although skeptics doubted many of these claims, Adams was given credit as the
co-discoverer of Neptune along with Leverrier. The true story was not uncovered until
October of 1998 when Olin Eggen died in Chile (he was the current assistant Royal
Astronomer and had taken secret documents with him to an observatory in Chile).
This "Neptune File" has shown that all of Airy's claims were untrue. Adams had
not made any accurate calculations. Adams was in close contact with Airy and Challis.
Challis had been actively searching for Neptune for months and did have the same star
charts available in Berlin. Challis did locate Neptune six days after Galle and d'Arrest,
but only when he searched around the location calculated by Leverrier!
While it is true that one of Adam's many calculated positions for Neptune was
very close, this cannot be considered a real prediction. In fact, Adam's best value prior
to Neptune's discovery was a position off by a whopping 12° and Adam's had so little
confidence in the result that he did not want to publish it.
The British, primarily Airy, were loathe to cede the glory of Neptune's discovery
to France and inflated the accuracy of Adam's calculations to rival those of Leverrier
and threw out a variety of excuses why no successful search had been carried out in
England. The British were even so brazen as to push their favored name for the planet
of Oceanus instead of Neptune.
Now that the truth is known, it is clear that only Leverrier deserves credit for the
discovery of Neptune and that Airy and his successors that perpetuated the lie deserve
condemnation.
1. Check the Info Panel for Neptune, what were the Ecliptic Longitude and
Latitude values of Neptune when discovered?
Ecliptic Lon.: 325 degrees 52' 35.7"
Ecliptic Lat.: -00 degrees 31' 57.3"
P & C – 2
A Trip to Mars
Select the File/Open Settings… menu, navigate through the folders Local Disk
(C:), Program Files, Carina Software, Voyager 4, 110 Settings, and open the "Inner
Planets" file. Start the animation. How do we get a spacecraft to go from Earth to Mars
when they're both moving? We could blast the rocket so fast that it gets to Mars before
Mars can ‘get away’. But we don’t have rockets like that and it would waste
tremendous amounts of fuel. There must be some more efficient route.
We want to get a spacecraft from Earth to Mars, the straight-line path is no good
because the planets move, and it would require too much fuel (even when they are
relatively close to each other). The most fuel-efficient trajectory from one orbit to
another (assuming no "gravitational slingshots" are available) is called a Hohmann
transfer. We can create such an orbit on screen. (A gravitational slingshot is a way of
getting a boost for a spacecraft by having fly past a planet, these are often used to get
spacecraft out to Jupiter, Saturn and beyond, but rarely used in trips to Mars.)
Stop the animation, click on the Now button on the Time Panel. Select the
Tools/Planet Report… menu, "Heliocentric Positions" should be selected in the pop-up
menu by default. Record the Longitude and Distance values for Earth.
Longitude =
184.0729
Distance (AU) =
0.99700
The longitude will be a value between 0 and 360° (you can round it to the nearest
degree) while the distance should be very close to 1. Close the Planet Report window.
Select the Tools/Define Orbiting Object… menu. Type in any name you want for
your spacecraft. Leave the number at 0, make the Type a spacecraft, leave the Primary
as Sun, and don't change the Diameter or Mag. Params.
In the "Orbit Size and Shape" area, we want the Perihelion Distance to be the
Distance value you wrote down above. Use 0.21 for the Eccentricity; this combined with
the perihelion distance will give an aphelion distance of about 1.53 AU, right around
the orbit of Mars. Leave the Drag Coefficient at 0.
In the "Orbit Orientation and Position in Orbit" area, leave the Equinox at 2000.
Enter 0 for the Inclination (so your spacecraft will stay in the ecliptic plane), 0 for
Longitude, and then for the Argument of Perihelion enter the Longitude value for Earth
that you wrote above (this means the place where the spacecraft is closest to the Sun –
perihelion – is exactly where Earth is now).
Change the Mean Anomaly to zero. For the Year, Month, and Day we want the
current year, month, and day; they are probably already correctly filled in. It should
P & C – 3
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look like the picture below but with the empty boxes filled in as above. Click OK.
Select the Window/Planet Panel menu and click
on the Spacecraft tab. You should see your spacecraft
on the list, click on the Name, Sym(bol), and Orbit
boxes to turn those on for your spacecraft. Close the
Planet Panel.
Your spacecraft should appear atop the Earth and you should see its green
orbital path arcing away from the Earth out to the vicinity of Mars' orbit. If it isn't
correct, select the Tools/ Select Satellites and Spacecraft… menu, click on your
spacecraft, then click Edit…. You'll need to select Perihelion Distance for the left pop-up
P & C – 4
menu, then double-check all the values for the spacecraft.
