## Research introduction for students

This summer I’m developing a fully-online course for transfer students to help them hit the ground running in our program (this is part of an entirely online program currently for psychology and business majors). The learning outcomes for the course focus on familiarizing students with the Hamline version of a liberal arts education and on research skills. It’s the latter part that this post is about because I’m seeking some feedback on the tentative plan.

The main approach will focus on having students investigate the research that debunks popular myths in their discipline. The rough outline is:

1. Explore popular (mis)uses of a particular myth.
1. This will intentionally have students use lower quality resources like the top 10 hits of a Google search.
2. The goal will be to flesh out how the myth is understood in society.
2. Form groups to dig deeper into each myth. Have them form a 10-100 explanation of the myth.
1. This is where the groups have to describe their myth using the 10 100 most used words in the English language (the word “thousand” isn’t in the 10 100 most used words). I’ve done this in courses before.
3. Learn how to seek out and evaluate higher quality resources.
1. We have some good modules for this from our library that first-year students use.
4. Review the lower quality sources from step one above to identify potential loopholes in the established research that might explain why the myth is so hard to dislodge from society’s general understanding.
5. Put together an annotated bibliography of sources
1. Include sources from steps 1, 3, and 4.
2. Typically the annotations will include how they discovered the source, what its general arguments are, and whether it targets a loophole.
6. Turn the annotated bibliography into a graphical one
1. Each source is represented in the graphic.
2. Physical space in the graphics is leveraged to communicate:
1. How the sources agree
2. How the sources reference each other
3. How the sources refer to the loopholes
4. Areas where new research could close or illuminate loopholes
7. The graphical bibliography will not be turned in as a graphic. Rather it will be the context of a video the groups will make to describe their project

• I’m excited to facilitate conversations around what high-quality research really says, as often it’s quite narrow. For example with (my favorite false myth) learning styles, a lot of the debunking research looks at particular features, like the learning comparison between being taught with a variety of approaches vs targeting the students’ claimed learning style. These are powerful pieces of evidence to argue against learning styles, but they are rarely comprehensive against every possible use.
• I love the 10-100 assignment as students really struggle to find the words that they’re allowed to use to explain something that they understand. This works as a great analogy for how researchers have to learn to communicate.
• I’m super intrigued by the graphical bibliography. I think that it could be a great way to illustrate the various connections among the sources.
• I like the notion of closing loopholes. If the groups can find that, say, some company is making money leveraging learning styles, maybe they can find that a new research study could help close that loophole, or identify a use for learning styles that’s legitimate.

• Letting students in the first week use just Google results might set a bad precedent.
• Access to full-text sources in a short term (the initial version will only be 8 weeks long).
• While I’m developing the class, I won’t always teach it, so I need to make sure that the assessments are solid and able to be performed by any qualified instructor.
• Psychology myths are pretty easy to find, but business myths have proven harder to track down.

What do you think? Here are some starters for you:

• This class sounds cool, my favorite part is . . .
• This class is a waste, what you should do is . . .
• Why not have the students develop a thorough research proposal for those loopholes?
• You should add in myths about online learning while you’re at it.
• What’s your beef with learning styles?
• Here’s a bunch of business myths for you: . . .
• I disagree with how you’re categorizing high-quality resources. It seems to me that they are comprehensive. Here’s an example . . .
• When I think about preparing students for research, I think of a bunch of different ideas than yours. Here’s twelve . . .
• If you love the 10-100 assignment so much why don’t you limit yourself to those words for your blog posts?
• There are tons of resources for graphical annotated bibliographies that you should be ashamed you didn’t mention. Here’s my favorite . . .
Posted in dean, research, teaching | 2 Comments

## Helicopter vs snowplow

Working in the dean’s office has given me all sorts of new respect for the Dean of Students office. Specifically, I’ve been very impressed at how they leverage students’ parents when helping the students get through tough times. My experience before the dean’s office was cloaking myself in FERPA and basically not talking to parents from the initial recruiting meetings (when I often try to keep eye contact with the potential student) to graduation, when I’m happy to both pose for and take pictures of the family.

But in the dean’s office, I am beginning to see how useful parents can be. If they reach out to us (or the dean of students office) with concern, they kick us into action, even though we’re unable to call them back. Typically we figure out if the student is ok and we convince them to call their parents to let them know.

