More orbital motion!! And: how do
we know the value of an AU, exactly?
IN-CLASS ACTIVITY: LET’S CALCULATE THE MASS OF JUPITER!
At its closest approach to the Earth, Jupiter is 588 million km
away.
We observe its moon Io every night and figure out its
maximum angular distance from Jupiter. From one side to the
other (blue line) is 4.94’ and we figure out that Io’s period of
revolution around Jupiter takes 1.77 Earth-days. How massive is
Jupiter?
IN-CLASS ACTIVITY: LET’S CALCULATE THE MASS OF JUPITER!
At its closest approach to the Earth, Jupiter is 588 million km
away.
We observe its moon Io every night and figure out its
maximum angular distance from Jupiter. From one side to the
other (blue line) is 4.94’ and we figure out that Io’s period of
revolution around Jupiter takes 1.77 Earth-days. How massive is
Jupiter?
d = ✓D
r ⇡ 421600km
IN-CLASS ACTIVITY: LET’S CALCULATE THE MASS OF JUPITER!
At its closest approach to the Earth, Jupiter is 588 million km
away.
We observe its moon Io every night and figure out its
maximum angular distance from Jupiter. From one side to the
other (blue line) is 4.94’ and we figure out that Io’s period of
revolution around Jupiter takes 1.77 Earth-days. How massive is
Jupiter?
r ⇡ 421600km
v=
r
GM
r
What is v?
2⇡r
v=
P
IN-CLASS ACTIVITY: LET’S CALCULATE THE MASS OF JUPITER!
At its closest approach to the Earth, Jupiter is 588 million km
away.
We observe its moon Io every night and figure out its
maximum angular distance from Jupiter. From one side to the
other (blue line) is 4.94’ and we figure out that Io’s period of
revolution around Jupiter takes 1.77 Earth-days. How massive is
Jupiter?
r
r ⇡ 421600km
P = 1.77days
2 3
4⇡ r
M= 2
P G
GM
2⇡ r
=
r
P
What is G?
G = 6.674 ⇥ 10
11
3
m kg
1
s
2
WE CALCULATED THE MASS OF JUPITER!
At its closest approach to the Earth, Jupiter is 588 million km
away.
We observe its moon Io every night and figure out its
maximum angular distance from Jupiter. From one side to the
other (blue line) is 4.94’ and we figure out that Io’s period of
revolution around Jupiter takes 1.77 Earth-days. How massive is
Jupiter?
WE CALCULATED THE MASS OF JUPITER!
At its closest approach to the Earth, Jupiter is 588 million km
away.
We observe its moon Io every night and figure out its
maximum angular distance from Jupiter. From one side to the
other (blue line) is 4.94’ and we figure out that Io’s period of
revolution around Jupiter takes 1.77 Earth-days. How massive is
Jupiter?
MODULE ASSESSMENT TIME! Woohoo!
FOLLOW UP:
In the previous problem we measured Io (4.94’ angular
diameter and 1.77 day period), and we now measure an
angular diameter of Europa’s orbit at 7.82’. Europa is
another moon of Jupiter. What is Europa’s period?
FOLLOW UP:
In the previous problem we measured Io (4.94’ angular
diameter and 1.77 day period), and we now measure an
angular diameter of Europa’s orbit at 7.82’. Europa is
another moon of Jupiter. What is Europa’s period?
We calculate 3.4 days, actual: 3.5 days
FOLLOW UP:
In the previous problem we measured Io (4.94’ angular
diameter and 1.77 day period), and we now measure an
angular diameter of Europa’s orbit at 7.82’. Europa is
another moon of Jupiter. What is Europa’s period?
We calculate 3.4 days, actual: 3.5 days
Important: once we are able to characterize orbits
around any one body, we have all information we
need to derive them all using Kepler’s 3rd law.
Let’s calculate the mass of the EARTH.
Ok, so we gather we can measure the mass of
astronomical objects if it has something in orbit around
it. How about our moon? Use what you know about the
moon (remember we calculated it’s distance on day 2 as
384500km) estimate the mass of the earth.
How do we calculate the mass of Mercury & Venus?
- Neither have natural satellites, i.e. moons
so we couldn’t use gravity…
How do we calculate the mass of Mercury & Venus?
- Neither have natural satellites, i.e. moons
so we couldn’t use gravity…
- We only had accurate measurements of their
mass after sending satellites in the past 40 years.
How do we calculate the mass of Mercury & Venus?
- Neither have natural satellites, i.e. moons
so we couldn’t use gravity…
- We only had accurate measurements of their
mass after sending satellites in the past 40 years.
- First estimated using density arguments (i.e.
we know Earth’s size and mass…)
How do we calculate the mass of Mercury & Venus?