Your spacecraft's orbit may have come up short or gone past Mars' orbit. That's
because Mars has a rather elliptical orbit and is sometimes closer or further than its
average distance. We could try to correct it by editing the spacecraft's Eccentricity value,
or we can just ignore it. All right, we’ll ignore it.
Animate. A spacecraft launched from Earth will already be moving in the same
direction as Earth with the same speed. The spacecraft already has that velocity for
'free', an efficient trip to Mars will just add a little extra to the velocity that the spacecraft
already possesses. In a Hohmann transfer, the rocket propelling the spacecraft
accelerates it in the same direction that the Earth (and it) were already moving. With
that extra speed, the spacecraft arcs into a wider orbit, reaching the orbit of Mars half an
orbit later.
Our goal was to get the spacecraft to Mars, not just to the orbit of Mars. Was
Mars anywhere near the spacecraft when the spacecraft reached Mars' orbit? You can
click Now to reset and animate again to check if you’re unsure. We need to launch the
spacecraft from Earth at a time such that when it reaches aphelion, Mars will be right
there. These Earth-to-Mars “launch windows” come roughly every two years.
Click once on your spacecraft to get its info Panel. The spacecraft goes from Earth
at perihelion to Mars (or at least Mars’ orbit) at aphelion, that’s half of a full orbit. So the
time to get from Earth to Mars will be half the spacecraft’s full period. How long will
the Earth-to-Mars trip take for your spacecraft in days (
again, this is half the Orbital
Period listed for the spacecraft
)? Also convert to months by dividing the days by 30.
Travel Time = 258.925 days = 8.63083 months
Open the Settings File "CSUB-Mars" in the 110 Settings folder. The left window
shows "Roadrunner-1" on the day it is launched from Earth. Note where Earth and Mars
are in their orbits, this is a correct launch window to get to Mars. Roadrunner-1 starts
out moving at Earth's fast speed plus some extra speed. It slows as it arcs out towards
Mars and is going slower than Mars when they meet. To go into orbit of Mars or land
on Mars, the spacecraft would have to accelerate to match orbits when it reaches Mars.
The two windows on the right show views from the spacecraft locked on Mars
and Earth. Start the animation and watch the journey. Stop when the spacecraft is close
to Mars. The two little white dots buzzing around Mars are its two small moons.
2. What are the names of the two moons of Mars?
Phobos and Deimos
Maximum Elongation
P & C – 5
Select the File/Open Recent Settings/Startup menu.
Click on the triangle button just to the right of the clock in
the Time Panel and select Sunset from that pop-up menu. Select the Center/ Planets/Sun
menu, then close the Sun's Info Panel. You should see Mercury either above or below
the horizon.
Think about it. We are seeing the Sun in the west at sunset, the Sun is moving
down below the horizon. If Mercury is above the Sun, then it will be visible in the
evening after the Sun has set (evening star!). If Mercury is below the Sun, then it was
already set before evening began but should rise the next morning before the Sun does
(morning star!).
3. Is Mercury currently an 'evening star' or a 'morning star'? What about Venus?
They are both a morning star and evening star
When Mercury reaches its highest above or below the Sun, that's called its greatest
eastern (above) or western (below) elongation. Select the Tools/Planet Report… menu
and select "Maximum Elongations" from the pop-up menu. Record the dates of the next
three eastern elongations (not western) for Mercury.
Mercury
Sep 14th, 2021
Maximum
Eastern
Jan 7, 2022
Elongation
Dates
Apr 29 2022
The dates you recorded should be separated by about four months, check them. If
not, maybe you incorrectly included western elongations or didn’t use three
consecutive eastern dates. These are dates a person could determine just by watching
Mercury in the real sky day after day. Close the Planet Report window.
The time for an event (like maximum eastern elongation) to recur for a planet is
called the planet's "synodic period". It will be different for each planet. Astronomers like
Copernicus, Tycho, and Kepler knew that the synodic period could be used to calculate
the planet's sidereal period – the time for it to make one orbit around the Sun.
Select the Chart/Set Time… menu, enter the first date of Maximum Eastern
Elongation from above. The hour and minute don't matter but the year does. Click on
the Julian Day tab, record the large number listed for the "Julian Day" (you can round it
off to the nearest integer).
P & C – 6
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2459472
(Julian Day #1)
The Julian Day number, or just Julian Day, is a continuous count of days and
fractions of days from the beginning of the year 4712 BC (arbitrarily selected, this date
has no special significance). We are using it here to simplify the calculation of days
elapsed between calendar dates.
Click the Local Time tab, set the time to your second maximum eastern elongation
date, and record the Julian Day for that date.
2459587
(Julian Day #2)
Repeat for the third date.