I was teaching a class once and I was curious how often students contacted their parents. I asked “who has contacted their parents once this week” and most hands went up. So I began to raise the contact number and the hands started to go down. Except for one student. Theirs stayed up, even as I went extreme just to get them to bring the exercise to the end. “Ten times?” Nope. “One hundred times?” Nope. Finally they said “my mom is my best friend and I’ve texted her twice during this class.” I used to tell that story to warn of ‘helicopter parents’ but I’m beginning to see it differently. That parent, as with most, knows their students quite well, and can often act as an emotional guide when tough college-based decisions come up. “Should I do my calc homework or audition for the play?” “Should I try for 3As and a C or 2As and 2Bs?” I don’t think there’s one answer for those sorts of questions, and often the answer is best a set of further questions. But what I’ve learned is that parents are really good at helping students in situations like that. If they need straight academic advising, I’m good at it. But holistic advising? I need help, and parents, even helicopter parents, can be a great tool to leverage.

So I think I can like the concept of ‘helicopter parent’. They are there for support when the student needs it, but, just like my July hobby of watching the Tour de France, they don’t impede.

However, there is a type of parent that I struggle with: the snowplow parent. This is the parent that pushes everything out of the way and chooses the path. That’s a very different beast than a helicopter parent.

I was recently talking to parents of new first year students and I realized that I had a good example of how I went from a snowplow research advisor to a helicopter one as I got better at it. When I first began working with research students (typically in the summer), I would be in my office working on something (typically my teaching plans for the fall) and a student would come in to say that the laser wasn’t aligned anymore. So I’d walk back to the lab with them and show them how to do it. Really I would elbow them out of the way and align it for them, thinking that they’d get more done with an aligned laser than not. I never really realized that I wasn’t doing a good job, but I should have realized how that connected to how they’d act during poster sessions. They’d happily tell anyone who stopped by all about their poster, but when asked a question, like “why this approach” or “have you thought of …” or “I would have …” they’d gaze around the room looking for me in the hopes I’d come and rescue them. Which I would, with answers like “oh, we do it this way because I knew it would work well” and “they’ve probably never heard of that approach but it’s a good idea” and “that’s a good idea, let’s you and me go talk about it.”

Then one summer I had to be gone for two weeks and I left my students behind with a long todo list. I broke my promise to check in with them via email so I was nervous the day I got back. As I walked into the lab, I immediately got furious. I was mostly mad at myself, but I’m sure it came off as furious with them. I walked up to the apparatus they’d built saying “there’s no way this will work” and I started taking it apart. As I was doing that I mentioned that another section of the apparatus was a mistake as well. Then they said “STOP!”. I said “What?” And then they said the two bravest words I’ve ever heard students say to me: “Get Out!” They explained that they’d tried lots of variations and they’d finally found a way to get the experiment to produce actionable data. As I went out to the hallway, still steaming, mind you, I finally realized that’s what success looks like.

So now I try to be a helicopter advisor. I set up an environment where they can learn and I hover, making sure they’re making good decisions. If they get stuck, I try to remember to ask good follow up questions hoping they’ll figure a way out, and I now realize that ending a summer with good data that I’m solely responsible for with a bunch of students who’ve watched me do it is a waste.

So, I think I’d like to be a helicopter advisor and I think I like helicopter parents. Just don’t be a snowplow.

Thoughts? Here are some starters for you:

• I like this. I especially like . . .
• I hate this, I especially hate . . .
• I think FERPA gets in the way. I wish I could call parents. Here’s what I’d say . . .
• I love FERPA, I never talk to parents and I hope I never have to. Here’s why . . .
• How can you defend helicopter parents? When they hover my students suffer and here’s why . . .
• I live in Florida, what is a snowplow?
• I think the lesson you should have learned is to not bother coming in in the summer.
• What are those students doing now?
Posted in dean, research, teaching | 4 Comments

## Form workflow using Google Apps Script

Now that I’m in the dean’s office (Associate Dean for undergraduates in the College of Liberal Arts if you must know) I serve on a lot of committees. Several of them accept submissions from faculty, staff, or students and have to vet them, adjudicate them, and let someone else know how to act on them. We use all kinds of systems to do this, including email submissions, Google forms, home-built web forms, and (gasp) paper. This post is for me to get some thoughts down for how to improve one such process using a combination of Google forms, Google docs, and Google Apps Script.

