Parikshit Padole

Hello again! Welcome to the follow-up blog from the previous one where we discussed How Solar Radiation Pressure Affects Satellites: Calculating Instantaneous Force and Acceleration. As the title suggests, this time we’ll dive deeper into understanding how to calculate long-term orbital perturbations due to solar radiation pressure.

This is a vast topic, but my goal is to simplify it as much as possible for everyone. These blogs are aimed at providing a foundational introduction to the fascinating realm of orbital mechanics.

Now, I have a confession—if you’re a mathematician or physicist, can we have a video call? Sometimes I wonder if I should have pursued Astrophysics instead of diving into AI and Robotics. 😅 Just kidding! I’m much better in this field, and I can’t imagine myself knee-deep in equations with Einstein-style hair. At least, I like to think I’m cool… Please don’t get offended by my nerdy jokes!

Alright, enough yapping—let’s get back to the topic. To quickly recap what we discussed in the last blog:

Solar radiation pressure is the force exerted by photons emitted by the Sun when they strike a satellite’s surface. The magnitude of this pressure depends on:

• The satellite’s cross-sectional area exposed to sunlight,
• The reflectivity of its surfaces,
• Its orientation relative to the Sun,
• And its distance from the Sun (though for low Earth orbit, this is usually assumed constant).

While the force from solar radiation pressure is small, over time it can cause significant changes in a satellite’s orbital elements. In the last blog, we focused on calculating the immediate effect of this pressure, but not how it accumulates and affects the spacecraft over time.

This blog will cover exactly that.

To make it more realistic and understandable, we’ll use EIRSAT-1 as a case study—one of the most well-known student-led CubeSat missions, launched by the students of University College Dublin as part of the European Space Agency’s Fly Your Satellite! programme.

I didn’t find extensive details about the mission (Wikipedia had some info, but we all know how trustworthy that is), so I gathered what I could from official publications and real-time data from the n2yo website. As always, I’ll provide links below for anyone who wants to fact-check.

Now, let’s get all the information down properly and begin!

EIRSAT-1 Satellite Data
Latitude	-41.18°
Longitude	74.18°
Altitude	481.19 km (299 mi)
Speed	7.62 km/s (4.73 mi/s)
Azimuth	232.0° SW
Elevation	-55°
Right Ascension	07h 51m 50s
Declination	-60° 20′ 10”
Launch Mass	2.305 kg
Dimensions	10.67 cm × 10.67 cm × 22.7 cm (4.20 in × 4.20 in × 8.94 in)
Satellite Period	94 minutes

 

Now we have the data let’s try to understand how and what steps are we going to take also before that let’s list down the factors to consider when calculating the effect of solar radiation pressure on an object:

Factors to Consider for Solar Radiation Pressure Calculations:

Orbital Elements:

• Semi-major axis
• Eccentricity
• Inclination
• Right Ascension of Ascending Node (RAAN)
• Argument of Perigee
• True Anomaly
• Mean Anomaly
• Eccentric Anomaly

Satellite Geometry:

• Cross-sectional area
• Reflectivity data

Material Properties:

• Absorptivity and emissivity
• Degradation rates over time

Orientation Data:

• The satellite’s attitude and orientation over time

Distance to the Sun:

• Updated regularly throughout the year to account for orbital variations

 

Step-by-Step Calculation

Solar Radiation Pressure Calculation

We already know the formula for calculating solar radiation pressure:

 

Determine the Cross-Sectional Area: Using the dimensions of

EIRSAT-1 (10.67 cm × 10.67 cm × 22.7 cm), the cross-sectional area facing the Sun (depending on its orientation) could be approximated. For the largest face:

 

Force on the Satellite Due to Solar Radiation Pressure

We’ll calculate the force exerted by solar radiation pressure using the following formula:

 

Instantaneous Acceleration on the Satellite Due to Solar Radiation Pressure

We can calculate the acceleration using Newton’s second law:

 

Atmospheric Drag Force Formula

F_D is the drag force in Newtons,

C_D is the drag coefficient (typically between 2.0 and 2.5 for satellites),

A is the cross-sectional area of the satellite (0.00024228 m²),

ρ is the atmospheric density at the satellite’s altitude,

v² is the satellite’s velocity relative to the atmosphere (7.69 km/s).

 

Now we don’t know what is Atmospheric Density for that point of time so I used

NRLMSISE 2.0 is an empirical, global reference atmospheric model of the Earth from ground to space. It models the temperatures and densities of the atmosphere’s components.

Now we get,

 

let’s substitute it in the formula

 

Acceleration due to Drag

 

 

Now Finally,

Lagrange’s Planetary Equations:

You might wonder why we still need to use Lagrange’s planetary equations when we’ve already calculated the forces acting on the satellite. These equations allow us to determine how the orbital elements of a satellite (such as semi-major axis, eccentricity, inclination, and others) change over time due to external forces.

Since the focus of this blog is to calculate long-term orbital perturbations, Lagrange’s planetary equations provide a structured way to analyse the cumulative effects of small forces like solar radiation pressure (SRP) and atmospheric drag. They give us a clear picture of how a satellite’s orbit evolves over extended periods.

 

Above are the formulas for the  Lagrange’s Planetary Equations where

ȧ is the Semi-major Axis

ė is the Eccentricity

i̇ is the Inclination

ω̇ is the Argument of Perigee

Ω̇ is the Right Ascension of the Ascending Node

Ṫ is the Time (Orbital Period)

I will be creating a separate blog dedicated to solving these equations in detail. If you have the necessary data for your own satellite, you’ll be able to input it into the equations and follow along to solve for long-term orbital changes.

