Questions tagged [mathematics]

For astronomy questions with a heavy mathematical component, more than simple formulas. For regular mathematics question, visit our sister site Mathematics Stack Exchange.

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How the heck does JPL actually do the numerical integration for their development ephemerides? (general questions)

After writing this answer to JPL Horizons - "highly accurate measurements of planetary positions" - how do they do it? which draws from Park, Folkner, Williams & Boggs 2021 The JPL ...
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How do I actually apply the small-angle approximation in this problem?

I'm asked to use the small-angle approximation to calculate the angular diameter (in degrees) of Jupiter as seen from Earth, if Jupiter were at the average distance of the Moon of 384,000 km. I know ...
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2 votes
2 answers
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Measure the deprojected distance between two points?

I have an image of a protoplanetary disk, from which I would like to measure the distance of various points from the centre. See below image for an example, where I would like to measure the distance ...
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Azimuth angle and elevation angle of a heliostat mirror

for the control of a heliostat I have a program for the calculation of azimuth and elevation. Now I am still missing the calculation of azimuth angle and elevation angle for setting the mirror. While ...
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Projection from Earth [closed]

If a Particle is projected from Equator due north with a velocity $v$ and Earth is rotating with a uniform angular velocity $\omega$. How can it be shown that the highest latitude reached will be ...
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1 vote
1 answer
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Using Saha's equation knowing the overall pressure

Knowing the pressure, $P_g$, how can I calculate $\frac{N_{I+1}}{N_I}$ using Saha's equation? If I assume that $P_g=(n_e+n_{H^+}+n_H)K_B T$ and that $n_e=n_{H^+}$, I can find the electronic pressure, $...
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What is the relationship between true anomaly and solar longitude?

So, my ultimate objective is to map solar longitude values onto a pseudo-Gregorian calendar. I thought this would be simple, but it has turned out to be anything but. I have asked several questions on ...
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Total scattered flux and polarization for a ring of dust particles close to a star

I have an important problem and maybe some of you can help me with the following question: I have a mathematical expression for the total scattered flux in Jansky and the total linear polarized flux (...
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1 answer
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How does this Toomre GI criteria have the period in the denominator?

I saw this equation in a literature review recently talking about the Toomre criterion for gravitational instability: Given here in section 2.1.1: https://arxiv.org/pdf/1801.06117.pdf, viz. But I am ...
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What kind of perspective projection is shown in this image?

I’m trying to develop software that overlays sky coordinates over 2d images of the sky taken by an iphone. I’ve done the sky coordinates part, but I’m trying to figure out what kind of perspective ...
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How to model the kinematics of gases expelled from comets and planetary atmospheres?

Once the ices of comets sublimate, where do they move to, and with which equations do you describe their motion in the interplanetary medium? Similarly, where do the gases that are stripped from the ...
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Accurate formula for calculating the mass of an exoplanet using the transit method

A friend and I did some work on exoplanets with the help of a research institute (IEEC in Spain) for a major high school project. What we did was to "redetect" the exoplanet XO-6b through ...
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N-body simulation still losing precision despite using arbitrary-precision arithmetic and symplectic integrator

A while ago I asked 2-body orbits expanding in nbody simulation?. One thing that I noticed was that JavaScript does not have arbitrary precision, and that I used RK4 as my integrator, while it is ...
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2 votes
1 answer
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What does "RV model including a GP with a quasi-periodic covariance structure" mean?

A candidate short-period sub-Earth orbiting Proxima Centauri says: 4.1 Simple periodogram analysis ]We first consider a very simple pre-whitening procedure applied to the CCF RVs in order to ...
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Finding the Laplacian of the deflection potential: I obtain 4𝜅(𝜃⃗ ), not 2𝜅(𝜃⃗ ). Why am I wrong?

From the review paper Bartelmann and schneider, 1999) Weak Gravitational Lensing, page 48. Like above, the Laplacian of the deflection function is convergence, $\kappa(\vec{\theta})$. I tried to ...
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Does a star shaped mirror present problems either optically, or in signal processing over a more nearly circular mirror?

Pure speculation about a future replacement for the JWST that I almost certainly won't live to see. The JWST's mirror (and then some) would fit unfolded and flat into the Starship's 8m diameter ...
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14,000 square degrees

The DESI Legacy Imaging Surveys says it produces a model of “the 14,000 square degrees of extragalactic sky visible from the northern hemisphere”. But I thought the whole celestial sphere (like any ...
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4 votes
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Probability of transit of an exoplanet in front of its star

I'm trying to calculate the probability of transit of an exoplanet in orbit around a star of diameter $d$, at a distance of $4d$ from its centre. Here's what I did: In this figure, we are only ...
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3 votes
1 answer
104 views

What's the Right Ascension & Declination of Galactic Left & Galactic Right?

