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Hydrostatic Shapes
7 Hydrostatic shapes It is primarily the interplay between gravity and contact forces that shapes the macroscopic world around us. The seas, the air, planets and stars all owe their shape to gravity, and even our own bodies bear witness to the strength of gravity at the surface of our massive planet. What physics principles determine the shape of the surface of the sea? The sea is obviously horizontal at short distances, but bends below the horizon at larger distances following the planet’s curvature. The Earth as a whole is spherical and so is the sea, but that is only the first approximation. The Moon’s gravity tugs at the water in the seas and raises tides, and even the massive Earth itself is flattened by the centrifugal forces of its own rotation. Disregarding surface tension, the simple answer is that in hydrostatic equi- librium with gravity, an interface between two fluids of different densities, for example the sea and the atmosphere, must coincide with a surface of constant ¡A potential, an equipotential surface. Otherwise, if an interface crosses an equipo- ¡ A ¡ A tential surface, there will arise a tangential component of gravity which can only ¡ ©¼AAA be balanced by shear contact forces that a fluid at rest is unable to supply. An ¡ AAA ¡ g AUA iceberg rising out of the sea does not obey this principle because it is solid, not ¡ ? A fluid. But if you try to build a little local “waterberg”, it quickly subsides back into the sea again, conforming to an equipotential surface. Hydrostatic balance in a gravitational field also implies that surfaces of con- A triangular “waterberg” in the sea. -
Astronomy 201 Review 2 Answers What Is Hydrostatic Equilibrium? How Does Hydrostatic Equilibrium Maintain the Su
Astronomy 201 Review 2 Answers What is hydrostatic equilibrium? How does hydrostatic equilibrium maintain the Sun©s stable size? Hydrostatic equilibrium, also known as gravitational equilibrium, describes a balance between gravity and pressure. Gravity works to contract while pressure works to expand. Hydrostatic equilibrium is the state where the force of gravity pulling inward is balanced by pressure pushing outward. In the core of the Sun, hydrogen is being fused into helium via nuclear fusion. This creates a large amount of energy flowing from the core which effectively creates an outward-pushing pressure. Gravity, on the other hand, is working to contract the Sun towards its center. The outward pressure of hot gas is balanced by the inward force of gravity, and not just in the core, but at every point within the Sun. What is the Sun composed of? Explain how the Sun formed from a cloud of gas. Why wasn©t the contracting cloud of gas in hydrostatic equilibrium until fusion began? The Sun is primarily composed of hydrogen (70%) and helium (28%) with the remaining mass in the form of heavier elements (2%). The Sun was formed from a collapsing cloud of interstellar gas. Gravity contracted the cloud of gas and in doing so the interior temperature of the cloud increased because the contraction converted gravitational potential energy into thermal energy (contraction leads to heating). The cloud of gas was not in hydrostatic equilibrium because although the contraction produced heat, it did not produce enough heat (pressure) to counter the gravitational collapse and the cloud continued to collapse. -
STARS in HYDROSTATIC EQUILIBRIUM Gravitational Energy
