Space and Physics
PUBLISHED
Every now and then on Reddit, somebody asks a fun question that demands an answer. This week it is the turn of user TheSum239, who asks, "If I'm in space and there is a body with less mass than me like a pen will the pen orbit me?"
The question is a good one. The answer seems obvious (you are a source of mass, as is the pen, so you will be attracted to each other via gravity) but the details are pretty interesting. Let's dive in.
The short answer is that yes, it is theoretically possible for the pen to begin orbiting you, creating your own human-pen system, like a moon that is manufactured by Bic. But you would need to be far from another source of mass for this to be achievable, and, unless you're packing a truly astonishing amount of mass for a human, you would need to nudge the pen at a truly teeny tiny velocity in order to achieve your noble goal of becoming your own solar system*. Even then, a stable and continued orbit is unlikely.
To answer the question properly, we need to simplify it somewhat. As physicists famously tend to do, we need to turn you and your pen into a sphere, whilst retaining their densities.
Humans are roughly as dense as water, sometimes in both senses, at around 1,000 kg/m3. We're going to make things easy on ourselves and pick a nice round number, and assume you are around 80 kg (176 pounds). To calculate your volume, as any high school revision website will tell you, we simply need to divide your mass by your density, giving us 0.08 m3.
Next, we use the formula for a volume of a sphere to get your human-ball radius, which is 27 centimeters or about 11 inches (again, with a few assumptions about your weight and density).
You will both be orbiting your shared center of mass, but given how much larger you are than the pen, this center of mass will be inside of you, given how tiny the pen's mass is in comparison to your own (no offence meant). But the pen, crucially, cannot be inside you, so we will assume that you want your pen/moon to orbit you at a distance of around 1 meter (3 feet) from your center to make it nice and neat.
Next, we can use the escape velocity formula. This bit too, now that everything is spherical, is nice and easy.
"The escape velocity formula is independent of the properties of the escaping object," Omni Calculator explains. "The only thing that matters is the mass and radius of the celestial body in question."
That gives us an escape velocity of 0.00010334 m/s, meaning that if you push your pen any faster than that, buddy you are losing yourself a pen, doomed to drift apart forever. In order to find a stable orbit, we need to rely on the hard work of physicists who came before us.
"Escape velocity decreases with altitude and is equal to the square root of 2 (or about 1.414) times the velocity necessary to maintain a circular orbit at the same altitude," Encyclopedia Britannica explains.
Performing that taxing calculation (dividing by 1.414) we get 0.00007308345 m/s. Should you, a spherical human stuck in space, be able to push the pen at that tiny speed you would turn yourself into a two-body system, with the pen orbiting around you nicely, until some other tiny force disturbs the system and sends it into chaos.
*yes, we know you technically aren't a sun.