For the purpose of being able to align my model to the galactic plane, I want to know the directions of the galactic center, anti-center, north, south, left & right. This gives me the answer for ...
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3 votes
1 answer
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Is it still called an orbital resonance if the ratio is irrational?

Previously, I asked At what point are orbital resonances no longer "ordered" but "chaotic?", and received an answer from @CarlWitthoft: Perhaps if the calculated fraction had an ...
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At what point are orbital resonances no longer "ordered" but "chaotic?"

Orbital resonances are typically in small valued integer ratios, like 2:1, 3:2, or 4:7. However, there are some resonances whose ratios have large reduced values, including the 73:69 Naiad:Thalassa ...
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2 votes
1 answer
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Sun Moon earth Angle

I want to do a theoretical prediction before trying out an experiment. I just noticed that the arc separating the illuminated side of moon and the shadowed side is an ellipse arc. So Is it possible to ...
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4 votes
1 answer
124 views

Scaling Relations

In this picture , Keplers third law is written in a scaling relationship with solar mass. $$a^3=\frac{G(m_1+m_2)}{4\pi^2}P^2 \rightarrow m_1+m_2 = \frac{4\pi^2a^3}{GP^2}$$ $$m_1+m_2=(1 M_{\odot})\, \...
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Converting a Satellite's position in Lat/Lon/Alt/Time to RA/Dec

I'm trying to convert a satellite's position given in latitude/longitude decimal degrees, altitude in meters, and time in milliseconds since epoch to right ascension and the declination. I have found ...
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Is it true that the 3 body problem can't be solved using the four basic functions, radicals, and integrals?

The two-body problem can be completely solved via two one-body problems, which only uses the four basic binary functions. However, the three-body problem cannot be solved with these functions and ...
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2 votes
1 answer
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Relative orientation in a 1:1 resonance of a planet and a satellite

To my understanding, in a two-body problem of a planet and a satellite, a 1:1 resonance means that the orbital period of the satellite is the same as its angular frequency (maybe not, so please ...
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6 votes
1 answer
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Physical Meaning of the Principal Tidal Fourier Modes and Their Integers

I'm trying to understand tides evolution of planets, and I often come across the Prinicipal Tidal Fourier Modes expressed as: $$ \omega_{lmpq}$$ For example, see the discussion after Eq. A15 in ...
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1 vote
1 answer
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Why is there a deviation between the ratio of $a^3$ and $T^2$ for the outer planets?

The Wikipedia article about Kepler's third law includes a nice table about the ratio between $a^3$ and $T^2$. However, for Mars, Jupiter, Saturn, Uranus, and Neptune, the ratio $\frac{a^3}{T^2}$ in $...
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3 votes
2 answers
235 views

Pendulum clock correction

I'm trying to solve this task: The pendulum clock was transported from the Earth's equator to Antarctica (in the vicinity of the southern geopole) for scientific experiments. Estimate the pendulum ...
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5 votes
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Calculate lines of equal solar altitude

I'm attempting to calculate lines of equal solar altitude, essentially a generalized case of the Earth terminator for solar altitude $ h $. For a given sun position with declination $ \delta $ and ...
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Where can I build my house on Mercury so that I can see double sunsets (and sunrises)? Can I do it near the poles where it's cooler?

CNN'S Mercury mission flies by closest planet to the sun for the first time says at the end: Mercury's unusual rotation and oval-shaped orbit around the sun (sic) means our star seems to quickly rise,...
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2 votes
1 answer
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How long does the "eclipse" last on a space station at the L1 point between a moon and a planet last when the moon blocks the sun in front?

The situation I am asking about is as depicted in the picture. Supposing I have a space station staying perpetually at the L1 point, the moon will completely block the space station from all sunlight ...
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2 votes
0 answers
87 views

What's the formula for change in argument of periapsis over time due to J2 perturbations for a geocentric satellite?

I'm trying to propagate the orbit of the ISS, but it became very clear to me very quickly that the J2 perturbations were nothing to scoff at. A quick google search gave me the equation for the change ...
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1 vote
0 answers
134 views

Is there a formula for a planet's orbit in a 3D space?