Disclaimer: I am not a mathematician or physicist; the information presented here is sourced from various online references, which I have linked below. If you have any suggestions or feedback, feel free to reach out to me at ppadole@imperial.ac.uk 😊

References

General Perturbation Theory – Lagrange’s Planetary Equations, LibreTexts

Lagrange’s Planetary Equations, ASU Control Systems

EIRSAT-1 Conferences, EIRSAT-1 Official Website

Development, Description, and Validation of the Operations Manual for EIRSAT-1, Journal of Astronomical Telescopes, Instruments, and Systems

NRLMSIS Atmosphere Model

Read Long-Term Orbital Perturbations of Satellites Due to Solar Radiation Pressure in full

First of all, hello again! Apologies for the long break—final year kept me busy, along with my part-time work, so I didn’t have much time to post. But now that things have settled down, I’m excited to start sharing the blogs I’ve been planning. I’ll be posting regularly, starting with today’s topic.
In this blog, we’ll dive into Solar Radiation Pressure and explore how it affects satellites.
Let’s begin!
In space, even the smallest forces can have significant effects on satellites over time. One of the key forces acting on a satellite is solar radiation pressure.

What is Solar Radiation Pressure?
Solar radiation pressure is the force exerted by photons from the Sun as they strike and reflect off the surface of a satellite. While seemingly insignificant, the pressure can considerably impact a satellite’s orbit, particularly its perigee height especially over long durations.

This force depends on:

1. The satellite’s distance from the Sun.
2. The satellite’s cross-sectional area facing the Sun.
3. The reflective properties of the satellite’s surface

In the May 1960 paper, “The Influence of the Solar Radiation Pressure on the Motion of an Artificial Satellite,” researchers focused on the Vanguard I satellite, launched into Earth orbit on 17 March 1958. This satellite, intended for geodetic measurements, was designed with a diameter of 16.5 cm and a mass of 1.47 kg. Early orbit analyses of Vanguard I revealed unexpected discrepancies between the observed and predicted perigee heights.

The researchers explored solar radiation pressure as a potential cause for these deviations. Their investigation, using analytical methods to model the effects of solar radiation pressure on satellite orbits, indicated that this pressure significantly contributed to the observed perturbations in Vanguard I’s perigee height. This finding highlighted that even though solar radiation pressure is a relatively small force, it could cause substantial orbital deviations, particularly for satellites with a high area-to-mass ratio like Vanguard I.

So, how do we calculate solar radiation pressure and its effect on a satellite’s motion? Below, we’ll walk through the key steps and formulas that help determine the instantaneous force and acceleration on a satellite caused by this pressure.

Let’s break it down:

Step 1: Calculating Solar Radiation Pressure

Solar radiation pressure at Earth’s distance from the Sun is given by the following formula:

Where

P is the solar radiation pressure.

L is the Sun’s luminosity (3.828 ×10^26 W) .

r is the distance from the Sun (Earth’s distance is 1.496 × 10^11 m )

c is the speed of light (3×10^8 m/s)

 

Substituting Earth’s distance into this equation gives us a solar radiation pressure at Earth:
P ≈4.56 × 10^(-6) N/m^2

This value will be used to calculate the force acting on a satellite.

 

Step 2: Calculating the Force on the Satellite

Once we know the solar radiation pressure, we can calculate the force on the satellite. The force depends on the satellite’s cross-sectional area (A) and its reflectivity(ρ). The formula for the force is:

Where:

F is the force exerted on the satellite.

A is the satellite’s cross-sectional area in square meters.

ρ is the reflectivity coefficient (ranging from 0 for a perfectly absorbing surface to 1 for a perfectly reflective surface).

 

Example Calculation:

Let’s say a satellite has a cross-sectional area of 10 m² and reflects 30% of the radiation

(i.e ρ=0.3 ). The force acting on this satellite would be:

So, the instantaneous force acting on the satellite due to solar radiation pressure is F=5.928 ×10^(-4) N

 

Step 3: Calculating the Acceleration on the Satellite

Next, we calculate the acceleration on the satellite due to the solar radiation pressure. Acceleration is simply the force divided by the satellite’s mass ( ):

Where:

a is the acceleration

F is the force due to solar radiation pressure.

m is the mass of the satellite in kilograms.

Example Calculation:

For a satellite with a mass of 500 kg, the acceleration would be:

Although the force and acceleration caused by solar radiation pressure are small, they are persistent. Over time, they can lead to measurable changes in the satellite’s trajectory, especially for satellites with large area-to-mass ratios. For lightweight satellites, like CubeSats, solar radiation pressure can become a dominant factor affecting their orbits.

In terms of application this force can be utilised for example in solar sails the force from solar radiation pressure can be used  for propulsion, enabling satellites to move without fuel.

Conclusion

Solar radiation pressure, while small, exerts a continuous force on satellites. By calculating the instantaneous force and acceleration due to this pressure, we can understand how sunlight can gradually change a satellite’s orbit.

In future blogs, we will explore how this force can affect a satellite’s orbit over time, including resonant orbits and eccentricity changes

 

Disclaimer: I am not a mathematician or physicist; the information presented here is sourced from various online references, which I have linked below. If you have any suggestions or feedback, feel free to reach out to me at ppadole@imperial.ac.uk 😊

Source List 

Comet Tails and Solar Radiation Pressure

NASA’s Explanation of Sunlight Exerting Pressure

Luminosity from Britannica

Photon Mass and Physics

Scientific Article on Solar Radiation Pressure

NASA Technical Report on Solar Radiation

Solar Radiation Pressure in Engineering

NSSDC Spacecraft Data on Vanguard I

SRP Image ESA

Read How Solar Radiation Pressure Affects Satellites: Calculating Instantaneous Force and Acceleration in full