Here’s the needed workflow:

1. Faculty member submits a form so that one of their courses can meet one of our general education requirements (we have a distributed system where any course in principle can meet any of the requirements).
2. The committee vets the submission against a rubric and discusses it. If it’s approved, the registrar (who sits on the committee) makes the change formal. The faculty member is also informed.
3. If there is a need for more information or a different tack to be approved, the committee sends the original faculty member a note explaining what is needed.
4. The faculty member can update their submission using the same form as step 1 above. Then steps 2 and 3 are repeated.

Problems with our current system:

• There’s no easy way to have the committee keep common digital notes on the submissions. That results in everything happening in the meeting with each individual keeping their own notes ahead of time.
• There is no real re-submission process. Right now both steps 3 and 4 and any repeats are done via email.
• The submissions are not formatted very well.

Why Google? What I crave the most is the ability for the committee to group comment on a google doc ahead of the meeting. This would both aid in getting us all on the same page and provide a great record for how we made decisions in the past (rather than just seeing what our decision was) for similar submissions. This single craving drives a lot of the workflow I have in mind:

1. Faculty member uses a single Google form that has multiple pages, having them jump to the appropriate section for each different type of general education requirement they’re applying for (I think I’ll make it that they’ll have to do it once for every different requirement).
2. Upon submission, a Google doc will be created based on a template (or several, I suppose, given the different nature of the various general education requirements). This will then be emailed and shared with the entire committee. Its name will include the faculty member, the course, and the version number. A link will go in the “version” column added to the results spreadsheet with the version number hyperlinked to the google doc.
1. Note that you can’t send emails using Google Apps Script based on a form submission. I gather this protects against massive amounts of email being sent. However, you can do a timed trigger and effectively have the system check once a day (or every five minutes, whatever you like).
3. The committee will use the commenting features of Google Docs to collect their thoughts ahead of the meeting. In the meeting a decision will be made and placed in the “decision” column that’ll be added to the automatic Google Form spreadsheet
1. “Approve”: This will fire a script to generate a final pdf version (that gets rid of the comments, of course) that gets emailed to the registrar and the original faculty member. A link for the pdf will go in the “final pdf” column of the spreadsheet.
2. “Feedback”: This won’t do anything, as the chair likes to tailor the feedback emails to folks. However, the chair will send the “edit submission” link to the original form submission so that any changes will be represented in the spreadsheet.
1. Note that we won’t turn on “send edit link” on the original form, as we don’t want it to be edited while we’re reviewing it. However, this awesome script allows you to gather these links upon the original submission and add them to the spreadsheet.
2. We’ll keep an additional column of “last timestamp” so that we can do a timed trigger looking for any rows where the submission timestamp differs from the “last timestamp” so we’ll know to do regenerate the google doc and change the “decision” column to “pending”
3. “Denied”: Frankly this doesn’t happen much, but we’ll have to add it for completeness. This won’t do anything either because the chair will want to tailor that email to the original faculty member
4. As noted in 2-2 above, if a re-submission comes in, a new google doc will get generated with the appropriate version number and the process repeats. In the spreadsheet you’d see a “version” column with multiple hyperlinks in it, one for each version.

I think I know how to do all of this. Here are the technical highlights:

1. Mix of triggers: There’s both “upon form submission” and “timed” triggers involved. Both are pretty straightforward to do.
2. Generate a Google doc based on a template and a form submission. This is pretty easy as well, especially if you use clear template placeholders like <<name>> or <<First learning outcome>>.
3. Generate a pdf from a Google doc. Easy
4. Email things to people. This gets harder with good formatting but not really a problem
5. Generate the “edit submission link”. The link above really figured that out for me.

So I think I’m in good shape. I’ll probably make a dummy system first to test, but I thought I’d post this in case people think I’m missing something or am thinking about things in a dumb way.