STARS IN HYDROSTATIC EQUILIBRIUM Gravitational energy and hydrostatic equilibrium We shall consider stars in a hydrostatic equilibrium, but not necessarily in a thermal equilibrium. Let us define some terms: U = kinetic, or in general internal energy density [ erg cm −3], (eql.1a) U u ≡ erg g −1 , (eql.1b) ρ R M 2 Eth ≡ U4πr dr = u dMr = thermal energy of a star, [erg], (eql.1c) Z Z 0 0 M GM dM Ω= − r r = gravitational energy of a star, [erg], (eql.1d) Z r 0 Etot = Eth +Ω = total energy of a star , [erg] . (eql.1e) We shall use the equation of hydrostatic equilibrium dP GM = − r ρ, (eql.2) dr r and the relation between the mass and radius dM r =4πr2ρ, (eql.3) dr to find a relations between thermal and gravitational energy of a star. As we shall be changing variables many times we shall adopt a convention of using ”c” as a symbol of a stellar center and the lower limit of an integral, and ”s” as a symbol of a stellar surface and the upper limit of an integral. We shall be transforming an integral formula (eql.1d) so, as to relate it to (eql.1c) : s s s GM dM GM GM ρ Ω= − r r = − r 4πr2ρdr = − r 4πr3dr = (eql.4) Z r Z r Z r2 c c c s s s dP s 4πr3dr = 4πr3dP =4πr3P − 12πr2P dr = Z dr Z c Z c c c s −3 P 4πr2dr =Ω. Z c Our final result: gravitational energy of a star in a hydrostatic equilibrium is equal to three times the integral of pressure within the star over its entire volume. -
1. A) the Sun Is in Hydrostatic Equilibrium. What Does
AST 301 Test #3 Friday Nov. 12 Name:___________________________________ 1. a) The Sun is in hydrostatic equilibrium. What does this mean? What is the definition of hydrostatic equilibrium as we apply it to the Sun? Pressure inside the star pushing it apart balances gravity pulling it together. So it doesn’t change its size. 1. a) The Sun is in thermal equilibrium. What does this mean? What is the definition of thermal equilibrium as we apply it to the Sun? Energy generation by nuclear fusion inside the star balances energy radiated from the surface of the star. So it doesn’t change its temperature. b) Give an example of an equilibrium (not necessarily an astronomical example) different from hydrostatic and thermal equilibrium. Water flowing into a bucket balances water flowing out through a hole. When supply and demand are equal the price is stable. The floor pushes me up to balance gravity pulling me down. 2. If I want to use the Doppler technique to search for a planet orbiting around a star, what measurements or observations do I have to make? I would measure the Doppler shift in the spectrum of the star caused by the pull of the planet’s gravity on the star. 2. If I want to use the transit (or eclipse) technique to search for a planet orbiting around a star, what measurements or observations do I have to make? I would observe the dimming of the star’s light when the planet passes in front of the star. If I find evidence of a planet this way, describe some information my measurements give me about the planet. -
Solar System Planet and Dwarf Planet Fact Sheet
Solar System Planet and Dwarf Planet Fact Sheet The planets and dwarf planets are listed in their order from the Sun. Mercury The smallest planet in the Solar System. The closest planet to the Sun. Revolves the fastest around the Sun. It is 1,000 degrees Fahrenheit hotter on its daytime side than on its night time side. Venus The hottest planet. Average temperature: 864 F. Hotter than your oven at home. It is covered in clouds of sulfuric acid. It rains sulfuric acid on Venus which comes down as virga and does not reach the surface of the planet. Its atmosphere is mostly carbon dioxide (CO2). It has thousands of volcanoes. Most are dormant. But some might be active. Scientists are not sure. It rotates around its axis slower than it revolves around the Sun. That means that its day is longer than its year! This rotation is the slowest in the Solar System. Earth Lots of water! Mountains! Active volcanoes! Hurricanes! Earthquakes! Life! Us! Mars It is sometimes called the "red planet" because it is covered in iron oxide -- a substance that is the same as rust on our planet. It has the highest volcano -- Olympus Mons -- in the Solar System. It is not an active volcano. It has a canyon -- Valles Marineris -- that is as wide as the United States. It once had rivers, lakes and oceans of water. Scientists are trying to find out what happened to all this water and if there ever was (or still is!) life on Mars. It sometimes has dust storms that cover the entire planet. -