- Neither have natural satellites, i.e. moons
so we couldn’t use gravity…
- We only had accurate measurements of their
mass after sending satellites in the past 40 years.
- First estimated using density arguments (i.e.
we know Earth’s size and mass…)
M
⇢=
V
⇢Earth = 3.82 ⇥ 10
rVenus = 6051 km
17
kg
km3
MVenus ⇡ 5 ⇥ 10
24
kg
How do we calculate the mass of Mercury & Venus?
- Neither have natural satellites, i.e. moons
so we couldn’t use gravity…
- We only had accurate measurements of their
mass after sending satellites in the past 40 years.
- First estimated using density arguments (i.e.
we know Earth’s size and mass…)
M
⇢=
V
⇢Earth = 3.82 ⇥ 10
rVenus = 6051 km
24
17
MVenus ⇡ 5 ⇥ 10 kg
kg
24
3
km MVenus = 4.87 ⇥ 10 kg
WAIT
but Dr Casey never showed us:
1.
How we know the size of Venus?
2. How we actually know the distance to the sun in km?
WAIT
but Dr Casey never showed us:
1.
How we know the size of Venus?
2. How we actually know the distance to the sun in km?
if we know one, we know the other…
Measuring planets’ size…
If we know the distance to the
planet we can use an angle to
measure size…
What is the distance to the sun?
i.e. what is an A.U. in km?
Archimedes,
~200BCE
✓
Dmoon
Archimedes measured ✓ to be 87o, which gave an
estimate to the distance, but it was actually way off,
because he could only estimate that angle roughly
looking at the Moon’s fractional illumination.
What is the distance to the sun?
i.e. what is an A.U. in km?
Archimedes,
~200BCE
✓
Dmoon
Archimedes measured ✓ to be 87o, which gave an
estimate to the distance, but it was actually way off,
because he could only estimate that angle roughly
looking at the Moon’s fractional illumination.
What is the distance to the sun?
Thanks to Kepler & Newton, we learned that once you know the
distance to one planet you know them all…
What is the distance to the sun?
Thanks to Kepler & Newton, we learned that once you know the
distance to one planet you know them all…
You would probably think the EARTH would be the easiest
distance to measure, but that’s actually not the case…
Not so! It’s actually easier to measure the distance to
the Sun using Venus than just Earth.
This is because Venus
transits the sun: i.e. a
Venus “eclipse”
And depending on your
latitude on Earth, the transit
of Venus will take longer/
shorter, appear higher/
lower on the face of the
sun.
Not so! It’s actually easier to measure the distance to
the Sun using Venus than just Earth.
This is because Venus
transits the sun: i.e. a
Venus “eclipse”
From far in the southern hemisphere
And depending on your
latitude on Earth, the transit
of Venus will take longer/
shorter, appear higher/
lower on the face of the
sun.
Not so! It’s actually easier to measure the distance to
the Sun using Venus than just Earth.
This is because Venus
transits the sun: i.e. a
Venus “eclipse”
From far in the southern hemisphere
From far in the northern hemisphere
And depending on your
latitude on Earth, the transit
of Venus will take longer/
shorter, appear higher/
lower on the face of the
sun.
Gather as many sightlines
of the Venus transit on
Earth as possible,
measuring their duration
and projected location on
the Sun, it’s just a
geometry problem.
Gather as many sightlines
of the Venus transit on
Earth as possible,
measuring their duration
and projected location on
the Sun, it’s just a
geometry problem.
Unfortunately measuring ↵ is quite hard
because it’s extremely small.
d
↵
DSV
DVE
Gather as many sightlines
of the Venus transit on
Earth as possible,
measuring their duration
and projected location on
the Sun, it’s just a
geometry problem.
Unfortunately measuring ↵ is quite hard
because it’s extremely small.
d
↵
DSV
DVE
But the duration of the transit is much easier
to measure, and that can tell you alpha.
A
B
30 min
intervals
A
B
30 min
intervals
A
B
30 min
intervals
Venus Transit Duration from location…
A is 6.25 hours
B is 5.5 hours
A
B
30 min
intervals
Venus Transit Duration from location…
A is 6.25 hours
B is 5.5 hours
A
B
30 min
intervals
Venus Transit Duration from location…
A is 6.25 hours
B is 5.5 hours
What are we trying to measure in this
diagram?
Discuss.
A
B
30 min
intervals
Venus Transit Duration from location…
A is 6.25 hours
B is 5.5 hours
What are we trying to measure in this
diagram?
R
R
R
h
A
✏
B
R
30 min
intervals
dEV
dVS
✏
dES
l✏ = dES ✏