Your thoughts? Here are some starters for you:

• I’m on this committee and can’t wait. My favorite part is . . .
• I’m on this committee and think this is dumb. The dumbest part is . . .
• Why use spreadsheets at all? Just do it all with the Google Apps Script Form API!
• At my school I could never do this because . . .
• Must be nice to be a Google school (it is!)
• What happens when you leave the dean’s office?
• I can’t believe you can’t do this with Mathematica!
• I thought you were a LAMP expert, why not write this from scratch with PHP etc? (quick answer: note the craving I laid out above. That’s hard to do with PHP)
• I can’t believe you think generating a pdf from a google doc is easy! It took me months to figure that out.
• Where’s the chair approval part?
• You do know that if you’re using an outline form you can’t have a single sub-point, right?
• Why are you reinventing the wheel? This is already done (and much better) here. (insert appropriate hyperlink, please!)
• Why not just have them submit the application as a google doc?
Posted in dean, programming, technology | 1 Comment

## Fair Asymmetric Dice, getting there

Ever since my son and I worked on whether different types of 20-sided dice were fair I’ve been thinking about whether there might be some oddly shaped dice that could still be fair.  I’ve watched these three numberphile videos and I’ve looked at these dice. All of which seems to suggest that if you have identically shaped sides, you’re in business. But I’ve still been wondering about other weird shapes.

So I’ve decided to see if I can search for some. This post lays down things I’ve discovered as I’ve geared up to a huge optimization run [SPOILER: I haven’t run it yet]. Here’s the list:

1. There is an angle you can drop a stick (without rotating) so that it’ll hit twice and bounce right back up without rotating.
2. Figuring out the correct response to a rigid body bounce on a hard surface was more complicated than I’d thought it would be.
3. Using Mathematica’s ConvexHullMesh was a cool way to make random multi-sided dice.
4. Getting Mathematica to use “WhenEvent” to determine impacts and then implement point 2 above was frustratingly difficult.
5. Doing bounces like I have in the past (treating the ground as a stiff half-spring) works well and lets me do some energy loss mechanisms that I couldn’t figure out in point 4 above.
6. Soft versus hard surfaces affect the roll (defined as the side that ends up down).
7. My kids have helped me a ton:
1. They think adding in translations in addition to rotations and bounces is important. I’m not sold, but it makes for cool animations.
2. They think hard surfaces look better.
3. They think most of the energy loss happens at the bounces (my first approach was to use a lot of air resistance and have the bounces be pure – especially when doing point 4 above).
8. Determining which side is “down” was trickier than I thought – though I’m open to other suggestions.
9. Ultimately I’d like to get feedback on the assumptions I’m building in so that when I run the optimization I can trust the results.

Given that it’s late on a Sunday night, I think I’ll just show some of my results (graphics and animations below) and see what sorts of questions folks have:

Rebound velocity of stick: This shows the rebound velocity of the center of mass of a stick that has symmetric masses on it. (think —0—-0— or 0———–0 or -0———-0- etc) where the position of the masses as measured from the center is the x-axis. Note that where the velocity crosses zero is the point where the stick would only rotate after the first bounce and therefore the motion would be totally symmetric (it would ultimately bounce right back up with the opposite angle with respect to the horizontal but without any rotational energy).

This shows a dumb mistake I was making: I was assuming that the center of mass of a polygon was equal to the center of mass of the vertices. As you can see below, that’s true for triangles but not for other polygons (COM_reg means center of mass of the region whereas COM_pts is the center of mass of the vertices).

Success bouncing a 2D shape. After finally giving up on the WhenEvent approach in Mathematica, I just said that any vertex that goes below the ground needs to experience an upward constand force. The horizontal line is the original height of the center of mass.

Showing that you really do get different rolls with a triangle. Note that now I’ve added in some friction.

Histogram of 100 rolls for that triangle. You can tell than 100 rolls isn’t good enough since surely eventually side 2 should land more often.

A 4-sided shape and accompanying histogram after 1000 rolls (note that each roll takes 0.5 seconds).

Combing through the data I found one of the rare ones where it lands on side 1:

Ok, I got 3D bouncing working!

And I can do lots of sides!

Here’s a 16-sider with some nice color added.

Here’s a 6-sider comparison showing a soft (top) and hard (bottom) floor. Note the difference in the ultimate side that ends up down.

Here’s a plot of the side that’s down as a function of time for the animation above. The blue curve is the soft surface and the red curve is the hard surface.