2459699
(Julian Day #3)
How many days elapsed from the first max elongation to the second? Just subtract
(Julian Day #1) from (Julian Day #2),
(Julian Day #2) – (Julian Day #1)
=
115 days
Also calculate the elapsed days from 2 to 3,
(Julian Day #3) – (Julian Day #2)
=
112 days
The two values should be nearly the same. Average your two values (add the two
together and then divide by two), this result is the synodic period of Mercury.
Synodic Period of Mercury = 113.5 days
[The textbook lists this value as 116 days.]
Copernicus Calculations
How can we use this synodic period of Mercury to get its sidereal period? Mercury
and Earth are both moving around the Sun, at different speeds. Mercury quickly catches
up with and overtakes Earth. It takes about 116 days for Mercury and Earth to finally
return to the same relative positions.
The mathematical relationship between the synodic period and the sidereal period
is not hard to work out. The result is
P & C – 7
1/S = 1/P
Merc
– 1/P
Earth
where
S = synodic period
P
Merc
= sidereal period of Mercury
P
Earth
= sidereal period of Earth
Not sure how to even begin? I’ll walk you through this calculation. We know P
Earth
= 365 days, so we will replace 1/P
Earth
with 1/365 = 0.002 739 726 . Similarly, replace the
1/S with 1/(your synodic period) = 0.008… . Now we have 0.008XXX = 1/P – 0.002739726.
Add 0.002739726 to both sides, you should now have 1/P = 0.01YYYY. If 1/x = N,
then x = 1/N. So, P
Merc
= 1 / 0.01YYYY, put your result here (you can round off the final
answer):
P
Merc
= 88.24 days
These measurements and calculations were done by Copernicus, Tycho, and
Kepler – without computers or calculators.
Sidereal Period of Jupiter
This method can be used for planets beyond the orbit of Earth as well. Open the
Planet Report window, select "Opposition Dates" from the pop-up menu. Record the
next two upcoming opposition dates for Jupiter.
August 19th 2021
September 26 2022
Opposition is when a planet appears opposite the Sun on the Celestial Sphere. The
planet will be rising in the east just as the Sun is setting in the west, and the planet will
appear highest in the sky around midnight.
We want to calculate the number of days elapsed between these two dates, we will
again use the Julian Day numbers. Set the date to the first opposition date and record
the Julian Day.
2459446.68666
(Julian Day opposition #1)
Record the Julian Day for this second opposition.
P & C – 8
2459484.68666
(Julian Day opposition #2)
Calculate the difference between these two Julian Day numbers, this difference is
the synodic period of Jupiter.
Synodic Period of Jupiter = 38 days
We can use this information to calculate the sidereal period of Jupiter, the formula
is
1/S = 1/P
Earth
– 1/P
Jup
Because Earth is now the inner planet of the pair, it appears in a different place in
the equation than before, the mathematics will be a little different. Use P
Earth
= 365 days
and the S value (synodic period of Jupiter) you obtained above; solve for P
Jup
.
Hint: You need to subtract the 1/P
Earth
= 0. 002739726 from both sides which will
make both sides negative, cancel out those negative signs.
Sidereal Period of Jupiter = P
Jup
= 71.03 days
Convert this to years using that 1 year = 365 days,
P
Jup
= 0.1949 years
4. Would Jupiter have the same synodic period if we were viewing from Venus?
No, If observed from Venus, which is closer to the Sun than Earth, the synodic
period of Jupiter would be different because Venus and Earth have different orbital
periods and different relative positions in their orbits.
Geocentric/Heliocentric
Using the techniques above, it is possible to measure the synodic periods of all the
planets, the values are
Synodic Period
Planet
(in days) Mercury
116
P & C – 9
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Venus
584
Mars
780
Jupiter
399
Saturn
378
To explain these values in the geocentric model requires using some odd numbers
for the deferents and epicycles of the planets. For example, Mercury appears to go
around its epicycle about 5 times faster than Venus does.
In the heliocentric model, Earth moves and these values are understood to depend
on the relative speed with which the planet and Earth orbit the Sun. In this model, as
we have done, one can use the synodic periods to calculate the sidereal periods – the
periods of orbit for the planets around the Sun. The results are
Sidereal Period
Planet
(in days) Mercury
88
Venus
225
Earth
365
Mars
687
Jupiter
4,330
Saturn
10,800
Note the pattern of increasing period with increasing distance from the Sun.
Copernicus discovered this and it gave him increased faith in the heliocentric model.
The planets orbit the Sun and planets closer to the Sun move faster.
How did Copernicus know that this was the correct order of the planets from the
Sun? From experiments done by Copernicus and others, we’ll learn about those in the
Distances computer lab.
