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

⇡ 421600kmr

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?

⇡ 421600kmr

v =

rGM

rWhat is v?

v =2⇡r

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? ⇡ 421600kmr

rGM

r=

2⇡

Pr M =

4⇡2r3

P 2G

P = 1.77days

What is G?

G = 6.674⇥ 10�11m3kg�1s�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…

- First estimated using density arguments (i.e. we know Earth’s size and mass…)

- 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…

- First estimated using density arguments (i.e. we know Earth’s size and mass…)

⇢ =M

V⇢Earth = 3.82⇥ 1017

kg

km3

rVenus = 6051 km

MVenus ⇡ 5⇥ 1024 kg

- 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…

- First estimated using density arguments (i.e. we know Earth’s size and mass…)

⇢ =M

V⇢Earth = 3.82⇥ 1017

kg

km3

rVenus = 6051 km

MVenus ⇡ 5⇥ 1024 kg

MVenus = 4.87⇥ 1024 kg

- We only had accurate measurements of their mass after sending satellites in the past 40 years.

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?

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.

✓

i.e. what is an A.U. in km?

What is the distance to the sun?

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.

✓

i.e. what is an A.U. in km?

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”

And depending on your latitude on Earth, the transit

of Venus will take longer/shorter, appear higher/lower on the face of the

sun.

From far in the southern hemisphere

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.

From far in the northern hemisphere

From far in the southern hemisphere

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.

DSV DVE

d

Unfortunately measuring is quite hard because it’s extremely small.

↵

↵

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.

DSV DVE

d

But the duration of the transit is much easier to measure, and that can tell you alpha.

Unfortunately measuring is quite hard because it’s extremely small.

↵

↵

30 min intervals

A

B

30 min intervals

A

B

30 min intervals

A

B

30 min intervals

A

B

Venus Transit Duration from location… A is 6.25 hours B is 5.5 hours

30 min intervals

A

B

Venus Transit Duration from location… A is 6.25 hours B is 5.5 hours

30 min intervals

A

B

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.

30 min intervals

A

B

Venus Transit Duration from location… A is 6.25 hours B is 5.5 hours

What are we trying to measure in this diagram?

h

✏

R�R�R� R�

✏dES

dEVdVS

l✏ = dES✏