Ok, so I know I haven’t put much detail in yet, but I wanted to at least get down some of what I’ve been working on. The next step is to check some of my assumptions with you fine readers and then to go ahead and do some long optimization runs. I was thinking of doing a genetic algorithm, but then I’d need a ton of runs (keeping in mind, of course, that each roll takes a while (0.5 seconds in 2D, 2 seconds in 3D).

So some questions for you:

1. 2D is way faster, is it worth my time to explore it?
2. I do friction in the following way (does it bother you?):
1. if it’s not in contact with the floor, there’s no energy loss
2. If it’s in contact with the floor, there’s a contribution to the force that’s proportional to the velocity for those variables related to the vertex that’s in the floor.
3. Note that means I’m not doing sliding friction but rather viscous friction.
3. By adding in translation, I have 2 more variables to keep track of (I’m already doing z(t) of the center of mass and the 3 Euler rotation angles). Worth it?
4. Doing a 1-d minimization would be much faster than something like a genetic algorithm. What would that variable be for, say, a 6 sided die?
5. What do you mean by a fair die? How much variation after, say, 1000 rolls is small enough for you? I’ve asked my kids what it would take for them to be suspicious of any dice they own (big D&D players) and they’ve got some interesting opinions.

So, can you help? Here are some starters for you:

• I’m in this class and . . . wait . . . nevermind
• Seriously, you haven’t posted since February and this is all you can come up with??!!
• I think this is cool. I think you should . . .
• I think this is dumb, I think you should . . .
• Once again that promise to switch to python wasn’t worth the breath it took to say it, I see.
• Here’s my address where you can send your new 3D printed dice to me.
• Even 1000 roles isn’t enough. This will never be done to my satisfaction.
• What about dice where a few sides almost never come up but the rest have near equal probability?
• Without even looking at your code I know what your WhenEvent problem is. Here’s how to fix it . . .
• I would have thought it would be obvious that a rigid body should bounce off an infinitely massive floor such that the local velocity of the part that hits the ground reverses its z-component and that there’s only one solution for the speed of the center of mass and the rate of change of the Euler angles that both conserves kinetic energy and correctly accounts for the expected change in angular momentum!
• I don’t understand why you say it’s hard to figure out which side is down. Just look at it!
Posted in fun, mathematica, physics | 7 Comments

## Critical Disagreement

I’m just wrapping up my time spent at a really great conference that’s all about the First-Year Experience for students in college. I’ve got lots of thoughts running through my head, including lots of cool ideas for a large part of my job: director of my university’s First Year Seminar program. This post is about just one of those.

I was in a pre-conference workshop that was about critical thinking. This is a hot topic in higher ed, especially ever since “Academically Adrift” was published, indicating research that many college student’s critical thinking skills actually got worse during their college years. Someone mentioned something that really has me thinking:

Students struggle to understand how two people who are both thinking critically can come to different conclusions.

Not surprisingly a few people in the workshop muttered about politics in the US to give a bunch of examples. For me, though, I realized that my main discipline doesn’t really suffer from this problem. If there’s a disagreement between two physicists, it usually means there’s just not enough evidence in yet. The two can be arguing about which theory best describes reality, but if they’re really arguing, it’s usually because both of their theories match all the available data. The reason they don’t fully agree is that the two theories make predictions about things that haven’t been detected yet.

This actually was a large part of my masters work. I was in a group that was really trying to understand how ultrashort pulse lasers work. [brief aside: These are lasers that blink. They’re only “on” for 0.0000000000001 s, then they wait for a few microseconds before they repeat.] My group had one theory, and another prominent group had another to describe exactly how these lasers developed the intense electric fields involved. The problem was that the standard measurement that people were able to do at the time did not distinguish the two theories. It was interesting to go to conferences and be involved in what felt like heated debates. But really they were just hopeful debates. Both sides wanted to be right, but both realized that at most only one of them was. My masters thesis was all about the development of a new measurement technique that could clearly distinguish the two theories.