The Trans-Neptunian Haumea
Astronomy & Astrophysics manuscript no. main c ESO 2018 October 22, 2018 High-contrast observations of 136108 Haumea? A crystalline water-ice multiple system C. Dumas1, B. Carry2;3, D. Hestroffer4, and F. Merlin2;5 1 European Southern Observatory. Alonso de Cordova´ 3107, Vitacura, Casilla 19001, Santiago de Chile, Chile e-mail: [email protected] 2 LESIA, Observatoire de Paris, CNRS. 5 place Jules Janssen, 92195 Meudon CEDEX, France e-mail: [email protected] 3 European Space Astronomy Centre, ESA. P.O. Box 78, 28691 Villanueva de la Canada,˜ Madrid, Spain 4 IMCCE, Observatoire de Paris, CNRS. 77, Av. Denfert-Rochereau, 75014 Paris, France e-mail: [email protected] 5 Universit Paris 7 Denis Diderot. 4 rue Elsa Morante, 75013 Paris, France e-mail: [email protected] Received 2010 May 18 / Accepted 2011 January 6 ABSTRACT Context. The trans-neptunian region of the Solar System is populated by a large variety of icy bodies showing great diversity in orbital behavior, size, surface color and composition. One can also note the presence of dynamical families and binary systems. One surprising feature detected in the spectra of some of the largest Trans-Neptunians is the presence of crystalline water-ice. This is the case for the large TNO (136 108) Haumea (2003 EL61). Aims. We seek to constrain the state of the water ice of Haumea and its satellites, and investigate possible energy sources to maintain the water ice in its crystalline form. Methods. Spectro-imaging observations in the near infrared have been performed with the integral field spectrograph SINFONI mounted on UT4 at the ESO Very Large Telescope. -
Taxonomy of Trans-Neptunian Objects and Centaurs As Seen from Spectroscopy? F
A&A 604, A86 (2017) Astronomy DOI: 10.1051/0004-6361/201730933 & c ESO 2017 Astrophysics Taxonomy of trans-Neptunian objects and Centaurs as seen from spectroscopy? F. Merlin1, T. Hromakina2, D. Perna1, M. J. Hong1, and A. Alvarez-Candal3 1 LESIA – Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France e-mail: [email protected] 2 Institute of Astronomy, Kharkiv V. N. Karin National University, Sumska Str. 35, 61022 Kharkiv, Ukraine 3 Observatorio Nacional, R. Gal. Jose Cristino 77, 20921-400 Rio de Janeiro, Brazil Received 4 April 2017 / Accepted 19 May 2017 ABSTRACT Context. Taxonomy of trans-Neptunian objects (TNOs) and Centaurs has been made in previous works using broadband filters in the visible and near infrared ranges. This initial investigation led to the establishment of four groups with the aim to provide the mean colors of the different classes with possible links with any physical or chemical properties. However, this taxonomy was only made with the Johnson-Cousins filter system and the ESO J, H, Ks filters combination, and any association with other filter system is not yet available. Aims. We aim to edit complete visible to near infrared taxonomy and extend this work to any possible filters system. To do this, we generate mean spectra for each individual group, from a data set of 43 spectra. This work also presents new spectra of the TNO (38628) Huya, on which aqueous alteration has been suspected, and the Centaur 2007 VH305. -
Atmospheres and Surfaces of Small Bodies and Dwarf Planets in the Kuiper Belt
EPJ Web of Conferences 9, 267–276 (2010) DOI: 10.1051/epjconf/201009021 c Owned by the authors, published by EDP Sciences, 2010 Atmospheres and surfaces of small bodies and dwarf planets in the Kuiper Belt E.L. Schallera Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA Abstract. Kuiper Belt Objects (KBOs) are icy relics orbiting the sun beyond Neptune left over from the planetary accretion disk. These bodies act as unique tracers of the chemical, thermal, and dynamical history of our solar system. Over 1000 Kuiper Belt Objects (KBOs) and centaurs (objects with perihelia between the giant planets) have been discovered over the past two decades. While the vast majority of these objects are small (< 500 km in diameter), there