Occultations
A conjunction is when two bodies appear close together in the sky. Planetary
conjunctions are common but it is very rare that the planets get so close in the sky that
one passes in front of another (an "occultation"). The last time a planetary occultation
was visible from Earth was 1818. The next planetary occultation visible from Earth will
come in the year 2065. Open the settings file "Venus and Jupiter in 2065", change the
Time Step to 5 sec, and animate. 5. What are the approximate start and end local times of the occultation? P & C – 10
Occultations of one planet by another are extremely rare because planets appear as
very tiny dots in the sky. Occultations and transits involving the Sun or Moon are much
more common because they occupy a much larger portion of our sky.
Occultations of one planet by another are extremely rare because planets appear as
very tiny dots in the sky. Occultations and transits involving the Sun or Moon are much
more common because they occupy a much larger portion of our sky.
Open the Settings File "Mars and Jupiter in 1387", read the Info text, and animate.
Some of Jupiter's smaller moons appear on screen along with Mars' two small moons.
6. Which is bigger, Jupiter's moon Thebe or Mars' moon Phobos? (Compare the
Diameter values in their Info panels.)
Conjunctions of Sun and Jupiter
Open the settings file entitled "Sinusoidal Sky". Animate, stop when the Sun's icon
appears to cover the icon for Jupiter. Write down the Julian Day (JD) number listed in
the Time Panel when this occurred.
First Conjunction:
JD 2460045.94006
Note that a close-up view of the Sun and Jupiter would likely reveal that Jupiter
passed above or below the Sun rather than being eclipsed. Continue the animation, stop
when the Sun has gone all the way around the celestial sphere and again appears to
cover Jupiter. Record the JD value.
Second Conjunction:
JD 2460448.94006
Take the difference between the two JD values. This time for Jupiter to return to
conjunction with the Sun is Jupiter's synodic period.
Time Difference = Jupiter's synodic period = 402 days
Jupiter's synodic period is about a year and a month. But why does it take that
amount of time? Especially when you consider that Jupiter takes 4330 days to orbit the
Sun.
On the following diagram, a green dot has been placed to represent the Earth’s
location in its orbit, a red dot has been placed for Jupiter’s location, and the asterisk at
P & C – 11
the center is the Sun. These locations correspond to a conjunction of the Sun and Jupiter
as seen from Earth. Note how a straight line going down from the Earth passes through
the Sun then Jupiter, Jupiter would be hidden behind the Sun as seen from Earth.
Let’s assume we are looking at the solar system from above the Earth’s north pole,
in that case Earth and Jupiter will orbit the Sun in the counterclockwise direction. Our
diagram above shows the Earth starting at the “12” position and Jupiter at “6”.
7. Keeping in mind that the Earth moves in the 12 > 11 > 10 … direction along its
orbit, at which clock position (1 to 12) would the Earth be at 400 days after the initial
position shown? Hint: 400 days is about 1 1/12
years.
In 400 days it'll be at around 12 – 1
8. Now tell me which clock position Jupiter will be at along its orbit after 400 days.
Hints: Jupiter is starting at 6 and advancing in the direction 5 > 4 > 3… It takes Jupiter
about 4330 days to make a full orbit, 400 days is around 1/12
th
of an orbit.
At the clock postition 5
Do you see that your answers to 7 and 8 form another conjunction with the Sun? If
the answers you chose do not have Jupiter directly behind the Sun as seen from Earth,
P & C – 12
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then you likely have wrong answers.
Conjunction Searches
Select the Tools/Conjunction
Search… menu. If you search for
'Planetary' conjunctions of the Sun
and Jupiter with a 'Separation' of
10 (degrees), you'll get a list of
dates spaced about 400 days apart.
Go ahead and do it.
If the 'Angle' listed for a conjunction in the results list is less than 0.25°, then
Jupiter will be seen to pass behind the Sun, that
happens in about 15% of the conjunctions. How often does it occur that Jupiter and
Saturn will be simultaneously behind the Sun? Let's check, do a Planetary search from
2000 to 2100 with a Separation of 1 (degree) and with the Sun, Jupiter, and Saturn all
selected.
9. How many times does this conjunction occur this century?
It doesn’t occur even once
Do a search for conjunctions of Neptune and Pluto this century with a Separation
of 60°. Because Neptune and Pluto are in a 3:2 orbital resonance, Neptune does 3 orbits
in the time that Pluto does 2, they never get very close to each other.
10. What is the closest angle between them this century based on the search
results? They get 16 AU closer
Selected Answers:
1.
Answer not shown.
2.
Phobos and Deimos
3.
Answers not shown (because they vary). There can be situations where you won’t be
able to tell whether it is a morning star or evening star. There can even be situations
in which the planet is both a morning and evening star; or neither!
P & C – 13
4.
Answer not shown.
5.
Answers not shown.
6.
Answer not shown.
7.
Answer not shown.
8.
At 5
9.
Zero. Even over 1000 years there is only one conjunction and it is not close enough
that both would be behind the Sun at once.
10. Answer not shown.
P & C – 14