Some Light Gravity

Jed Rembold

January 28, 2026

Announcements

Stated goal was to find 90% of asteroids 1 km or larger with near-Earth orbits
How do we know when that goal is reached?
- Crater comparisons
- Rediscovery analysis
- Theoretical models

Kepler told us that there must be these relationships, but he couldn’t say why
Newton found that, if the direction that something is moving changes, then it must have experienced some force
- Newton’s big connection (one of them) was determining that the necessary force turns out to be from gravity (at least for most astronomic objects)

Gravity is the universal attractor
- Anything with mass attracts anything else with mass
- Strength of force increases with the amount of mass involved
- Strength of force decreases rapidly with distance between the masses

Kepler already had worked out \[ \frac{a^3}{p^2} = \text{same value for all planets orbiting Sun} \]
Newton worked out, starting with the force, that two objects held in orbit by gravity would obey: \[ \frac{a^3}{p^2} = \frac{G(M_1 + M_2)}{4\pi^2} \] where:
- $M_1, M_2$ are the masses of the objects in kilograms
- $a$ is the average separation between the objects in meters
- $p$ is the orbital period in seconds
- $G$ is the gravitational constant ($6.67\times10^{-11}\,\tfrac{m^3}{kg\,s^2}$)

Put in more convenient units, Newton’s formulation of Kepler’s 3rd boils down to: \[ \frac{a^3}{p^2} = (M_1 + M_2)_\odot \] where
- $M_1, M_2$ are the masses of objects in solar masses (multiples of the Sun’s mass)
- $a$ is the average separation of the objects in AU
- $p$ is the orbital period in years
For the Sun and most planets, $M_1 + M_2 \approx 1 M_\odot$
If you can measure $a$ and $p$, then you can work out the mass of the objects!

Apparent brightness is the intensity of radiation (or reflected radiation) from a celestial body
- As measured by the observer, so generally from the Earth’s surface
- Measured in units of watts per meter squared ($W/m^2$)
For our Sun, this is about $ 1400\ W/m^2$
- Clearly, the apparent brightness of other stars is going to be much, much less
It is frequently useful to thus use a different scale, where instead we talk about apparent magnitude

System introduced around 150 BC!
Hipparchus divided stars into six groups:
- Brightest were “1st magnitude”
- Faintest (that he could see) were “6th magnitude”
These days we are much more precise, but have defined things to still largely adhere to these same ideas
- Measured on an inverted logarithmic scale
  - Brighter objects have small magnitudes, and they can be negative
  - A factor of 100 in brightness corresponds to a difference of 5 in magnitude

\[ m = -2.5\log\left(\frac{B_{obj}}{B_{Vega}}\right)\]

I won’t usually give you much of time in class to work on homework
But in this case:
- To try to get content to you sooner with the Friday deadlines, I’d squeezed and compressed some ideas forward
- I suspect there are questions about the homework that I can address