So that really has me thinking. Does this approach to natural science distinguish how it employs “critical thinking” from other disciplines? I’ve begun to explore the political arguments I’ve been involved in, especially those where I feel like both sides are “thinking critically.” We have access to the same facts, but we feel that the best moves for the future are nearly diametrically opposed. It seems to me this happens for at least two reasons:

1. We disagree on base assumptions about something.
2. We prioritize particular future events differently.

I’ve been a participant in arguments that have fizzled because both parties have realized that either 1 or 2 above is what’s happening. That fizzle takes the form of “oh, well then I guess we just disagree then” or “well, then you’re just an uncaring SOB, I guess.”

In physics, the arguments come to an end when new data comes to light. People can be disappointed that their theory wasn’t right, but they don’t kick themselves for being wrong. Their theory matched the data that was known. They just bet wrong. Moving forward they’re happy to use the correct theory.

One interesting physics example is how to interpret quantum mechanics. There’s tons of disagreement about what’s real, what a measurement does, how many universes there are, etc. However, to participate in the argument, you have to back a theory that matches all the data. When pseudoscience folks try to join or say things like “well anything goes”, they’re usually pretty easily shot down when their theories are shown to not match particular measurements. The argument is really about what reality is, not how to make calculations or predictions.

So what do you think? Here are some starters for you:

• This is interesting. Here’s an argument that happens in my field  . . .
• Have you ever heard of reading?! Here are 10 things you should read before blathering on like this  . . .
• I think you’re not being entirely honest here. After reading your paper I see that your new technique vindicated your group’s theory. Seems fishy to me.
• I remember when FROG was invented. It opened up so many new ways to think about our ultrashort lasers, thanks!
• If someone disagrees with me, it’s clearly because they’re not thinking critically.
• Quantum mechanics is just a spherical-earth conspiracy.
• What arguments do you see your students having in lab?
• Are you saying that if disagreement lingers the participants aren’t thinking critically?
• I wrote “Academically Adrift” and now I think I should go and write a whole new chapter about how the critical thinking abilities of bloggers goes downhill.
Posted in dean, physics, teaching | 2 Comments

## No connections

Driving home today I heard a great story on NPR. I liked it so much that I thought I’d put it here to remind myself about it. I might have forgotten some of the details, but I think I still remember the gist.

### Teachers without their PLCs

Principals have been realizing lately how hard it is to get their teachers to do good work. Too many of them have been spending time talking to each other to find better ways to teach. That takes away from the time that they could be interacting with their students. Now a few schools around the country have started to use a new approach: locking each teacher away from all others.

When the teachers get to school, they can be seen smiling and chatting with their friends in the parking lot. But when they get to the school, they’re met with a phalanx of administrators just inside the door who put all the teachers in bags and cart them off to their classrooms where they’re locked in. At the end of the day they are brought back out, and only then are they able to interact with their co-workers.

“It really sucks because it’s usually so great to get good ideas from people in similar situations that I can incorporate into my teaching,” said one teacher, clearly pining for her friends. Another added “I can’t believe that they’re painting us all with the same brush, assuming that all we want to do it talk with our friends instead of teaching. That’s not fair. I love to teach!”

But the principals have come to understand that any sort of access to professional development that might bring in new ideas can only take away from the tried and true approaches to working with students. While their teachers are complaining, the principals are sure this is the right approach.

Some brave teachers have pointed out that the lack of access to their professional coworkers has changed their behavior. Now if they can’t figure out how to work with a student, they just keep trying other things that they come up with off the top of their head. “Before I would find a friend at lunch to brainstorm ways to help, but now I have way more time to really focus on the problem.”

The bag and lock technology is being pioneered by BeAlone, whose founder realized just how bad things were when he asked how his daughter was doing in school and only got a response after the teacher checked in with all his daughter’s teachers. “I couldn’t believe how long that took! I just wanted to know if she was getting an A+, not whether she was developing lifelong learning strategies.” The company’s clients also include comedians and musicians who like to make sure their audience is only listening to them. The cost is $200 per bag or schools can rent them for$30 per teacher for the school year.

Your thoughts? Here are some starters for you

• That’s weird, I listened to NPR on the way home today and I didn’t hear that story.
• Wait, I feel like you’re being sarcastic, but I can’t put my finger on it.
• You forgot about my favorite part: …
• This is dumb. Principals who do this just don’t realize how creative teachers can be when they can work together.
• This is great. I think we should seek to have this in all schools!
Posted in fun, teaching, technology | 1 Comment

## Mass changing orbits

A few weeks ago my good friend John Burk posted some intriguing questions about what happens to planetary orbits as the sun loses mass (all that heat has to come from somewhere!). I’ve been thinking about it ever since and finally got around to doing some modeling to see if I could answer any of the questions.

