Project on Lightcurves of Solar System Bodies

Level 3 Project AP4

Contact: Pedro Lacerda (


This project consists of using photometric data of small solar system objects to extract their rotational lightcurves. The lightcurve properties (spin period and photometric range) will be used to find the approximate shape of the objects and set limits on their densities.

Skills/Knowledge developed: Numerical data analysis

Skills/Courses required: Python programming experience would be an advantage.

Kuiper belt objects

Kuiper Belt objects (KBOs) are leftovers from the formation of the solar system that survive in orbits beyond planet Neptune (for this reason they are sometimes called trans-Neptunian objects, or TNOs). Kept at very low temperatures (about 40 kelvin), KBOs are believed to be very icy in composition. This makes them different from the asteroids which have a more rocky composition. The Kuiper belt was only discovered in 1992, so there is still much to learn. In this projects we will focus on the rotations, shapes and densities of KBOs.


Small solar system bodies are seen in reflected sunlight. Generally, they display periodic brightness variations as their non-spherical bodies rotate and change the reflecting cross-section. A plot of brightness versus time is called a lightcurve. As we will see, lightcurves are powerful tools to infer properties of small solar system objects.

Figure: How lightcurves are produced. A spinning, egg-shaped object varies in apparent cross-section area. As the object is visible only through reflected sunlight back to Earth (just like the Moon), its brightness varies proportionately with the cross-section.

Project Description

In this project, we will use photometric data on a few Kuiper belt objects (KBOs), also known as trans-Neptunian objects (TNOs). For each object there will be a table of observation time, magnitude, and uncertainty. See nelow

For each object, the data might be taken on different nights. An example is plotted below:


The data reflect an underlying lightcurve caused by the rotation of the object, which is what should be investigated.


We will use the lightcurve data to find the underlying period, P, of the lightcurve. Assuming a given period, we can “fold” the data from different nights onto a single rotation, producing a phased lightcurve.

The “folding” consists of calculating the rotational phase, phi_i, associated with each observation time, t_i,

phi_i = mod(t_i, P)/P

and then plotting the lightcurve in magnitude vs. rotational phase.

If the correct period was used, the lightcurve will look like a well-behaved periodic curve (left panel below). Instead, choosing a wrong period will produce a very scattered phase plot (right panel below).


Once the correct period is identified, we can proceed to measure the peak-to-peak range of photometric variation, ∆m. This is the range of magnitudes sampled by the lightcurve.


Using the photometric range found above, we will estimate the shape of the object, approximating them as prolate ellipsoids spinning around a short axis. The maximum and minimum cross sections of such an ellipsoid, which correspond to the lightcurve maximum and minimum are given by

A_max = pi * a * b
A_min = pi * b * b


The ratio of these is the ratio of the maximum to minimum flux from the object. Because the object brightness is given in magnitudes, the flux ratio needs to be extracted from the photometric range using:

mag_1 - mag_2 = -2.5 * log10(flux_1/flux_2)

The photometric range can then be related to the shape of the object in terms of the axis ratio a/b.

Once the shape and spin period are known, we can set limits on the object’s density by requiring that it can hold its shape against rotational break-up. This constraint is equivalent to balancing the gravitational and centripetal accelerations on a test particle sitting on the equator at the tip of the ellipsoid,

a_g = G M m / a^2
a_c = m v^2 / a

which can be written in terms of spin period, density, shape (a/b).

Finally, we can compare the obtained values of spin period, shape and density with those of asteroids in a similar size range.


A file containing the data for this project can be downloaded following this link.

The data file contains four columns:

  1. the name of the KBO;
  2. the time (Julian date) of the observation;
  3. the magnitude of the KBO at that time;
  4. the uncertainty in the magnitude.
1998WH24  2451492.0220  20.631     0.03
1998WH24  2451492.0267  20.637     0.03
1998WH24  2451492.0314  20.651     0.03

A python programme to read the data file into a computer is available from this link.


  3. List of Known Trans-Neptunian Objects
  5. Lightcurves of Small Solar System Bodies
  6. Romanishin, W., & Tegler, S. C. 1999, Nature, 398, 129
  7. Dworetsky, M. M. 1983, MNRAS, 203, 917
  8. Sheppard, S. S., Lacerda, P., & Ortiz, J. L. 2008, The Solar System Beyond Neptune, 129
  10. Julian Date Converter
  11. Asteroid Fact Sheet