are now many objects known that are massive enough to attain hydrostatic equilibrium (and are therefore considered dwarf planets) including Pluto, Eris, MakeMake, and Haumea. The discoveries of these large objects, along with the advent of large (> 6-meter) telescopes, have allowed for the first detailed studies of their surfaces and atmospheres. Visible and near-infrared spectroscopy of KBOs and centaurs has revealed a great diversity of surface compositions. Only the largest and coldest objects are capable of retaining volatile ices and atmospheres. Knowledge of the dynamics, physical properties, and collisional history of objects in the Kuiper belt is important for understanding solar system formation and evolution. 1 Introduction The existence of a belt of debris beyond Neptune left over from planetary accretion was proposed by Kuiper in 1951 [1]. Though Pluto was discovered in 1930, it took over sixty years for other Kuiper belt objects (KBOs) to be detected [2] and for Pluto to be recognized as the first known member of a larger population now known to consist of over 1000 objects (Fig. -
Astronomy 241: Review Questions #1 Distributed: September 20, 2012 Review the Questions Below, and Be Prepared to Discuss Them I
Astronomy 241: Review Questions #1 Distributed: September 20, 2012 Review the questions below, and be prepared to discuss them in class on Sep. 25. Modified versions of some of these questions will be used in the midterm exam on Sep. 27. 1. State Kepler’s three laws, and give an example of how each is used. 2. You are in a rocket in a circular low Earth orbit (orbital radius aleo =6600km).Whatve- 1 locity change ∆v do you need to escape Earth’s gravity with a final velocity v =2.8kms− ? ∞ Assume your rocket delivers a brief burst of thrust, and ignore the gravity of the Moon or other solar system objects. 3. Consider a airless, rapidly-spinning planet whose axis of rotation is perpendicular to the plane of its orbit around the Sun. How will the local equilibrium surface temperature temperature T depend on latitude ! (measured, as always, in degrees from the planet’s equator)? You may neglect heat conduction and internal heat sources. 3 4. Given that the density of seawater is 1025 kg m− ,howdeepdoyouneedtodivebefore the total pressure (ocean plus atmosphere) is 2P0,wherethetypicalpressureattheEarth’s 5 1 2 surface is P =1.013 10 kg m− s− ? 0 × 5. An asteroid of mass m M approaches the Earth on a hyperbolic trajectory as shown " E in Fig. 1. This figure also defines the initial velocity v0 and “impact parameter” b.Ignore the gravity of the Moon or other solar system objects. (a) What are the energy E and angular momentum L of this orbit? (Hint: one is easily evaluated when the asteroid is far away, the other when it makes its closest approach.) (b) Find the orbital eccentricity e in terms of ME, E, L,andm (note that e>1). -
Way-Out World
Non-fiction: Way-Out World Way-Out World By Hugh Westrup What strange satellite circles the visible edge of the solar system? The detonation of a single nuclear bomb can do catastrophic damage. So imagine the power of more than one bomb—not just two or 10 or even 10 million, but 10 billion. Astronomers have evidence that a collision with enough force to equal the explosion of 10 billion nuclear bombs once happened in the solar system. Out of that crack-up was born one of the oddest things in space. Its name is Haumea. “There is so much to learn about this newfound object, and we keep finding surprises,” says Mike Brown, an astronomer at the California Institute of Technology. “It’s just crazy.” ODD Balls The solar system is always changing. What astronomers know about it is changing even faster. Advances in telescope technology keep deepening their view of space, continually revealing new objects and new features on old objects. One example of that change in perspective is Pluto. For more than 70 years, astronomers considered it the ninth planet. Then, in 2007, the International Astronomical Union reclassified it as a dwarf planet. Like a planet, a dwarf planet orbits the sun. It also has enough mass, and therefore enough gravity, to give it a rounded shape. But it lacks pull; its gravity isn’t strong enough to clear its neighborhood of most smaller objects the way that planets do. A year later Pluto was reclassified again. Now it’s a plutoid. A plutoid is simply a dwarf planet that exists beyond Neptune, the eighth planet. -