I'm trying to model the orbits of Earth and Mars and their positions at a given point in time. But I can only find 2D equations of elliptic orbits, which only contain the x and y variables. I have to ...
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1 vote
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How to determine how often a planet's center will be collinear with its moon's?

Expanding on the question: Let's say I have a planet orbiting some star (earth-like, sun-like, for the sake of example). If this planet has two moons, M1 with orbital period of 30 earth-days and M2 ...
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8 votes
2 answers
5k views

What if Earth gained 1 km/s orbital velocity?

How would Earth's orbit be affected if we (hypothetically) added $1 {\rm km/s}$ to its orbital velocity? Would Earth reach close to Mars' orbit? Could Earth get gravitational assist from Mars and go ...
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4 votes
1 answer
140 views

Confusion when calculating the length of the day

I'm trying to solve this task: Tula and Moscow are located on the same meridian. In which city length of the day is bigger on 22 of June? 22 of December? I was thinking like that: we need to ...
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3 votes
2 answers
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What is the rate of change by day for a given right ascension?

I am writing a small Julia app to get my feet wet in that language. And I want to calculate "what's up tonight" based on a table of right ascension (RA)/Dec(lination) (table of stars), a ...
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10 votes
3 answers
491 views

Role of power laws in astronomy?

I often see astronomers fitting data to power laws. What about power laws makes them so useful in astronomy? Why are so many astronomical observations well-fit by power laws? I know it's a relation ...
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5 votes
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How did pressure evolve in the (early) universe?

I am trying to derive how the cosmological pressure $p(t)$ evolved over time in the universe, especially in the radiation and matter dominated epochs. There are some very nice explanations how $H(t)$ ...
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2 votes
2 answers
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Basic, learning examples of applications of CFD and MHD in astrophysics, and current research directions using these techniques?

I am an undergraduate in Physics, and I have an opportunity to work with a some of my seniors to learn Computational Fluid Dynamics (CFD) and Magnetohydrodynamics (MHD). I would like to choose my ...
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3 votes
1 answer
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Orbital resonances - expansion of disturbing function

I want to study the orbital resonance type 3:1 between an asteroid and Jupiter. For this purpose, I found the expansion of the disturbing function, $R$, in Celletti A., Stability and Chaos in ...
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16 votes
2 answers
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At what distance from the Sun can planetary moons exist?

Mercury and Venus are theorized to have no moons because they are so close to the sun. Is there a theoretical distance in which moons tend to exist based on simulations?
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2 votes
1 answer
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Flux vs. Velocity representation of spectra?

I'm looking for a basic explanation of the flux vs. velocity representation of a spectrum and how it's obtained from the regular flux vs. wavelength representation. A good example of this is in Arav ...
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4 votes
1 answer
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How galactic density falls using Oort constants

I am given the observations that Oort constants $A$ and $B$ are, respectively: $14.5 \text{km s}^{-1} \text{kpc}^{-1}$ and $-12 \text{km s}^{-1} \text{kpc}^{-1}$. From these, I am supposed to conclude ...
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1 answer
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Poisson's equation-like formula in axis-symmetric Galactic potential in terms of Oort constants

I am tackling this problem, from a 2016 Cambridge Astrophysics Tripos past paper: Let $\Phi(R, z)$ be the axi-symmetric Galactic potential. At the Solar location, $(R, z) = (R_0, 0)$, prove that $$\...
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10 votes
2 answers
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What is the difference between the two terms named "Eccentricity" and "Ellipticity"?

I don't get the difference between the two terms named "Eccentricity" and "Ellipticity", especially, in astronomy. I understand eccentricity as a measure of the curvature of a ...
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0 votes
0 answers
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Stellar temperature ~ frequency relationship?

According to Wikipedia, 76% of all stars are of spectral type M, 12% are of K, 7.5% are of G, etc. (see this section on Stellar Classification). This is very helpful, but is too vague, as the ...
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8 votes
4 answers
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Why can't we use the semi-minor axis in Kepler's third law?

Why can't we use the semi-minor axis in Kepler's third law? The formula for the third law is: $$\dfrac{a^3}{T^2} = \text{const.} $$ Instead of semi-major axis (a) why can't we use the semi-minor axis (...
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2 votes
2 answers
536 views

How to estimate the inclination of a galaxy between edge on and face on?

I have the images of a galaxy and the field of view and I would like to determinte the inclination of a galaxy between edge-on and face-on. Would that information be enough to do the calculation? Or ...
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