My first thought was to see if there were ways to simplify the differential equation solving approach. What John was doing was the full 3D version assuming the sun’s mass stays significantly above the mass of the planet. So he’s already doing some simplifications because he’s not bothering with the center of mass frame. I think he’s right to do that because after a page or so of notes, I’ve realized that the center of mass frame gets pretty ugly when one of the participants is losing mass.

So what else can be simplified? The great thing about central force problems is that you can reduce it all the way down from six variable to one (in fact that’s one of my standards when I teach Theoretical Mechanics):

1. Each of the two masses has 3 variables (x, y, and z) so you start with six.
2. The center of mass approach lets you model the problem as a fictitious mass (=m1m2/(m1+m2)) that’s the same distance from a fictitious force center as the two actual masses are from each other. Now you’re down to three.
3. If it’s a central force, the angular momentum is conserved. That means the fictitious particle has to stay in a plane. Now you’re down to two variables.
4. If the angular momentum is conserved, you can treat the rotational part of the kinetic energy as an effective potential energy, leaving only the radius variable. Now you’re down to one.

You can model the complex 6-dimensional problem as a single (reduced) mass experiencing a potential energy function given by:

$U_\text{effective}=U_\text{actual}+\frac{l^2}{2 \mu r^2}$

where $\mu$ is the reduced mass and U is the potential energy that is only a function of r. To get the force this fictitious one dimensional particle feels, you just need to take a derivative (and add a negative sign).

So I gave it a try. The first thing I did was try to see how far into the future I could integrate using Mathematica. It turns out I could go quite a ways! Here’s a plot of the radius as a function of time.

As you can see, I was able to go out several billion seconds of integration time. This turned out to be around a billion “years” given the simple parameters I chose. If the radius grows, we expect the years to take longer. Here’s a plot of the instantaneous “year time” over the simulation:

as expected!

So it seems that a very slow loss of the sun’s mass would just slowly increase both the circle radius and the year time of the planet.

Another question that John asked was whether there might be an analytical solution to this. I quickly tried DSolve instead of NDSolve in Mathematica and got no joy (I wasn’t overly hopeful). I did ask a good friend of mine who’s a real expert in differential equations whether he knew of a particular decay function I could use that might have an analytical solution. He couldn’t think of one, but did point out that if you had the mass changes be discrete you could “easily” build up the solution since in between the mass changes you’ve got a simple inverse square law orbit that does have an analytical solution.

What he means is that if you start with, say, a circular orbit, you can predict exactly where the planet will be and what direction it’s traveling (and what speed) when the first mass change happens. When it does, you now have initial conditions for a slightly different inverse square law problem. Because the sun has lost mass, it doesn’t pull as hard as before so the circular orbit becomes an ellipse. That ellipse is fully analytical and you can figure out everything you need to know about the planet at the next mass drop. Repeat this to your hearts content and you’ve got a piecewise analytical solution.

This sounded intriguing, but I started to wonder what the physical differences would be between the two approaches. I figured I could check on a relatively small time scale and look for differences. So I coded up both a continuous and a discrete mass loss model, where they connect with each other after each mass loss. Here’s a plot of both mass loss functions:

Here’s the animated result (the blue dots are the points of the mass change for the discrete (blue) model:

As you can see, there’s a pretty noticeable difference in the orbits. Admittedly this is only because the mass jumps are pretty big, but it still makes me nervous.

Here’s a plot of the radius function for both:

Note how the continuous one seems to never get any close to the sun while the blue one is clearly showing more of an elliptical motion (it gets closer and further from the sun during every orbit). To see that more clearly, here’s a plot of r'[t] or the rate of change of the radius:

It sure looks like the orange line doesn’t go negative (indicating the planet never gets closer to the sun). Here’s a zoom in of the orange one to see it better:

Yep, never negative!

Your thoughts? Here are some starters for you

• This is cool, but I’d like to hear more about . . .