Ice Ontnos: Focus on 136108 Haumea
Ice onTNOs: Focus on 136108 Haumea C. Dumas Collaborators: A. Alvarez, A. Barucci, C. deBergh, B. Carry, A. Guilbert, D. Hestroffer, P. Lacerda, F. deMeo, F. Merlin, C. Snodgrass, P. Vernazza, … Haumea Pluto (dwarf planets) DistribuTon of TNOs Largest TNOs Icy bodies in the OPSII context • Reservoir of volales in the solar system (H2O, N2, CH4, CO, CO2, C2H6, NH3OH, etc) • Small bodies populaon more hydrated than originally pictured – Main-belt comets (Hsieh and JewiZ 2006) – Themis asteroids family (Campins et al. 2010, Rivkin and Emery 2010) • Transport of water to the inner terrestrial planets (e.g. talk by Paul Hartogh) Paranal Observatory 6 SINFONI at UT4 7 SINFONI + NACO at UT4 8 SINFONI = MACAO + SPIFFI (SINFONI=Spectrograph for INtegral Field Observations in the Near Infrared) • AO SYSTEM: MACAO (Multi-Application Curvature Adaptive Optics): – Similar to UTs AO system for VLTI – 60 elements curvature sensing bimorph mirror – NGS or LGS – Developed by ESO • NEAR-IR SPECTRO: SPIFFI (SPectrometer for Infrared Faint Field Imaging): – 3-D spectrograph, 32 image slices, 1-2.5µm – Developed by MPE:Max Planck Institute for Extraterrestrial Physics + NOVA: Netherlands Research School for Astronomy SINFONI - Main characteristics • Location UT4 Cassegrain • Wavelength range 1-2.5µm • Detector 2048 x 2048 HAWAII array • Gratings J,H,K,H+K • Spectral resolution 1500 (H+K-filter) to 4000 (K-band) (outside OH lines) • Limiting magnitude (0.1”/spaxel) K~18.2, H+K~19.2 in hr, SNR~10 • FoV sampling 32 slices • Spatial resolution 0.25”/slice (no-AO), 0.1”(AO), 0.025” (AO) • Resulting FoV 8”x8”, 3”x3”, 0.8”x0.8” • Modes noAO, NGS-AO, LGS-AO SINFONI - IFS Principles SINFONI - IFS Principles (Cont’d) SINFONI - products Reconstructed image PSF spectrum H+K TNOs spectroscopy Orcus (Carry et al. -
Pluto and Its Cohorts, Which Is Not Ger Passing by and Falling in Love So Much When Compared to the with Her
INTERNATIONAL SPACE SCIENCE INSTITUTE SPATIUM Published by the Association Pro ISSI No. 33, March 2014 141348_Spatium_33_(001_016).indd 1 19.03.14 13:47 Editorial A sunny spring day. A green On 20 March 2013, Dr. Hermann meadow on the gentle slopes of Boehnhardt reported on the pre- Impressum Mount Etna and a handsome sent state of our knowledge of woman gathering flowers. A stran- Pluto and its cohorts, which is not ger passing by and falling in love so much when compared to the with her. planets in our cosmic neighbour- hood, yet impressively much in SPATIUM Next time, when she is picking view of their modest size and their Published by the flowers again, the foreigner returns gargantuan distance. In fact, ob- Association Pro ISSI on four black horses. Now, he, serving dwarf planet Pluto poses Pluto, the Roman god of the un- similar challenges to watching an derworld, carries off Proserpina to astronaut’s face on the Moon. marry her and live together in the shadowland. The heartbroken We thank Dr. Boehnhardt for his Association Pro ISSI mother Ceres insists on her return; kind permission to publishing Hallerstrasse 6, CH-3012 Bern she compromises with Pluto allow- herewith a summary of his fasci- Phone +41 (0)31 631 48 96 ing Proserpina to living under the nating talk for our Pro ISSI see light of the Sun during six months association. www.issibern.ch/pro-issi.html of a year, called summer from now for the whole Spatium series on, when the flowers bloom on the Hansjörg Schlaepfer slopes of Mount Etna, while hav- Brissago, March 2014 President ing to stay in the twilight of the Prof.